Hyperbolicity properties of algebraic varieties 9782856299517

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Hyperbolicity properties of algebraic varieties B. Claudon, P. Corvaja, J.-P. Demailly, S. Diverio, J. Duval, C. Gasbarri, S. Kebekus, M. P˘ aun, E. Rousseau, N. Sibony, B. Taji, C. Voisin Edited by S. Diverio

Panoramas et Synthèses Numéro 56

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Comité de rédaction Claire LACOUR Olivier BENOIST Quentin MÉRIGOT Serge CANTAT Anne MOREAU Fabienne CASTELL Bertrand RÉMY Indira CHATTERJI Séverine RIGOT Anne-Laure DALIBARD Sergio SIMONELLA Anne-Sophie de SUZZONI Todor TSANKOV Diego IZQUIERDO Bertrand RÉMY (dir.)

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ISSN 1272-3835 ISBN 978-2-85629-951-7

Directeur de la publication : Fabien Durand

PANORAMAS ET SYNTHÈSES 56

HYPERBOLICITY PROPERTIES OF ALGEBRAIC VARIETIES B. Claudon, P. Corvaja, J.-P. Demailly, S. Diverio, ˘ J. Duval, C. Gasbarri, S. Kebekus, M. Paun, E. Rousseau, N. Sibony, B. Taji, C. Voisin Edited by S. Diverio

Société mathématique de France

Benoît Claudon Univ Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France E-mail : [email protected] Pietro Corvaja Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Udine, Via delle Scienze, 206, 33100 Udine, Italy Jean-Pierre Demailly Université Grenoble Alpes, Institut Fourier Laboratoire de Mathématiques, CNRS UMR 5582, 100 rue des Maths 38610 Gières Email : jean-pierre.demailly@univ-grenoble -alpes.fr Simone Diverio Dipartimento di Matematica “Guido Castelnuovo” SAPIENZA Università di Roma Piazzale Aldo Moro 5 I-00185 Roma E-mail : [email protected]

Stefan Kebekus Mathematisches Institut, Albert-LudwigsUniversität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany and Freiburg Institute for Advanced Studies (FRIAS), Freiburg im Breisgau, Germany and University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France E-mail : [email protected] Mihai Păun Universität Bayreuth, Germany E-mail : [email protected] Erwan Rousseau Institut Universitaire de France & Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France E-mail : [email protected] Nessim Sibony Université Paris-Saclay, CNRS,Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France and Korea Institute for Advanced Study, Seoul, 130-722 South Korea E-mail : [email protected]

Julien Duval Laboratoire de Mathématiques, Université ParisSaclay, 91405 Orsay cedex, France [email protected]

Behrouz Taji School of Mathematics and Statistics F07, The University of Sydney, NSW 2006 Australia E-mail : [email protected]

Carlo Gasbarri Carlo Gasbarri, IRMA, UMR 7501, 7 rue RenéDescartes, 67084 Strasbourg Cedex, France

Claire Voisin CNRS, IMJ-PRG, 4 Place Jussieu, 75005 Paris, France E-mail : [email protected]

Classification mathématique par sujets. (2010) — 14D23, 14E05, 14E30, 14F10, 14J10, 14J17, 32Q45, 32Q26. Mots-clés et phrases. — Hyperbolicité au sens de Kobayashi, lemme de Brody, théorie de la distribution des valeurs, conjecture de Lang, hyperbolicité algébrique, inégalités à la Bogomolov-MiyaokaYau, espace de modules, hyperbolicité au sens de Viehweg, hypersurface projective, différentielle de jets, conjecture de Kobayashi, courbure négative, point rationnel, approximation diophantienne, fibration. Keywords and phrases. — Kobayashi hyperbolicity, Brody’s lemma, value distribution theory, Lang’s conjecture, algebraic hyperbolicity, Bogomolov-Miyaoka-Yau type inequality, moduli space, Viehwegs hyperbolicity, projective hypersurface, jet differential, Kobayashi’s conjecture, negative curvature, rational point, Diophantine approximation, fibration.

HYPERBOLICITY PROPERTIES OF ALGEBRAIC VARIETIES B. Claudon, P. Corvaja, J.-P. Demailly, S. Diverio, J. Duval, C. Gasbarri, ˘ S. Kebekus, M. Paun, E. Rousseau, N. Sibony, B. Taji, C. Voisin

Abstract. — Since its introduction in the 70’s, the notion of Kobayashi hyperbolicity has attracted a lot of attention in the mathematical community. Besides its aspects exclusively belonging to the several complex variables world, an extremely fascinating theme is that of its interactions with the algebraic, arithmetic, and differential geometric properties of algebraic varieties. These interactions are essentially what this book is about. Some of the issues addressed are: distribution and distribution of values of entire curves, algebraic analogues of hyperbolicity, hyperbolicity properties of projective hypersurfaces and of varieties of general type, hyperbolicity of moduli spaces, relationships between hyperbolicity and negative curvature, distribution of rational points on hyperbolic (arithmetic) varieties, and interplay of different kinds of natural fibrations on algebraic varieties and hyperbolicity. The volume has the ambition to make a point of the state of the art, each chapter treating a different aspect of the subject, trying to keep the language friendly enough to encourage in particular PhD students as well as young researchers in complex geometry to get into the most recent advances in the study of hyperbolicity properties of algebraic varieties. Résumé (Propriétés d’hyperbolicité des variétés algébriques). — Depuis son introduction dans les années 70, la notion d’hyperbolicité au sens de Kobayashi a attiré beaucoup d’attention dans la communauté mathématique. À côté de ses aspects plus purement d’analyse complexe en plusieurs variables, un thème très fascinant est celui de ses interactions avec les propriétés algébriques, arithmétiques, et géométro-différentielles des variétés algébriques. L’étude de ces interactions est essentiellement l’objectif de ce livre. Parmi les thématiques abordées figurent : distribution et distribution des valeurs des courbes entières, analogues algébriques de la notion d’hyperbolicité, propriétés d’hyperbolicité des hypersurfaces projectives et des variétés de type général, hyperbolicité des espaces de modules, relations entre hyperbolicité et courbure négative,

distribution des points rationnels dans les variétés (arithmétiques) hyperboliques, et connexions entre plusieurs types de fibrations naturelles dans les variétés algébriques et hyperbolicité. Ce volume a pour ambition de dresser l’état de l’art, chaque chapitre traitant un aspect différent du sujet, tout en essayant de tenir un langage assez simple qui puisse encourager en particulier les thésards ainsi que les jeunes chercheurs en géométrie complexe à rentrer dans les développements les plus récents de l’étude des propriétés d’hyperbolicité des variétés algébriques.

CONTENTS

Résumés des articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Julien Duval — Around Brody lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Brody lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 11

Mihai Păun & Nessim Sibony — Value Distribution Theory for Parabolic Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Basics of Nevanlinna Theory for Parabolic Riemann Surfaces . . . . . . . . . . . 4. The Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Parabolic Curves Tangent to Holomorphic Foliations . . . . . . . . . . . . . . . . . . 7. Brunella index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 18 22 29 35 41 65 70

Erwan Rousseau — On algebraic hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The approach via foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Orbifold Bogomolov-Miyaoka-Yau inequalities . . . . . . . . . . . . . . . . . . . . . . . . 4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 75 76 83 86

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CONTENTS

Jean-Pierre Demailly — A simple proof of the Kobayashi conjecture on the hyperbolicity of general algebraic hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Hyperbolicity concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Semple tower associated to a directed manifold . . . . . . . . . . . . . . . . . . . . . . . 3. Jet differentials and Green-Griffiths bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Existence of hyperbolic hypersurfaces of low degree . . . . . . . . . . . . . . . . . . . . 5. Proof of the Kobayashi conjecture on the hyperbolicity of general hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 131

Carlo Gasbarri — McQuillan’s approach to the Green-Griffiths conjecture for surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Curves of bounded genus on surfaces with big cotangent bundle . . . . . . . . . 3. Entire curves on surfaces with big cotangent bundle . . . . . . . . . . . . . . . . . . . 4. An approach to the general case of Green-Griffiths conjecture for surfaces References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 136 140 164 166

Benoît Claudon & Stefan Kebekus & Behrouz Taji — Generic positivity and applications to hyperbolicity of moduli spaces . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Fractional semipositivity and application to hyperbolicity . . . . . . . . . . . 3. Logarithmic differentials with fractional pole order . . . . . . . . . . . . . . . . . . . . 4. Fractional tangents and foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Fractional semipositivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Application to hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II. Proof of the semipositivity result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Positivity of relative dualising sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Failure of semipositivity, construction of morphisms . . . . . . . . . . . . . . . . . . . 9. Proof of the semipositivity result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 170 173 182 182 187 193 193 197 197 201 205 205

Simone Diverio — Kobayashi hyperbolicity, negativity of the curvature and positivity of the canonical bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Complex differential geometric background and hyperbolicity . . . . . . . . . . . 3. Motivations from birational geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. An example by J.-P. Demailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Wu-Yau theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Kähler case, and the quasi-negative holomorphic sectional curvature case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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89 89 90 95 102 112

209 209 212 224 228 233 248

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CONTENTS

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

Pietro Corvaja — Some arithmetic aspects of hyperbolicity . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Heights, Diophantine approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Higher dimensional Diophantine approximation . . . . . . . . . . . . . . . . . . . . . . . 4. A proof of Siegel’s theorem for integral points on curves . . . . . . . . . . . . . . . 5. The generalized Fermat equation and triangle groups . . . . . . . . . . . . . . . . . . 6. Algebraic groups and the S-unit equation theorem . . . . . . . . . . . . . . . . . . . . 7. Integral points on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253 263 278 282 286 295 306 315

Claire Voisin — Fibrations in algebraic geometry and applications . . . . . . . . 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Fibrations and holomorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Fibrations from families of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. MRC and Γ-fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Special varieties and the core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319 319 321 328 333 344 351

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

RÉSUMÉS DES ARTICLES

Autour du lemme de Brody Julien Duval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Le lemme de Brody est un outil fondamental en hyperbolicité. Il fournit une courbe entière, c’est-à-dire l’image d’une droite complexe par une application holomorphe non constante, en-dehors d’une suite divergente de disques holomorphes. Ainsi le lemme de Brody caractérise l’hyperbolicité en termes d’absence de courbes entières. Nous présentons des applications directes du lemme de Brody, notamment le théorème de Green (hyperbolicité du complément de 5 droites dans le plan projectif) et un exemple de surface hyperbolique de degré 6 dans l’espace projectif. Nous décrivons aussi une variante du lemme de Brody permettant de mieux localiser la courbe entière produite. Comme sous-produit de cette variante, l’hyperbolicité est caractérisée en termes d’inégalité isopérimétrique linéaire pour les disques holomorphes. Théorie de la distribution des valeurs pour les surfaces de Riemann hyperboliques Mihai Păun & Nessim Sibony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Nous donnons des versions de résultats classiques en théorie des distributions de valeurs quand l’espace source est une surface de Riemann parabolique, c’està-dire une surface sans fonctions sous-harmoniques bornées non constantes. Nous obtenons un théorème d’annulation pour de telles applications, impliquant une version des théorèmes classiques de Bloch et d’Ax-Lindemann. Dans ce contexte, des résultats de Brunella et McQuillan sont aussi étendus, au moyen de preuves relativement simples.

x

RÉSUMÉS DES ARTICLES

Sur l’hyperbolicité algébrique Erwan Rousseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Nous étudions des propriétés de nature algébrique qu’on présume en rapport avec l’hyperbolicité. Un résultat classique de Demailly fournit des minorations pour le genre des courbes dans les variétés hyperboliques. Les inégalités de Demailly sont étroitement liées aux conjectures géométriques de Lang–Vojta affirmant que les courbes pour les paires logarithmiques de type général devraient satisfaire des inégalités similaires. Partant du résultat classique de Bogomolov, qui démontre de telles inégalités pour des surfaces de type général à deuxième nombre de Segre positif, nous nous concentrons sur une preuve alternative de Miyaoka, qui rend l’inégalité effective (puisque les constantes peuvent être choisies comme fonctions des nombres de Chern de la surface). La preuve est présentée comme une illustration de la théorie des orbifolds de Campana : les minorations du genre des courbes sont obtenues comme conséquences d’inégalités orbifold générales de type Bogomolov–Miyaoka–Yau. Une preuve simple de la conjecture de Kobayashi sur l’hyperbolicité des hypersurfaces algébriques génériques Jean-Pierre Demailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Nous étudions une célèbre conjecture de Shoshichi Kobayashi (1970), suivant laquelle une hypersurface algébrique de dimension n et de degré d ≥ dn suffisamment grand dans l’espace projectif complexe Pn+1 est hyperbolique. Par une caractérisation classique de Brody, une telle variété ne contient pas de courbe entière holomorphe non constante. Comme on le sait bien depuis les travaux de Green et Griffiths, un ingrédient crucial est la structure géométrique de certains fibrés de jets et des différentielles de jets associées. Plus précisément, on utilise la tour de Demailly–Semple, qui est une tour tordue de fibrés projectifs liés aux différentielles de jets invariants par reparamétrisation. D’après un résultat d’annulation fondamental, les différentielles globales de jets à valeurs dans des fibrés en droites négatifs fournissent des équations différentielles algébriques que toutes les courbes entières doivent satisfaire. Si le lieu de base de ces équations différentielles est suffisamment petit, autrement dit s’il y a assez d’équations différentielles indépendantes, alors toutes les courbes entières doivent être constantes. Au début des années 2000, Yum-Tong Siu a proposé une stratégie quelque peu différente qui a finalement conduit à une preuve en 2015. La preuve de Siu, qui est basée sur des arguments de théorie de Nevanlinna combinée avec l’usage de champs de vecteurs tordus apparaît longue et délicate. En 2016 la conjecture a été démontrée d’une autre façon par Damian Brotbek, faisant un usage direct des opérateurs différentiels wronskiens et des idéaux multiplicateurs associés. Peu de temps après, Ya Deng a montré que l’approche

PANORAMAS & SYNTHÈSES 56

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pouvait être mise en oeuvre pour fournir une valeur explicite de dn . Nous donnons une preuve courte basée sur une simplification radicale de leurs idées, ainsi qu’une amélioration de la borne de Deng, à savoir dn = b(en)2n+2 /5c. Nous montrons que la même technique fournit des exemples d’hypersurfaces algébriques lisses de Pn+1 de petit degré d = O(n2 ), suivant une approche due à Shiffman et Zaidenberg. Approche de McQuillan pour la conjecture de Green-Griffiths pour les surfaces Carlo Gasbarri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Après un retour sur la preuve par Bogomolov du fait que sur les surfaces lisses telles que c21 > c2 , les courbes de degré géométrique borné forment un famille bornée, nous expliquons les étapes principales de la preuve, due à McQuillan, des conjectures de Green–Griffiths pour ces surfaces. Vues de loin, les deux preuves suivent la même stratégie, mais la seconde requiert une analyse bien plus profonde des outils utilisés. Afin de décrire la preuve de McQuillan nous expliquons la construction des courants d’Ahlfors associés aux courbes entières dans une variété et nous montrons comment ceux-ci peuvent être utilisés pour produire des substituts des nombres d’intersection. Une preuve de l’inégalité tautologique à la fois dans le cas standard et dans le cas logarithmique est donnée. Nous expliquons comment les hypothèses nous permettent de supposer que la courbe entière considérée (qui a posteriori ne pourra pas exister) est la feuille d’un feuilletage. Afin de simplifier certains points techniques de la preuve, nous imposons des restrictions sur les singularités de ce feuilletage (la preuve générale nécessite une analyse beaucoup plus poussée mais les idées principales apparaissent déjà sous ces restrictions). Dans la dernière section, nous décrivons très brièvement une stratégie possible (proposée par McQuillan) pour la preuve du cas général de la conjecture, avec une explication des principales difficultés à surmonter. Positivité générique et applications à l’hyperbolicité des espaces de modules Benoît Claudon & Stefan Kebekus & Behrouz Taji . . . . . . . . . . . . . . . . 169 Les espaces de modules de variétés algébriques comprennent naturellement les dégénérescences de variétés en famille. Par exemple, l’obstruction aux dégénérescences non triviales de certaines familles de variétés peut être décrite par la non existence de certaines sous-variétés dans les espaces de modules associés. Nous étudions le rôle conjectural de la dimension de Kodaira en tant qu’invariant pouvant être utilisé pour décrire de telles obstructions. Des progrès récents dans la compréhension des propriétés de positivité de faisceaux tangents de variétés non uniréglées, et les généralisations naturelles au cadre logarithmique à de paires (X, D), s’est révélé être un outil puissant dans l’étude de la dimension de Kodaira des bases de familles de variétés. Nous faisons une recension détaillée de la généralisation due à Campana et Păun des résultats de semi-positivité générique de Miyaoka. Dans sa forme la

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plus simple ce résultat affirme que les quotients du fibré cotangent logarithmique Ω1X (log D) ont des pentes semi-positives par rapport à tout diviseur ample, pourvu que le diviseur log-canonique KX + D soit pseudo-effectif. Une conséquence-clé relie ensuite la « grosseur » de KX + D aux propriétés de positivité du faisceau des formes pluri-(log-)différentielles. Ceci, avec un résultat de Viehweg et Zuo sur la « grosseur » du faisceau des formes pluri-differentielles des champs de modules des variétés canoniquement polarisées, conduit à la preuve d’une célèbre conjecture de Viehweg sur l’hyperbolicité algébrique de ces espaces : les sous-variétés des champs de modules des variétés canoniquement polarisées sont toutes de type log-général. Hyperbolicité au sens de Kobayashi, négativité de la courbure et positivité du fibré canonique Simone Diverio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Nous fournissons une recension détaillée de la percée récente de Wu et Yau, généralisée peu de temps ensuite par Tosatti et Yang (ainsi que par Diverio et Trapani). Cette percée se situe à l’intersection de la géométrie différentielle complexe et de l’hyperbolicité au sens de Kobayashi. Plus précisément, une conjecture ancienne de Kobayashi, énoncée dès le début de la théorie, prévoit qu’une variété projective complexe (ou plus généralement une variété kählérienne compacte) hyperbolique au sens de Kobayashi doit être à fibré canonique ample. D’une part, il est connu également depuis le début de la théorie qu’une variété compact complexe avec une métrique hermitienne dont la courbure sectionnelle holomorphe est négative, est hyperbolique au sens de Kobayashi. D’autre part, on sait – par les célèbres travaux d’Aubin et Yau – qu’une variété kählérienne compacte à fibré canonique ample admet une métrique kählérienne à courbure de Ricci (constante) négative. Le théorème de Wu et Yau établit que si une variété lisse projective admet une métrique kählérienne à courbure sectionnelle holomorphe négative, alors elle admet aussi une métrique kählérienne (éventuellement différente) à courbure de Ricci négative. Le résultat peut donc être vu comme une confirmation faible de la conjecture de Kobayashi ci-dessus, puisqu’il aboutit à la même conclusion, avec une hypothèse plus forte sur la courbure sectionnelle holomorphe. Une présentation soigneusement détaillée de la preuve du théorème de Wu–Yau est aussi l’occasion de présenter les notions de base de la géométrie différentielle complexe et de plusieurs résultats, positifs et négatifs, à propos du lien entre courbure et hyperbolicité au sens de Kobayashi. Des questions ouvertes naturelles sont aussi discutées. La preuve du théorème de Wu-Yau présentée ici suit les idées principales originales, mais la conclusion de la preuve est simplifiée quelque peu grâce à l’approche pluri-potentielle de Diverio et Trapani.

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Quelques aspects arithmétiques de l’hyperbolicité Pietro Corvaja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Nous proposons un survol de l’étude des points entiers et de certains aspects de l’approximation diophantienne sur les variétés algébriques, et nous traitons des analogues arithmétiques de la notion d’hyperbolicité pour celles-ci. Suivant une conjecture due à Lang et Vojta, les variétés (quasi-projectives) définies sur des corps de nombres dont les points complexes forment une variété hyperbolique (au sens analytique complexe) devraient admettre des ensembles dégénérés de points entiers ou rationnels. En dimension 1, d’après les travaux de Siegel et Faltings, on sait que les notions analytique et arithmétique d’hyperbolicité sont équivalentes. Nous montrons, en nous concentrant principalement sur le cas de la dimension 2, que beaucoup de problème diophantiens apparemment sans rapport peuvent être ramenés à des questions portant sur la distribution des points entiers sur certaines surfaces algébriques. Des exemples significatifs sont les suivants. Le théorème de Darmon et Granville sur l’équation de Fermat généralisée xp + y q = z r est démontré ici d’une façon légèrement simplifiée et le lien avec l’hyperbolicité du triplet d’exposants (p, q, r) est développé en détail. Une conjecture sur les dénominateurs des points rationnels sur les courbes elliptiques est relié à la conjecture de Vojta, et une version plus faible est établie inconditionnellement. Un outil fondamental des preuves de finitude ou de dégénérescence des points entiers sur les variétés est fourni par l’approximation diophantienne. Cette théorie est aussi liée aux questions d’hyperbolicité, et en particulier un nouveau « gap principle » pour les points rationnels sur les courbes elliptiques est démontré et on montre que sa formulation est liée à une condition d’hyperbolicité. Fibrations en géométrie algébrique et applications Claire Voisin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Nous faisons un survol de différentes méthodes pour construire des fibrations rationnelles sur les variétés algébriques (c’est-à-dire des applications rationnelles dominantes de variétés normales, qui induisent des extensions algébriquement closes de corps de fonctions) et des applications de celles-ci. Ces fibrations sont un outil majeur dans la théorie de la classification des variétés algébriques. Les plus importantes d’entre elles sont la fibration d’Iitaka, la fibration MRC et la fibration Gamma. Nous les présentons avec plusieurs façons concrètes de les utiliser. Nous discutons enfin la fibration « core », introduite récemment par Campana, qui est un lien conjectural entre ces fibrations définies algébriquement et l’hyperbolicité.

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Around Brody lemma Julien Duval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Brody’s lemma is a basic tool in hyperbolicity. It provides an entire curve, i.e., a non constant holomorphic image of the complex line, out of a diverging sequence of holomorphic discs. Consequently Brody’s Lemma characterizes hyperbolicity in terms of absence of entire curves. We present direct applications of Brody’s Lemma, including the Green theorem (hyperbolicity of the complement of 5 lines in the projective plane) and an example of a hyperbolic surface of degree 6 in the projective space. We also describe a variant of Brody’s lemma aiming to better localize the entire curve it produces. As a byproduct of this variant, hyperbolicity is characterized in terms of a linear isoperimetric inequality for holomorphic discs.

Value Distribution Theory for Parabolic Riemann Surfaces Mihai Păun & Nessim Sibony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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We give versions of classical results in value distribution theory when the source space is a parabolic Riemann Surface, i.e., a surface with no non-constant bounded subharmonic functions. A vanishing theorem for such maps is obtained, of which a version of the classical Bloch Theorem and Ax-Lindemann Theorem are consequences. Results by Brunella and McQuillan are also extended in this context, with quite simple proofs.

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On algebraic hyperbolicity Erwan Rousseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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We study properties of algebraic nature that are expected to be related to hyperbolicity. A classical result of Demailly establishes lower bounds for the genus of curves in hyperbolic manifolds. The inequalities of Demailly are closely related to geometric Lang-Vojta’s conjectures claiming that curves on logarithmic pairs of general type should satisfy similar inequalities. Starting with the classical result of Bogomolov, which proves such inequalities for surfaces of general type with positive second Segre number, we focus on the alternative proof of Miyaoka, which makes the inequality effective (since constants can be chosen to be functions of Chern numbers of the surface). The proof is presented as an illustration of the theory of orbifolds of Campana: lower bounds on the genus of curves are obtained as consequences of some general orbifold Bogomolov-Miyaoka-Yau inequalities. A simple proof of the Kobayashi conjecture on the hyperbolicity of general algebraic hypersurfaces Jean-Pierre Demailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 We investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension n and of sufficiently large degree d ≥ dn in the complex projective space Pn+1 is hyperbolic. By a classical characterization due to Brody, such a variety does not possess non-constant entire holomorphic curves. As is well-known since the work of Green and Griffiths, one crucial ingredient is the geometric structure of certain jet bundles and their associated jet differentials. More precisely, one makes use of the so-called Demailly-Semple tower, which is a twisted tower of projective bundles related to jet differentials that are invariant by reparametrization. According to a fundamental vanishing theorem, global jet differentials with values in negative line bundles provide algebraic differential equations that all entire curves must satisfy. If the base locus of these differential equations is small enough, which is to say, if there are enough independent differential equations then all entire curves must be constant. In the early 2000’s Yum-Tong Siu proposed a somewhat different strategy that ultimately led to a proof in 2015. Siu’s proof, which based on Nevanlinna theory arguments combined with the use of slanted vector fields, appears to be long and delicate. In 2016 the conjecture was settled in a different way by Damian Brotbek, making direct use of Wronskian differential operators and associated multiplier ideals. Shortly afterwards Ya Deng showed how the approach could be completed to yield an explicit value of dn . We provide a short proof based on a drastic simplification of their ideas, along with a further improvement of Deng’s bound, namely dn = b(en)2n+2 /5c.

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We show that the same technique provides examples of smooth algebraic hypersurfaces of Pn+1 of low degree d = O(n2 ), following an approach due to Shiffman and Zaidenberg. McQuillan’s approach to the Green-Griffiths conjecture for surfaces Carlo Gasbarri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 After reviewing the proof by Bogomolov of the fact that, on smooth projective surfaces with c21 > c2 , curves of bounded geometric genus form a bounded family, we explain the main steps of the proof, given by McQuillan, of the Green-Griffiths conjectures for these surfaces. Viewed from afar the two proofs follow the same strategy, but the second requires a much deeper analysis of the tools involved. In order to describe McQuillan’s proof we explain the construction of the Ahlfors currents associated to entire curves in a variety and we show how these can be used to produce a substitute for intersection numbers. A proof of the tautological inequality in both the standard case and the logarithmic case is given. We explain how the hypotheses allow us to suppose that the involved entire curve (which a posteriori should not exist) is a leaf of a foliation. In order to simplify some technical points of the proof we impose some restrictions on the singularities of this foliation (the general case requires a much more involved analysis but the main ideas of the proof are already visible under this restriction). In the last section we give a very brief description of a possible strategy (proposed by McQuillan) for the proof of the general case of the conjecture, together with an explanation of the main difficulties that must be overcome. Generic positivity and applications to hyperbolicity of moduli spaces Benoît Claudon & Stefan Kebekus & Behrouz Taji . . . . . . . . . . . . . . . . 169 Moduli theory of algebraic varieties naturally includes the study of the degeneration of varieties in families. For example, the obstruction to non-trivial degeneration of certain families of varieties can be described as non-existence of some subvarieties in their associated moduli spaces. We study the conjectural role of the Kodaira dimension as an invariant that can be used to describe such obstructions. Recent advances in our understanding of the positivity properties of tangent sheaves of non-uniruled varieties, and the natural generalizations to the logarithmic setting for pairs (X, D), has proved to be a powerful tool in the study of Kodaira dimension of base spaces of varying families of manifolds. We give a detailed account of Campana and Păun’s generalization of the generic semi-positivity results of Miyaoka. In its simplest form this result asserts that quotients of the logarithmic cotangent bundle Ω1X (log D) have semi-positive slopes with respect to any ample divisor, as long as the log-canonical divisor KX + D is pseudo-effective. A key consequence then relates bigness of KX + D to positivity properties of the sheaf of pluri-(log-)differential forms. Together with a result of Viehweg

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and Zuo on the bigness of the sheaf of pluri-differential forms for moduli stacks of canonically polarized manifolds, one is led to a proof of a celebrated conjecture of Viehweg on the “algebraic hyperbolicity” ’ of such spaces: subvarieties of moduli (stacks) of canonically polarized manifolds are all of log-general type. Kobayashi hyperbolicity, negativity of the curvature and positivity of the canonical bundle Simone Diverio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 We give a detailed account of a recent breakthrough by Wu and Yau, generalized shortly afterwards by Tosatti and Yang (and also by Diverio and Trapani). The breakthrough sits at the crossroad of complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture of Kobayashi, stated at the very beginning of the theory, predicts that a complex projective (or more generally compact Kähler) Kobayashi hyperbolic manifold should have ample canonical bundle. On the one hand it is also known since the beginning of the theory that a compact complex manifold with a Hermitian metric whose holomorphic sectional curvature is negative is Kobayashi hyperbolic. On the other hand a compact Kähler manifold with ample canonical bundle is known—by the celebrated work of Aubin and Yau—to admit a Kähler metric with (constant) negative Ricci curvature. Wu and Yau’s theorem states that if a smooth projective manifold admits a Kähler metric with negative holomorphic sectional curvature, then it also admits a possibly different Kähler metric whose Ricci curvature is negative. The result can be therefore seen as a weak confirmation of Kobayashi’s conjecture above, since it gives the same conclusion but with the stronger hypothesis about the holomorphic sectional curvature. Beside a careful, fully detailed presentation of the proof of the Wu-Yau theorem, we take the opportunity to give some basic background material on complex differential geometry and several results, positive and negative, about the link between curvature and Kobayashi hyperbolicity. Some natural open questions are also discussed. The proof of the Wu-Yau theorem presented here closely follows the original main ideas by Wu and Yau, but the conclusion of the proof is simplified somewhat by using the pluripotential approach of Diverio and Trapani. Some arithmetic aspects of hyperbolicity Pietro Corvaja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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We give a survey of the study of integral points and some aspects of Diophantine approximation on algebraic varieties, and we treat arithmetic analogues of the notion of hyperbolicity for algebraic varieties. According to a conjecture by Lang and Vojta, those (quasi projective) algebraic varieties, defined over number fields, whose complex points form a hyperbolic

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manifold (in the complex analytic sense) should admit only degenerate sets of integral or rational points. In dimension one, after the work of Siegel and Faltings, it is known that the analytic and arithmetic notions of hyperbolicity are equivalent. We show, mainly focusing on the two-dimensional case, that many apparently unrelated Diophantine problems can be reduced to questions about the distribution of integral points on certain algebraic surfaces. Significant examples are the following. The theorem of Darmon and Granville on the generalized Fermat equation xp + y q = z r is proved here in a slightly simplified way and its connection with the hyperbolicity of the triple of exponents (p, q, r) is developed in detail. A conjecture about the denominators of rational points on elliptic curves is linked to Vojta’s conjecture, and a weaker version is unconditionally established. A main tool in the proofs of finiteness or degeneracy results for integral points on varieties is provided by Diophantine approximation. The theory of Diophantine approximation is also linked to questions of hyperbolicity, and in particular a new “gap principle” for rational points on elliptic curves is proved and its formulation is shown to be directly linked to a hyperbolicity condition. Fibrations in algebraic geometry and applications Claire Voisin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 We give a survey of various methods for constructing rational fibrations on algebraic varieties (i.e., dominant rational mappings of normal varieties that induce an algebraically closed extension of function fields), and their applications. These fibrations are a major tool in the classification theory of algebraic varieties. The most important among them are the Iitaka fibration, the MRC fibration, and the Gamma fibration. We present them together with several concrete modes of use. We discuss finally the core fibration, introduced more recently by Campana, which is a conjectural bridge between these algebraically defined fibrations and hyperbolicity.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

INTRODUCTION

Kobayashi hyperbolicity is ubiquitous in complex geometry. Despite its innocentsounding definition, given about 50 years ago by the Japanese mathematician S. Kobayashi, the notion of hyperbolicity of complex spaces has proved to be of a great depth (1). The most prominent evidence for this depth is surely the enormous number of (albeit often conjectural) counterparts in several branches of mathematics, including algebraic geometry, arithmetic geometry, complex analysis and complex differential geometry. In particular, on complex algebraic varieties Kobayashi hyperbolicity, which is definitely an analytic property of the variety, is expected to be completely characterized by algebro-geometric positivity properties of the canonical bundle of the variety and of all of its subvarieties. Moreover, whenever the variety is defined over a number field, one expects hyperbolicity to be characterized in terms of arithmetic properties of the variety, e.g., by finiteness of rational points. These conjectural characterizations were first proposed by S. Lang in the mid-eighties. To a large extent, they are the guiding light for a big amount of research done in the last 40 years. In fact, the Lang conjectures have led to a number of spectacular results. Let us be more precise. Given a complex manifold X, which we assume to be smooth for simplicity, and a tangent vector v ∈ TX , one can intrinsically measure the length FX (v) of v as follows:  FX (v) = inf ||u|| | u ∈ T∆ , df (u) = v , where ∆ ⊂ C is the complex unit disk, ||u|| is the length of the tangent vector u measured by the Poincaré metric of ∆, and the infimum is taken over all holomorphic maps f : ∆ → X and u ∈ T∆ such that df (u) = v. We say that X is (infinitesimally) (1)

My colleague, collaborator, and friend E. Rousseau told me some years ago that he had the honor to meet in person S. Kobayashi during a conference in Japan, before he passed away in 2012. In that occasion, obviously he could not resist and asked him: “Can I ask you how your definition of complex hyperbolicity came up to your mind?”. Apparently, Kobayshi answered that he was just trying naïvely to dualize Carathéodory notion of complex hyperbolicity to see whether it gave something interesting. . .

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Kobayashi hyperbolic if FX is positive definite on each fiber of TX and satisfies uniform lower bounds FX (v) ≥ ε||v||ω with respect to some (and hence any) Hermitian metric ω on X, when v ∈ TX,x and x ∈ X describes a compact subset of X. Thus, locally, Kobayashi hyperbolicity of a given complex space X is the obstruction to mapping into that space holomorphic disks of arbitrarily large radius whose origin passes through a given point with a given velocity (see J. Duval’s chapter for much more on this). When the complex space is in addition compact, one possible characterization of Kobayashi hyperbolicity is given by what is usually called Brody hyperbolicity, i.e., the absence of non constant holomorphic images of the complex line in X. In addition to this basic concept, we now review in an elementary way several notions of a different nature, that are a posteriori deeply intertwined with Kobayashi hyperbolicity. We shall use as leitmotif the case of compact Riemann surfaces. The central theme of this monograph is to understand the aforementioned intertwining in higher dimensions. Consider a compact Riemann surface X. From a topological point of view X is described by a single natural number: its topological genus g = g(X) ∈ N. We may distinguish at a first glance two noteworthy classes of compact Riemann surfaces: those of genus zero and those whose genus is at least one. Equivalently, by a topological uniformization theorem these two classes can be distinguished as simply connected and non-simply connected, and in the latter case the universal cover is always homeomorphic to the Euclidean plane. But now the complex structure (or, equivalently, the choice of an orientation plus a conformal class of riemannian metrics) comes into the picture. This extra datum yields further refinement, distinguishing genus zero, genus one and genus greater than one. Indeed, once the topological universal cover is endowed with the unique complex structure that makes the covering map a local biholomorphism, as a complex manifold it becomes biholomorphic to either the complex projective line P1 (C) (for g = 0), the complex plane C (for g = 1), or the unit disk ∆ (for g ≥ 2). The latter result is the celebrated Poincaré-Koebe Uniformization Theorem and in some sense this is where our story begins. Of course, it was shown much earlier by Liouville that, far from being biholomorphic to the unit disk, the complex plane admits no non-constant holomorphic maps to the unit disk! Since any holomorphic map F : C → X can be holomorphically lifted to the universal cover of X, by this discussion we immediately see that a compact Riemann surface is Kobayashi hyperbolic if and only if it has topological genus greater than or equal to two. We can characterize hyperbolic compact Riemann surfaces by other means as well. Let us start from the perspective of algebraic geometry. Any compact Riemann surface is an algebraic curve over the field of complex numbers. If we take a canonical divisor on X, i.e., the divisor of any meromorphic section of the canonical line bundle Vdim X ∗ ∗ KX = TX = TX , then its degree is 2g(X) − 2 by the Hurwitz Formula. Thus,

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a compact Riemann surface is hyperbolic if and only if the degree of its canonical bundle is positive. It is not hard to check, for instance by using Riemann-Roch formula, that a line bundle on a curve has positive degree if and only if it is ample, i.e., the complete linear system associated to some high tensor power of this line bundle provides an embedding in some projective space. We can thus characterize hyperbolic compact Riemann surfaces as those having ample canonical line bundle. Again by RiemannRoch, if KX is ample then the growth rate of the dimension of the space of global ⊗m holomorphic sections of KX is linear in m. A projective manifold X for which ⊗m 0 dim X dim H (X, KX ) ∼ m is said to be of general type. Therefore a compact Riemann surface is hyperbolic if and only if it is of general type. One possible way to prove uniformization begins with the construction on X of a Hermitian metric whose underlying Riemannian metric has constant curvature (we are speaking here of Gaussian curvature, though in complex dimension one practically all concepts of curvature—Riemannian sectional, Ricci, holomorphic sectional, holomorphic bisectional, and so forth—coincide). It is always possible (but highly non trivial if g ≥ 2) to construct such a metric, and the Gauß-Bonnet Theorem prescribes the sign of the curvature, which must be negative if and only if g ≥ 2. Thus we obtain yet another characterization: a compact Riemann surface is hyperbolic if an only if it carries a Hermitian metric of (constant) negative curvature. Last but not least, suppose that X is defined over a number field k, say X can be embedded in some complex projective space in such a way that it is cut out by a certain number of homogeneous polynomial equations whose coefficients are in k. For F/k a field extension, denote by X(F ) the set of points of X whose homogeneous coordinates can be chosen to be all in F . Then X is hyperbolic if and only if X(F ) is a finite set for every finite field extension F/k (the hardest part of this result, the “only if” part, is the content of the celebrated proof of Mordell’s Conjecture by Faltings). Thus we obtain a further characterization of the hyperbolicity of X, this time in terms of its arithmetic properties. At the end of this certainly incomplete list of characterizations it is worth mentioning, perhaps as an aside, a very important phenomenon. Even when one wants to consider naturally-defined families of compact Riemann surfaces, rather than a single surface, hyperbolicity manifests itself! Speaking very imprecisely, it is possible to prove that the moduli space of smooth compact Riemann surfaces of genus g is Kobayashi hyperbolic. As the reader might guess by this point, the extraordinary richness and depth of the connections described above gets considerably more subtle as the dimension increases, and almost all of the aforementioned characterizations remain largely conjectural (or even false!) in overly general settings. For instance, in dimension greater than one there is certainly no way to characterize hyperbolicity merely from a topological point of view. An example is given by smooth projective hypersurfaces of degree d: any two such surfaces are homeomorphic (and

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even diffeomorphic) since they belong to a same smooth proper family. It is now known that if d is large enough and the hypersurface is generic then it is Kobayashi hyperbolic (this challenging conjecture, made by Kobayashi in 1970, was proved in full generality only recently; see J.-P. Demailly’s chapter for more on this). On the other hand, for each d, the Fermat hypersurface of degree d always contains projective lines, hence non constant holomorphic images of the complex line, and therefore is not Kobayashi hyperbolic. For a Riemann surface the (holomorphic) cotangent bundle and the canonical bundle coincide, but this is of course no longer true as soon as the dimension gets bigger than one. If we try to study hyperbolicity in terms of some ampleness property in higher dimensions, we quickly realize that ampleness of the cotangent bundle is a sufficient but not a necessary condition for hyperbolicity while, at the other extreme, ampleness of the canonical bundle is not a sufficient condition (but was conjectured to be necessary by Kobayashi, again at the early stage of the theory in 1970—a conjecture that remains open in general). And looking again at the example of projective hypersurfaces of high degree, we know that the generic one is Kobayashi hyperbolic but has non-ample cotangent bundle, whereas the Fermat hypersurface is not hyperbolic but has ample canonical bundle. Moreover, Kobayashi hyperbolicity is inherited by subvarieties, while ampleness of the canonical bundle (or more generally bigness, i.e., being of general type) is not. One could reasonably speculate that the hereditary nature of Kobayahsi hyperbolicity may well have led S. Lang to state, in his prescient 1986 article “Hyperbolic and Diophantine Analysis,” the following key conjecture. Conjecture (Lang ’86). – A complex projective manifold X is Kobayashi hyperbolic if and only if X and all of its subvarieties are of general type. Lang’s conjecture, which remains wide open, has inspired a huge amount of mathematics. In particular, the chapter by C. Gasbarri, and in part—but from a somehow different point of view—the chapter by N. Sibony and M. Păun, deal with one direction: a projective manifold of general type should not admit any holomorphic map from the complex line whose image is Zariski dense. For the other direction, which is known in dimension at most two, a folklore approach is described in S. Diverio’s chapter and involves, besides standard conjectures coming from birational geometry and especially the Minimal Model Program, the evergreen idea of “decomposing” the problem into simpler pieces by means of fibrations. Moreover, the systematic use of fibrations in hyperbolicity-type problems and beyond is the core of the chapter by C. Voisin. It is also plausible that in order to check Kobayashi hyperbolicity algebraically in the same spirit of Lang’s conjecture, one may look merely at algebraic curves inside X, but require more from these curves than being of general type. In the mid-nineties Demailly singled out a property of Kobayashi hyperbolic projective manifolds that he has called algebraic hyperbolicity, whose definition is as follows. Having fixed an ample line bundle A → X, the manifold X is said to be algebraically hyperbolic if

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there exists an ε0 > 0 such that for every finite map ν : C → X from a compact Riemann surface C to X one has the inequality 2g(C) − 2 ≥ ε0 (C · ν ∗ A). Demailly conjectured that this property is equivalent to hyperbolicity. Aside from its connection to Kobayashi hyperbolicity via Demailly’s conjecture, algebraic hyperbolicity is an interesting property in its own right, and is the object of study in the chapter by E. Rousseau, especially in connection with quantitative estimates for the ε0 , say when A = KX , in terms of Chern numbers of the manifolds. When one translates the aforementioned ampleness and/or hyperbolicity properties into the language of complex differential geometry, several different notions of curvature come into the picture. Suppose that our complex projective manifold X is endowed with a Hermitian metric ω, which we take for simplicity to be Kähler. If the Riemannian sectional curvature of the underlying metric is constantly equal to −1, we get the classical Riemannian notion of real hyperbolic manifold, while if it is bounded above by a negative constant, say k < 0, we get the metric hyperbolicity-type notion of CAT(k) space. The holomorphic sectional curvature is the complex counterpart of the Riemannian sectional curvature and is nothing more that the Riemannian sectional curvature computed only on the 2-planes of the tangent space which are invariant under the complex structure. The holomorphic sectional curvature is of course the meaningful object when one wants to control the holomorphic curves traced into X (whose real tangent spaces inside TX have this invariance property) and so it should not come as a surprise that having negative holomorphic sectional curvature implies Kobayashi hyperbolicity. Unfortunately, the negativity of the holomorphic sectional curvature is far from being a characterization of hyperbolicity (and from the point of view of Lang’s Conjecture the latter observation parses well, since the notion of holomorphic sectional curvature has no natural algebro-geometric counterpart). On the other hand the Ricci curvature, a notion which has to do with volume, is nothing but the negative of the curvature of (the Chern connection for the metric induced by the Kähler volume on) the canonical bundle KX . As such, the Ricci curvature is directly connected to the ampleness of the canonical bundle. In light of the aforementioned Kobayashi conjecture and Yau’s solution of the Calabi conjecture, it is therefore reasonable to expect that a Kobayashi hyperbolic projective manifold admits a Kähler metric with negative Ricci curvature. More around this topic can be found in S. Diverio’s chapter. The arithmetic side of the story, treated in part in P. Corvaja’s chapter, is perhaps the most difficult to understand in higher dimensions. In the arithmetic setting there is also a precise conjecture by S. Lang that mimics the one-dimensional situation, stating that a complex projective manifold defined over a number field k is Kobayashi hyperbolic if and only if it has only a finite number of points with coordinates in any given number field extending k. There is also a sort of dictionary, essentially due to P. Vojta, that conjecturally translates statements from arithmetic geometry (especially diophantine approximation) into complex analysis (especially Nevanlinna’s value distribution theory, cf. N. Sibony and M. Păun chapter) and vice versa.

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Finally, moduli spaces of higher dimensional (canonically polarized) varieties also have tendency to possess hyperbolicity properties of sorts, as explained in the chapter by B. Claudon, S. Kebekus, and B. Taji. The situation is expectedly much more subtle than its one-dimensional counterpart, the moduli spaces of curves, and appropriate notions of positivity and hyperbolicity must be worked out to in order to properly understand the phenomenon. In summary, the present monograph is an attempt to collect a number of results, theories, and developments around this fascinating subject. Each chapter is written by a different author (or authors), and tries to focus on some particular aspect of Kobayashi hyperbolicity that has seen important progress in the decades since its conception. In this sense we try to provide a (most likely partial) state of the art of several relevant features of the story, addressed to as broad a population of readers as possible, and with particular attention paid to graduate students and young researchers in this population.

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The idea of making a monograph about the state of the art around Kobayashi hyperbolicity and algebraic varieties came to Tien-Cuong Dinh’s mind several years ago (at least nine, but maybe more. . . ), as a member of the editorial board of “Panoramas et Synthèses”. I am grateful to him for proposing me to edit this volume, especially for the trust he gave me considering that at the time I was quite young (but unfortunately I am not anymore now). I would like to warmly thanks all the colleagues who accepted enthusiastically to participate to this project, giving their invaluable contribution. For me it was a true pleasure but above all a great honor to work with them for this book. In particular, let me thank Jean-Pierre Demailly, Erwan Rousseau, and Claire Voisin for a very pleasant early brainstorming about the possible content of this book, as well as about a possible choice of contributors. At the same time I would like to sincerely apologize with those friends and colleagues who would have also been a natural choice as contributors here, but that I eventually could not consider: the available space is limited and at some point one has to make unpleasant choices! My gratitude also goes to Bertrand Remy who treated the submission and handled contacts with the various referees, and to Bertrand Remy and Serge Cantat who gave me a lot of crucial advices and suggestions to improve the exposition. A special thank goes to Odile Boubakeur, for the excellent editorial work and for the precise and constant control over the whole process. To finalize the present monograph took infinitely much more time than expected. While I apologize with all the people involved and thank them for their extreme patience, I also welcome that this delay allowed to include some very recent and spectacular results. Last but not least, I friendly thank Dror Varolin who helped me to give the introduction (and unfortunately the introduction only) of this book—which was initially written in my hybrid “Italian-English” language—a genuine English shape.

Ad Angela, Alessandra, e Caterina.

Roma, October 25, 2021 Simone Diverio

Panoramas & Synthèses 56, 2021, p. 1–12

AROUND BRODY LEMMA by Julien Duval

Abstract. – Brody’s lemma is a basic tool in hyperbolicity. It provides an entire curve, i.e., a non constant holomorphic image of the complex line, out of a diverging sequence of holomorphic discs. Consequently Brody’s Lemma characterizes hyperbolicity in terms of absence of entire curves. We present direct applications of Brody’s Lemma, including the Green theorem (hyperbolicity of the complement of 5 lines in the projective plane) and an example of a hyperbolic surface of degree 6 in the projective space. We also describe a variant of Brody’s lemma aiming to better localize the entire curve it produces. As a byproduct of this variant, hyperbolicity is characterized in terms of a linear isoperimetric inequality for holomorphic discs.

1. Brody lemma Let X be a compact complex manifold. An entire curve in X is a non constant holomorphic map f : C → X. It is a Brody curve if its derivative kf 0 k is bounded, where the norm is computed with respect to the standard metric on C and a given riemannian metric on X. Brody curves arise naturally as limits of sequences of larger and larger holomorphic disks, thanks to Brody lemma [5]. Brody lemma. – Let fn : D → X a sequence of holomorphic maps from the unit disk to a compact complex manifold. Suppose kfn0 (0)k unbounded. Then there exist affine reparametrizations rn of C such that fn ◦ rn converges locally uniformly toward a Brody curve, after extracting a subsequence. 2010 Mathematics Subject Classification. – 32Q45. Key words and phrases. – Brody curve, Ahlfors current, hyperbolicity.

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Proof. – We may suppose fn smooth up to the boundary (replacing fn (z) by fn ( z2 )). Denote by δ(z) the euclidean distance from z to ∂D. As the function δkfn0 k vanishes on ∂D it reaches its maximum inside D, say at an . This is where we will reparametrize fn . Let Dn be the disk D(an , δ(a2n ) ). We have kδfn0 kDn ≤ δ(an )kfn0 (an )k. But δ ≥ δ(a2n ) on Dn so kfn0 kDn ≤ 2kfn0 (an )k. z . let Dn0 be the preimage Define the reparametrization by rn (z) = an + kf 0 (a n )k n 0 of Dn by rn . Its radius is unbounded as δ(an )kfn (an )k ≥ kfn0 (0)k. We may suppose that Dn0 increases toward C after extracting. Moreover k(fn ◦ rn )0 kDn0 ≤ 2 and k(fn ◦ rn )0 (0)k = 1. By Ascoli theorem we may extract a subsequence of fn ◦ rn which converges locally uniformly toward a holomorphic map f : C → X such that kf 0 kC ≤ 2 and kf 0 (0)k = 1. It is a Brody curve. As a consequence we get a characterization of Kobayashi-hyperbolicity. Recall that the Kobayashi pseudometric of X at p in the direction v is K(p, v) = inf{r > 0 | ∃f : D → X holomorphic, f (0) = p, f 0 (0) = vr }. It measures the size of the holomorphic disks passing through a point in a given direction (the larger the disk through p in direction v the smaller K(p, v)). The manifold X is Kobayashi-hyperbolic if its pseudometric is non degenerate. Criterion. – A compact complex manifold X is Kobayashi-hyperbolic if and only if there is no entire curve in it. Indeed the vanishing of K(p, v) gives rise to a sequence of holomorphic disks fn : D → X such that fn (0) = p and fn0 (0) is unbounded in the direction of v. By Brody lemma we get a Brody curve in X. Conversely the Kobayashi pseudometric has to vanish along any entire curve. In the sequel we will say that U ⊂ X is hyperbolic if U does not contain any entire curve. For instance D is hyperbolic by Liouville theorem. Hyperbolicity is invariant under étale covering. Indeed if U → V is such a covering, any entire curve in U may be pushed down in an entire curve in V . Conversely any entire curve in V can be lifted to an entire curve in U . For instance compact complex curves of genus ≥ 2 are hyperbolic as they are uniformized by D. Another consequence of Brody lemma is that hyperbolicity is an open property. Openness. – Let X be a compact complex manifold and F ⊂ X a closed subset. If F is hyperbolic then so is any sufficiently small neighborhood of F . Otherwise we would get an entire curve in each -neighborhood F of F . This would allow us to construct a sequence of holomorphic disks fn : D → F n1 such that kfn0 (0)k is unbounded, giving at the limit a Brody curve in F . We discuss now some examples of hyperbolic complex surfaces. We start with the simplest hyperbolic complement, generalizing Picard theorem (the hyperbolicity of P1 (C)\3 points).

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Green theorem [9]. – Let L be a collection of five lines in general position in P2 (C). Then P2 (C) \ L is hyperbolic. Here general position means there is no triple point in the configuration. Proof. – Embedd P2 (C) into P4 (C) by z 7→ [l1 (z) : · · · : l5 (z)] using the equations of the lines. Call P its image and P ∗ = P \ H the complement of the collection H of coordinate hyperplanes in P4 (C). We want to prove the hyperbolicity of P ∗ . Let Fn be the self-map of P4 (C) given by z 7→ [z1n : · · · : z5n ]. It induces an étale covering from P4 (C) \ H to itself, so the hyperbolicity of P ∗ and Fn−1 (P ∗ ) are equivalent. The point is that Fn−1 (P ) converges toward a polyhedron whose hyperbolicity is easily checked by Liouville theorem. This will conclude the proof by openness. Let us make it precise. By the general position assumption, P avoids the coordinate lines (zi = zj = zk = 0). T So P is contained in X = {i,j,k} (max(|zi |, |zj |, |zk |) ≥ kzk) for some small  > 0. Here kzk = max|zi |. Now Fn−1 (X ) = X n1 decreases toward the polyhedron X = X1  when n goes to infinity. Note that on X kzk is reached on one out of three arbitrary components of z, meaning that kzk is always reached on three components at least. So X is alternatively seen as a finite union of faces Xi,j,k = (|zi | = |zj | = |zk | = kzk). Let us check its hyperbolicity. Let f : C → P4 (C) be a holomorphic map. It lifts to (C5 )∗ (essentially because H 1 (C, O ∗ ) = 0, see [10]) so f (z) = [f1 (z) : · · · : f5 (z)] where fi is holomorphic. Now if f (C) ⊂ X it has to spend some time in one of the faces, say X1,2,3 . This implies by analytic continuation that |f1 | = |f2 | = |f3 | everywhere. But kf k = max(|f1 |, |f2 |, |f3 |) by definition of X. Hence |f1 | dominates the other components everywhere, meaning that f is bounded in the chart (z1 = 1) thus constant by Liouville theorem. Therefore X does not contain any entire map. Remark. – This argument is dynamical in essence as X is nothing but an intermediate Julia set for F2 which is known to attract backward iterates of a generic plane. It stems from the proof of Picard theorem by A. Ros [17] and works in any dimension [2]. We focus now on examples of hyperbolic surfaces of low degree in P 3 (C). This fits into the framework of Kobayashi conjecture which predicts that a generic surface of degree ≥ 5 in P3 (C) is hyperbolic. It holds true for degree ≥ 18 [16] but few examples of hyperbolic surfaces of smaller degree are known, according to the motto “the lower the degree the harder the hyperbolicity”. Here we adapt a deformation method due to M. Zaidenberg [19] to produce examples of degree 6 by reduction to Green theorem. This also works for higher degrees and higher dimensions [11]. Note that the following remains open. Question. – Find a hyperbolic quintic surface in P3 (C). We will use Brody lemma in the following form.

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Sequences of entire curves. – Let X be a compact complex manifold. Then any sequence of entire curves in X can be made converging toward an entire curve after reparametrization and extraction. Indeed let (fn ) be the sequence of entire curves. By translating we may suppose that fn0 (0) does not vanish and by dilating that actually kfn0 (0)k is unbounded. It remains to apply Brody lemma. We will also invoke the following fact. Stability of intersections. – Let X be a complex manifold and H ⊂ X an analytic hypersurface. Suppose that a sequence (fn ) of entire curves in X converges toward an entire curve f . If f (C) is not contained in H then f (C) ∩ H ⊂ lim fn (C) ∩ H. Indeed let z be a point in f −1 (H) and h a local equation of H near f (z). As h ◦ f does not vanish identically near z, h ◦ fn has to have a zero in any small neighborhood of z for large n by Rouché theorem. We construct now our example of hyperbolic sextic surface as a suitable small deformation of a union of six planes (see also [6], [7] for other examples). A hyperbolic sextic surface. – Let (Pi = (pi = 0)) be a collection of six planes in general position in P 3 (C). Then we can find a sextic surface S = (s = 0) such that the surface Σ = (Πpi = s) is hyperbolic for  6= 0 sufficiently small. Here general position means there is no quadruple point in the configuration. Moreover S will be in general position with respect to the Pi , in the sense that it will avoid the triple points of the configuration. Note that by definition Σ ∩ Pi ⊂ S. Proof. – The first step reduces the problem to the hyperbolicity of complements. It is the heart of Zaidenberg’s method (see [19]) and goes as follows. If Σn is not hyperbolic for n going to zero we have entire curves fn : C → Σn . By Brody lemma we get at the limit an entire curve f : C → Σ0 = ∪Pi . It lands inside one of the planes. We analyze now its position with respect to the other planes. The crucial remark is the following. If f (C) is not contained in Pi then f (C) ∩ Pi ⊂ S. Indeed by stability of intersections f (C) ∩ Pi ⊂ lim fn (C) ∩ Pi ⊂ lim Σn ∩ Pi ⊂ S. We infer that f (C) cannot land into a double line of the configuration of planes. If it were the case f (C) would have to avoid the 4 triple points on the line by the remark and the general position of S, contradicting Picard theorem. We end up with f (C) contained in one plane and avoiding the others except at points of S, again by the remark. So f (C) is in a complement of the form S Pi \ ( j6=i Pj \ S). Hence we are finished if we are able to find a sextic surface S such that all these complements are hyperbolic. The second step consists in constructing this sextic surface. Note that the situation is similar to Green theorem. We have plane complements of five lines on which a few points are deleted. To create S we proceed by deformation in order to remove

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these points on more S and more double lines. Our starting point is the collection of complements Pi \ ( j6=i Pj ) which are hyperbolic by Green theorem. We want to remove points on a double line, say L = P1 ∩ P2 , keeping the hyperbolicity. For this consider a sextic surface S0 = (s0 = 0) in general position with respect to the Pi and deform it toward the union of the remaining planes by taking S1 = (p3 p4 p25 p26 = 0 s0 ) for a small 0 6= 0. In restriction to L, this pushes the points of S the sextic surface toward the triple points. So the complement P1 \( i6=1 Pi \(L∩S1 )) is S close to P1 \ ( i6=1 Pi ), hence hyperbolic by a suitable openness argument. This is a particular case of the following lemma which we apply inductively to conclude. Lemma. – Let ∆k be a collection of k double lines, L = Pi1 ∩ Pi2 an extra one and ∆k+1S= ∆k ∪L. Assume Sk = (sk = 0) already constructed such S that the complements Pi \ ( j6=i Pj \ (∆k ∩ Sk )) are hyperbolic. Then so are Pi \ ( j6=i Pj \ (∆k+1 ∩ Sk+1 )) where Sk+1 = (pi3 pi4 p2i5 p2i6 = k sk ) for any small enough k 6= 0. Note that Sk+1 is still in general position with respect to the Pi if Sk was. Remark also that the geometry does not change on ∆k . We have ∆k ∩ Sk = ∆k ∩ Sk+1 . Proof of the lemma. – Take L = P1 ∩ P2 for simplicity. If we cannot find such an k , we have a sequence of sextic surfaces Sk+1,n converging toward P3 ∪ P4 ∪ P5 ∪ P6 and entire curves fn (C) sitting in one of the corresponding complements, say S P1 \ ( j≥2 Pj \ ((∆k ∩ Sk ) ∪ (L ∩ Sk+1,n ))). We get at the limit an entire curve f (C) in P1 . As before it cannot degenerate inside a double line. By stability of intersections, for j ≥ 2 we have f (C) ∩ Pj ⊂ limfn (C) ∩ Pj ⊂ (∆k ∩ Sk ) ∪ lim L ∩ Sk+1,n . If j ≥ 3 we infer that f (C) ∩ Pj ⊂ ∆k ∩ Sk as Pj ∩ L ∩ Sk+1,n is empty by general S position. Note now that lim L ∩ Sk+1,n consists in the triple points of L hence sits in j≥3 Pj . Then thanks to S the previous case we also have f C) ∩ P2 ⊂ ∆k ∩ Sk . Therefore f (C) lands in P1 \ ( j≥2 Pj \ (∆k ∩ Sk )) contradicting the hypothesis. 2. A variant A drawback of Brody lemma is the lack of information about the location of the entire curve it produces. It might land far away from the points where the disks blow up. Here is a simple example due to J. Winkelmann (see also [18], and [10] for background on blow-ups). Example. – Let A = C2 /(Z ⊕ iZ)2 be the standard torus and π : A˜ → A the blowup of A at a point p. Take a dense injective line L in A, say L = (z2 = λz1 ) for λ irrational. Consider the sequence of disks fn (D) on L given by fn (z) = (nz, λnz). Let f˜n the strict transform of fn . If f˜ is a Brody curve obtained from the f˜n by reparametrization, as in the Brody lemma, then f˜(C) is contained in the exceptional divisor E.

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Indeed if f˜(C) is not contained in E it projects down to a (non constant) Brody curve f (C) in A. It is linear by Liouville theorem (the derivative of the lift of f in C2 is bounded), and parallel to L by construction. So it is again a dense injective line L0 . In particular the derivative of π −1 | 0 = f˜◦ f −1 is bounded because f 0 is constant. On L the other hand consider an open cone of vertex p transversal to the direction of L0 . By density of L0 it cuts out a sequence of smaller and smaller disks Dn on L0 converging toward p. But π −1 (Dn ) converges toward a non constant disk in E (the basis of the cone) by definition of a blow-up. This contradicts the boundedness of (π −1 | 0 )0 . L We want to address this problem by constructing entire curves where the disks accumulate in terms of area. To formulate this we introduce the notion of Ahlfors current. We start by briefly recalling what a current is (see [10] for more background on currents). Let X be a compact complex manifold of dimension n endowed with a hermitian metric. Denote by ω its area form. A current (of bidimension (1, 1)) in X is a continuous linear form on the space of differential forms (of bidegree (1, 1)) of X. It is positive if it is non negative on positive forms (those whose restriction to any complex tangent line is an area form), and closed if it vanishes on exact forms. Instances of such currents are (1) the current of integration on any complex curve Z [C] : α 7→ = α, C

(2) the current given by a positive closed form β of bidegree (n − 1, n − 1), Z α 7→ = β ∧ α. X

In general a positive current may be seen as a form of bidegree (n − 1, n − 1) with measure coefficients. A useful fact is the compactness of the set of positive currents of R mass 1 (here the mass of T is or X T ∧ ω if we see T as a form with measure coefficients). Let now fn : D → X be a sequence of holomorphic disks smooth up to the boundary. We make the standing assumption ln (A) → 0, an where an = area (fn (D)) and ln = length (fn (∂D)). In other words the boundaries of the disks become asymptotically negligible. By compactness we may suppose that the sequence of normalized currents of integration [fna(D)] converges toward a positive n current T of mass 1. It is closed because of (A). This is the Ahlfors current associated to the sequence of disks. Let us present our variant of Brody lemma. Variant [8]. – Let T be an Ahlfors current in X and K ⊂ X a compact set charged by T . Then there exists an entire curve passing through K.

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R Here K charged by T means K T ∧ ω > 0. In other words the proportion of area of fn (D) near K remains bounded away from zero. As in Brody lemma the entire curve f : C → X intersecting K is obtained by reparametrizing the holomorphic disks. By the very construction described in [8] it cuts K out on a set of positive area. So by analytic continuation we have more when K is analytic. If Y ⊂ X is an analytic subset charged by T then Y contains an entire curve. In particular if C is a compact complex curve charged by T then genus(C) ≤ 1. As a consequence we get another characterization of hyperbolicity with a flavor of negative curvature (see also [12] for another proof). Criterion. – Let X be a compact complex manifold. Then X is hyperbolic if and only if its holomorphic disks satisfy a linear isoperimetric inequality. This means that there exists a constant C such that for any holomorphic disk f : D → X smooth up to the boundary we have area(f (D)) ≤ C length(f (∂D)). Indeed if it does not hold we get a sequence of holomorphic disks satisfying (A), therefore an Ahlfors current and so an entire curve in X by the theorem. Conversely the presence of an entire curve forbids a linear isoperimetric inequality by the following lemma, due to Ahlfors. Ahlfors lemma [1]. – Let f : C → X be an entire curve. Then there exists a sequence rn → +∞ such that the corresponding sequence of holomorphic disks fn : D → X, z 7→ f (rn z) satisfies (A). Proof. – Denote by l(r) the length of f (∂(rD)) and a(r) the area of f (rD) (r > 0). If λ(z)|dz| is the pull-back by f of the metric on X then in polar coordinates Z 2π Z 2π l(r) = λ(r, θ)rdθ, a0 (r) = λ2 (r, θ)rdθ. 0

0

By Cauchy-Schwarz inequality we get l (r) ≤ 2πr a0 (r). Dividing by 2πr a2 (r) and integrating we infer that Z +∞ 2 Z +∞ 0 l (r) dr 1 a (r) ≤ dr ≤ < +∞. 2 (r) 2πr 2 (r) a a a(1) 1 1 2

This gives a sequence rn → +∞ such that

l(rn ) a(rn )

→ 0.

Ahlfors lemma also gives a description of rational curves (maybe singular) as entire curves of bounded area. Rational curves. – Let f : C → X be an entire curve such that area(f (C)) < +∞. Then f extends to a holomorphic map from P 1 (C) to X, a rational curve.

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Indeed by Riemann removable singularity theorem it is enough to extend f continuously at infinity. Because of the bounded area, applying Ahlfors lemma we get a sequence rn such that length(f(∂(rn D))) → 0. We may suppose that f (∂(rn D)) converges toward a point p. It suffices to show that the annuli An = f (rn D \ rn−1 D) also converge toward p. Note that area(An ) → 0. If a point qn of An remains far from p say at distance > , then for large n An ∩B(qn , ) is a proper holomorphic curve in B(qn , ) passing through qn . By Lelong inequality we have area(An ∩ B(qn , )) ≥ c 2 (c depending only on X). This is the contradiction. Lelong inequality [13]. – Let C be a proper holomorphic curve in B(0, ) ⊂ Ck , passing through 0. Then area(C) ≥ π 2 . Proof. – For 0 < r <  put Cr = C ∩ B(0, r) and a(r) = area(Cr ). Note that ∂Cr = C ∩ ∂B(0, r) by properness. We claim that a(r) r 2 is increasing. Assuming this we get

as C goes through a(s) a(r) 1 − 2 = 2 s2 r s Z =

area(C) a(r) a(r) = lim 2 ≥ lim 2 ≥ π r→ r r→0 r 2 0. As for the claim if 0 < r < s <  we have Z Z Z Z 1 1 1 c 2 c 2 c 2 dd kzk − 2 dd kzk = 2 d kzk − 2 dc kzk2 r Cr s ∂Cs r ∂Cr Cs Z Z c 2 c 2 d log kzk − d log kzk = ddc log kzk2 ≥ 0.

∂Cs

∂Cr c

Cs \Cr

2

Here we used the fact that dd kzk is the euclidean metric and that ddc log kzk2 is a non negative form, as well as Stokes theorem. A consequence is the compactness of holomorphic disks under an area bound in absence of rational curves. Compactness of disks. – Assume X does not contain any rational curve. Let fn : D → X be a sequence of holomorphic disks of bounded area. Then it converges locally uniformly toward a holomorphic disk after extracting. Indeed if (fn ) does not admit any converging subsequence, then kfn0 k has to be unbounded on a smaller disk (if not apply Ascoli theorem). So by Brody lemma we get an entire curve of bounded area, contradicting the absence of rational curve. Let us now sketch a proof of our variant under this hypothesis (absence of rational curves) for simplicity. Proof of the variant. – Recall that we are given an Ahlfors current T = lim [fna(D)] n with alnn → 0, charging K. Note that the hypothesis forces an to be unbounded. Let U be a neighborhood of K. For simplicity we will just construct an entire curve passing through U out of these disks. The full statement would follow in the same

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way taking a sequence of neighborhoods shrinking to K and extracting diagonally. By assumption for large n we have area(fn (D) ∩ U ) ≥ . an The strategy for constructing our entire curve is to look for germs of it, small disks passing through U , try and double them as long as we can and pass to the limit using an area bound. Let us make this scheme more precise. Consider collections F n of round disks centered in D depending on n. They are card( F n ) disjoint if the disks in F n are disjoint, and consistent if is bounded from an below, say by δ. Let us call disjoint consistent collections simply families. Families automatically contain subfamilies with an area bound. Indeed for at least half of the disks D of F n we have area(fn (D ∩ D)) ≤ 2δ . Let us explain how to get families of germs through U . For each point z of D let Dz be the smallest disk centered at z such that area(fn (Dz ∩ D) ∩ U ) = 1. By Besicovich’s covering theorem (see [14] p.30) we may extract from this collection a fixed number N of disjoint subcollections covering together D. Take the one with most area in U . This gives the families F n of germs. We will double them now. Given a disk D write 2D for the disk centered at the same point and of radius twice. Consider the collections 2 F n = {2D, D ∈ F n }. If we are able to extract families from them, then we say that the doubling process works. In this case as already seen we get an area bound. Moreover we may suppose that the disks 2D stay mostly inside D. If not we would get families of such disks for which on one hand 2D ∩ ∂D is relatively big and on the other hand 2ln length(fn (2D ∩ ∂D)) ≤ δa (same argument as for the area bound). But alnn → 0 by n assumption and areafn (D ∩ D)) ≥ 1 by construction. This would therefore contradict the following lemma (after reparametrization). Lemma. – Let hn : D+ → X be a sequence of holomorphic maps from the upper half unit disk D+ to X, smooth up to ]−1, 1[ and of bounded area. Suppose that length(hn (]−1, 1[)) → 0. Then area(hn (rD+ )) → 0 for r < 1. Before proving this lemma we sketch the end of the argument. Assume that we are able to iterate this doubling process indefinitely. Then we have families 2k F n for each k (decreasing in k). By definition we know that card(2k F n ) ≥ δk an meaning that 2k F n is non empty for n big enough. As before we get an area bound Ck for the images of its disks. Consider now such a disk 2k D in 2k F nk . Reparametrizing D by D we infer that gk = fnk ◦ rk : 2k D → X satisfies the following bounds area(gk (2l D)) ≤ Cl for 0 ≤ l ≤ k. After extracting we may pass to the limit because of these local area bounds and get an entire map f : C → X such that area(f (C) ∩ U ) ≥ 1. But in general the doubling process may fail at some step, at the first for instance. Then a combinatorial argument shows that there exist in F n arbitrary long chains of

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disks D1 , . . . , Dk of fast decreasing size, such that 2Dk ⊂ 2Dk−1 ⊂ · · · ⊂ 2D2 ⊂ 2D1 . Now instead of reasoning with disks, we involve the annuli 2Di \ 2Di+1 . Actually we have families of such annuli. With roughly the same arguments as before for the disks, by reparametrizing the middle annulus of these chains and passing to the limit, we get this time a holomorphic map f : C∗ → X such that area(f (C∗ ) ∩ U ) ≥ 1. See [8] for more details. Proof of the lemma. – Recall that X does not contain any rational curve. So after extracting we may suppose that hn → h locally uniformly. We may also assume that hn (]−1, 1[) → p and h∗n ω → µ weakly, where p is a point, ω the hermitian form on X and µ a positive measure. We first prove that h ≡ p. This is simple if hn (D+ ) remains in a chart near p. Indeed in this case we may assume hn with values in the unit ball of Ck and p = 0. Consider, for a in rD+ (r < 1), the Green function ga with respect to rD+ with pole at a. As log khn k is subharmonic we have Z Z c log khn (a)k ≤ log khn kd ga ≤ log khn kdc ga → −∞. ∂(rD+ )

]−r,r[

So hn (a) → 0. The general case follows by analytic continuation. Indeed take a point a in ]−1, 1[ with small mass for µ. So the area of hn (2d+ ) is small, where d+ is a small half disk centered at a. Then a length-area estimate similar to the one used in Ahlfors lemma implies that hn (∂(tn d+ )) remains near p for some 1 < tn < 2. Hence hn (d+ ) also remains near p by Lelong inequality. Therefore the previous argument applied to d+ gives h| + ≡ p which concludes by analytic continuation. d Actually we have a bit more. Namely hn (rD+ ) → p. This can be seen by reparametrization. If not after extracting we would get a sequence of points an (= xn + iyn , yn → 0) in rD+ such that hn (an ) remains far from p. Then gn (z) = hn (xn + 2yn z) would share exactly the same assumptions as the hn , but with gn ( 2i ) remaining far from p. This would contradict the previous paragraph. Finally we prove that area(hn (rD+ )) → 0. Take r < r0 < 1. As hn (r0 D+ ) → p we may suppose as before that hn takes its values in the unit ball of Ck and p = 0. We use an argument from value distribution theory. We have Z r0  Z Z r0  Z   dt r0 + c 2 dt area(hn (rD )) log( ) ≤ dd khn k = dc khn k2 r t t r tD+ r ∂(tD+ ) 0 Z r0  Z π Z Z r    dt d = khn (teiθ )k2 dθ dt + dc khn k2 . dt 0 t r r [−t,t] Rπ Rπ The first integral reads as 0 khn (r0 eiθ )k2 dθ − 0 khn (reiθ )k2 dθ which goes to 0 as hn → 0. The second integral is controlled by length(hn (]−1, 1[)) which also goes to 0.

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Bloch principle. – Let us mention to finish an application to Bloch principle due to M. McQuillan. In the classical setting Bloch principle reads as follows. Consider a configuration of four lines in general position in P 2 (C). It defines three extra lines joining opposite double points, the diagonals. Borel [4] remarked that the complement of the configuration is hyperbolic modulo the diagonals. Any entire curve avoiding the configuration has to be contained in a diagonal. Bloch [3] translated this in terms of disks. A diverging sequence of holomorphic disks avoiding the configuration has to converge toward the diagonals. In other words diverging sequences of disks localize precisely where entire curves are. In [15] McQuillan extends this principle to general logarithmic surfaces by using the variant. Thanks to the referee for his suggestions. References [1] L. Ahlfors – “Zur Theorie der Überlagerungsflächen”, Acta Math. 65 (1935), p. 157– 194. [2] F. Berteloot & J. Duval – “Sur l’hyperbolicité de certains complémentaires”, Enseign. Math. 47 (2001), p. 253–267. [3] A. Bloch – “Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires”, Ann. Sci. École Norm. Sup. 43 (1926), p. 309–362. [4] E. Borel – “Sur les zéros des fonctions entières”, Acta Math. 20 (1897), p. 357–396. [5] R. Brody – “Compact manifolds and hyperbolicity”, Trans. Amer. Math. Soc. 235 (1978), p. 213–219. [6] C. Ciliberto & M. Zaidenberg – “Scrolls and hyperbolicity”, Internat. J. Math. 24 (2013), 1350026, 25. [7] J. Duval – “Une sextique hyperbolique dans P3 (C)”, Math. Ann. 330 (2004), p. 473– 476. [8]

, “Sur le lemme de Brody”, Invent. math. 173 (2008), p. 305–314.

[9] M. L. Green – “Some Picard theorems for holomorphic maps to algebraic varieties”, Amer. J. Math. 97 (1975), p. 43–75. [10] P. Griffiths & J. Harris – Principles of algebraic geometry, Wiley-Interscience, 1978. [11] D. T. Huynh – “Examples of hyperbolic hypersurfaces of low degree in projective spaces”, Int. Math. Res. Not. 2016 (2016), p. 5518–5558. [12] B. Kleiner – “Hyperbolicity using minimal surfaces”, preprint. [13] P. Lelong – “Propriétés métriques des variétés analytiques complexes définies par une équation”, Ann. Sci. École Norm. Sup. 67 (1950), p. 393–419. [14] P. Mattila – Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math., vol. 44, Cambridge Univ. Press, 1995. [15] M. McQuillan – “The Bloch principle”, preprint arXiv:1209.5402. [16] M. Păun – “Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity”, Math. Ann. 340 (2008), p. 875–892.

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[17] A. Ros – “The Gauss map of minimal surfaces”, in Differential geometry, Valencia, 2001, World Sci. Publ., River Edge, NJ, 2002, p. 235–252. [18] J. Winkelmann – “On Brody and entire curves”, Bull. Soc. Math. France 135 (2007), p. 25–46. [19] M. Za˘ıdenberg & B. Shiffman – “New examples of Kobayashi hyperbolic surfaces in CP3 ”, Funktsional. Anal. i Prilozhen. 39 (2005), p. 90–94.

Julien Duval, Laboratoire de Mathématiques, Université Paris-Saclay, 91405 Orsay cedex, France [email protected]

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Panoramas & Synthèses 56, 2021, p. 13–72

VALUE DISTRIBUTION THEORY FOR PARABOLIC RIEMANN SURFACES by Mihai Păun & Nessim Sibony

Abstract. – We give versions of classical results in value distribution theory when the source space is a parabolic Riemann Surface, i.e., a surface with no non-constant bounded subharmonic functions. A vanishing theorem for such maps is obtained, of which a version of the classical Bloch Theorem and Ax-Lindemann Theorem are consequences. Results by Brunella and McQuillan are also extended in this context, with quite simple proofs.

1. Introduction Let X be a complex hermitian manifold. S. Kobayashi introduced a pseudodistance, determined by the complex structure of X. We recall here its infinitesimal version, cf. [19]. Given a point x ∈ X and a tangent vector v ∈ TX,x at X in x, the length of v with respect to the Kobayashi-Royden pseudo-metric is the following quantity kX,x (v) := inf{λ > 0; ∃f : D → X, f (0) = x, λf 0 (0) = v}, where D ⊂ C is the unit disk, and f is a holomorphic map. Let |v| denote the hermitian length of the tangent vector v. If kX,x (v) ≥ c|v|, with c > 0 as function on the tangent space of X, we say that X is Kobayashi hyperbolic. In general, we remark that it may very well happen that kX,x (v) = 0. However, if X is compact, then thanks to Brody re-parametrization lemma this situation has a geometric counterpart, as follows. If there exists a sequence (xn , vn ) such that |vn | = 1 and such that lim kX,xn (vn ) = 0, then one can construct a holomorphic non-constant map f : C → X. The point x = lim xn is not necessarily in the image of f .

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Therefore, if any entire curve drawn on X is constant, then the pseudo-distance defined above is a distance, and we say that X is Brody hyperbolic, or simply hyperbolic (since most of the time we will be concerned with compact manifolds). As a starting point for the questions with which we will be concerned with in this article, we have the following result. Proposition 1.1. – Let X be a compact Kobayashi hyperbolic manifold. Let C be a Riemann surface. Let E ⊂ C be a closed, countable set. Then any holomorphic map f : C \ E → X admits a holomorphic extension fe : C → X. In particular, in the case of the complex plane we infer that any holomorphic map f : C \ E → X, must be the restriction of an application fe : P1 → X (under the hypothesis of Proposition 1.1). We will give a proof and discuss some related statements and questions in the first paragraph of this paper. Observe however that if the cardinal of E is at least 2, then C \ E is Kobayashi hyperbolic. Our next remark is that the surface C\E is a particular case of a parabolic Riemann surface. We recall here the definition. A Riemann surface Y is parabolic if any bounded subharmonic function defined on Y is constant. This is a large class of surfaces, including e.g., Y \ Λ, where Y is a compact Riemann surface of arbitrary genus and Λ ⊂ Y is any closed polar set. It is known (cf. [1], [35], page 80) that a non-compact Riemann surface Y is parabolic if and only if it admits a smooth exhaustion function σ : Y → [1, ∞[ such that τ := log σ is harmonic in the complement of a compact set of Y . Moreover, we impose the normalization Z (1) ddc log σ = 1, Y

c

where the operator d is defined as follows √ −1 c d := (∂ − ∂). 4π On the boundary S(r) := (σ = r) of the parabolic ball of radius r we have the induced measure dµr := dc log σ | . S(r)

The measure dµr has total mass equal to 1, by the relation (1) combined with Stokes formula. Since we are dealing with general parabolic surfaces, the growth of the Euler characteristic of the balls B(r) = (σ < r) will appear very often in our estimates. We introduce the following notion.

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Definition 1.2. – Let ( Y , σ) be a parabolic Riemann surface, together with an exhaustion function as above. For each t ≥ 1 such that S(t) is non-singular we denote by χσ (t) the Euler characteristic of the domain B(t), and let Z r χσ (t) dt Xσ (r) := t 1 be the (weighted) mean Euler characteristic of the ball of radius r. Actually, we will mostly use a related function X+ σ (r), whose definition is a bit more technical than 1.2, cf. Definition 3.4. If Y = C, then Xσ (r) is bounded by log r. The same type of bound is verified if Y is the complement of a finite number of points in C. If Y = C \ E where E is a closed polar set of infinite cardinality, then things are more subtle, depending on the density of the distribution of the points of E in the complex plane. However, an immediate observation is that the surface Y has finite Euler characteristic if and only if (2)

Xσ (r) = O(log r).

In the first part of this article we will extend a few classical results in hyperbolicity theory to the context of parabolic Riemann surfaces, as follows. We will review the so-called “first main theorem” and the logarithmic derivative lemma for maps f : Y → X, where X is a compact complex manifold. We also give a version of the first main theorem with respect to an ideal J ⊂ O X . This will be a convenient language when studying foliations with singularities. As a consequence, we derive a vanishing result for jet differentials, similar to the one obtained in case Y = C, as follows. Let P be a jet differential of order k and degree m on a projective manifold X, with values in the dual of an ample bundle (see [11]; we recall a few basic facts about this notion in the next section). Then we prove the following result. Theorem 1.3. – Let Y be a parabolic Riemann surface. We consider a holomorphic map f : Y → X such that we have (†)

X+ σ (r). = 0. r→∞ Tf,ω (r)

lim sup

Let P be an invariant jet differential of order k and degree m, with values in the dual of an ample line bundle. Then we have P

(jk (f )) = 0

identically on Y .

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For example, the requirement above is satisfied if Y has finite Euler characteristic and infinite area. We will call the image of f : Y → X a parabolic curve. In the previous statement we denote by jk (f ) the k th jet associated to the map f . If Y = C, then this result is well-known, starting with the seminal work of A. Bloch cf. [3]; see also [12], [34] and the references therein, in particular to the work of T. Ochiai [27], Green-Griffiths [14] and Y. Kawamata [16]. It is extremely useful in the investigation of the hyperbolicity properties of projective manifolds. In this context, the above result says that the vanishing result still holds in the context of Riemann surfaces of (possibly) infinite Euler characteristic, provided that the growth of the function X+ σ (r), is slow when compared to Tf,ω (r). It also holds when the source is the unit disk, provided that the growth is large enough. As a consequence of Theorem 1.3 we obtain the following result (see Section 4, Corollary 4.6). Let X be a projective manifold, and let D = Y1 + · · · + YN be an effective snc (i.e., simple normal crossings) divisor. We assume that there exists a logarithmic jet differential P on (X, D) with values in a bundle A−1 , where A is ample. Let f : C → X be an entire curve which do not satisfy the differential equation defined by P . Then we obtain a lower bound for the number of intersection points of f (Dr ) with D as r → ∞, where Dr ⊂ C is the disk of radius r. Concerning the existence of jet differentials, we recall Theorem 0.1 in [13], see also [23]. Theorem 1.4. – Let X be a manifold of general type. Then there is a couple of integers m  k  0 and a (non-zero) holomorphic invariant jet differential P of order k and degree m with values in the dual of an ample line bundle A. Thus, our Result 1.3 can be used in the context of the general type manifolds. As a consequence of Theorem 1.3, we obtain the following analogue of Bloch’s theorem. It does not seem to be possible to derive this result by using e.g., AhlforsSchwarz negative curvature arguments. Observe also that we cannot use a BrodyGreen type argument, because the Brody reparametrization lemma is not available in our context. Theorem 1.5. – Let CN /Λ be a complex torus, and let Y be a parabolic Riemann surface of finite Euler Characteristic. We consider a holomorphic map f : Y → CN /Λ. Then the smallest analytic subset X containing the closure of the image of f is either the translate of a sub-torus in CN /Λ, or there exists a map R : X → W onto a general type subvariety of an abelian variety W ⊂ A such that the area of the curve R ◦ f ( Y ) is finite. In the second part of this paper our aim is to recast some of the work of M. McQuillan and M. Brunella concerning the Green-Griffiths conjecture in the parabolic setting. We first recall the statement of this problem.

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Conjecture 1.6 ([14]). – Let X be a projective manifold of general type. Then there exists an algebraic subvariety W ( X which contains the image of all non-constant, holomorphic curves f : C → X. It is hard to believe that this conjecture is correct for manifolds X of dimension ≥ 3. On the other hand, it is very likely that this holds true for surfaces (i.e., dim X = 2), on the behalf of the results available in this case. Given a map f : Y → X defined on a parabolic Riemann surface Y , we can associate a Nevanlinna-type closed positive current T [f ]. If X is a surface of general type and if Y has finite Euler characteristic, then there exists an integer k such that the k-jet of f satisfies an algebraic relation. As a consequence, there exists a foliation F by Riemann surfaces on the space of k-jets Xk of X0 , such that the lift of f is tangent to F . In conclusion we are naturally led to consider the pairs (X, F ), where X is a compact manifold, and F is a foliation by curves on X. We denote by T F the so-called tangent bundle of F . R We derive a lower bound of the intersection number X T [f ] ∧ c1 (T F ) in terms of a Nevanlinna-type counting function of the intersection of f with the singular points of F . As aRconsequence, if X is a complex surface and F has reduced singularities, we show that X T [f ] ∧ c1 (T F ) ≥ 0. For this part we follow closely the original argument of [21]. When combined with a result by Y. Miyaoka, the preceding inequality shows that the classes {T [f ]} and c1 (T F ) are orthogonal. Since R the class of the current T [f ] is nef, we show by a direct argument that we have X {T [f ]}2 = 0, and from this we infer that the Lelong numbers of the diffuse part R of T [f ] are equal to zero at each point of X. This regularity property of R is crucial, since it allows to show R –via the Baum-Bott formula and an elementary fact from dynamics– that we have X T [f ] ∧ c1 (N F ) ≥ 0, where N F is the normal bundle of the foliation, and c1 (N F ) is the first Chern class of N F . We then obtain the next result, in the spirit of [21]. Theorem 1.7. – Let X be a surface of general type, and consider a holomorphic map f : Y → X, where Y is a parabolic Riemann surface such that X+ σ (r) = o(Tf (r)) . We assume that f is tangent to a holomorphic foliation F , then the dimension of the Zariski closure of f ( Y ) is at most 1. In the last section of our survey we give a short proof of M. Brunella index theorem [9]. Furthermore, we show that that this important result admits the following generalization.

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? Let L be a line bundle on a projective surface X, such that S m TX ⊗ L has a nonidentically zero section u. Let f be a holomorphic map from a parabolic Riemann surface Y to X, such that such that X+ σ (r) = o(Tf (r)). Then we show that we have Z c1 (L) ∧ T [f ] ≥ 0. X

Brunella’s theorem corresponds to the case m = 1 and L = N F : indeed, a folia? tion on X can be seen as a section of TX ⊗ N F (or in a dual manner, as a section of TX ⊗ T F? ). If X is a minimal surface of general type, such that c21 > c2 , we see that this implies Theorem 1.7 directly, i.e., without considering the T [f ]-degree of the tangent of F . In particular, we do not need to invoke Miyaoka’s generic semi-positivity theorem, nor the blow-up procedure of McQuillan. This gives a more direct proof. It is a very interesting problem to generalize the inequality above in the framework of higher order jet differentials, cf. Section 7 for a precise statement. Acknowledgements. – It is our pleasure thank the referee for her/his valuable suggestions. We also thank D. Brotbeck, J. Noguchi, E. Rousseau and Min-Ru for careful reading and pointing out minor inaccuracies in the previous versions of this survey, which permitted to avoid several slips. Finally, the slight delay with which this article appears was very helpful for the improvement of its quality!

2. Preliminaries 2.1. Motivation: an extension result. – We first give the proof of Proposition 1.1. We refer to [19] for further results in this direction. Proof of Proposition 1.1. – A first observation is that it is enough to deal with the case where E is a single point. Indeed, assume that this case is settled. We consider the set E0 ⊂ E such that the map f does not extends across E0 ; our goal is to prove that we have E0 = ∅. If this is not the case, then Baire’s Theorem implies that E0 contains at least an isolated point, since it is countable, closed and non-empty and this leads to a contradiction provided that we are able to deal with the case of an isolated point. Thus we can assume that we have a holomorphic map f : D? → X, where D? is the pointed unit disk. Let g P be the Kobayashi metric on X; we remark that by hypothesis, g P is non-degenerate. By the distance decreasing property of this metric we infer that 1 |f 0 (t)|2g P ≤ 2 |t| log2 |t|2

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for any t ∈ D? . On the other hand, X is hyperbolic, so we have |f 0 (t)|2gX ≤ C|f 0 (t)|2g P . Here gX is a metric on X and C is a positive constant (depending on the metric) independent of the point t ∈ D? . We thus have C (3) |f 0 (t)|2gX ≤ 2 . |t| log2 |t|2 This is a crucial information, since now we can argue as follows. The inequality (3) implies that the area of the graph associated to our map Γ0f ⊂ D? × X defined by Γ0f := {(t, x) ∈ D? × X : f (t) = x} is finite. By the theorem of Bishop-Skoda (cf. [33] and the references therein) this implies that there exists an analytic subset Γ ⊂ D × X whose restriction to D? × X is precisely Γ0f . Hence we infer that the fiber of the projection Γ → D on the second factor is a point. Indeed, if this is not the case, then the area of the image (via f ) of the disk of radius ε is bounded from below by a constant independent of ε > 0. This of course cannot happen, as one can see by integrating the inequality (3) over the disk of radius ε. Thus, we know that the volume is bounded near E. We then consider the positive closed current defined by the graph ΓE f := {(t, x) ∈ D \ E × X : f (t) = x} and given that E × X is closed, complete pluripolar, the current extends as a positive closed current see for e.g [33]. As above the set of points where the Lelong number is one gives the holomorphic extension of the graph, and our proposition is proved. In the same spirit, we quote next a classical result due to A. Borel. Let D ⊂ Cn be a bounded symmetric domain, and let Γ ⊂ Aut( D) be an arithmetically defined torsion-free group, cf. [6] for definitions and references. The quotient Ω := D/Γ admits a projective compactification, say X, called the Baily-Borel compactification. Using Proposition 1.1, the result in [6] can be stated as follows. Theorem 2.1. – [6] Let φ : C \ E → D/Γ be a holomorphic non-constant map, where E is countable. Then ϕ extends across E and defines a holomorphic map φe : C → X. In connection with this result, we recall the following conjecture proposed in [26]. Conjecture 2.2. – Let X be a Kobayashi hyperbolic compact manifold of dimension n. We denote by B the unit ball in Cp , and let E be a closed pluripolar subset of B. Then any holomorphic map f :B\E →X extends across E. In the case where X is the quotient of a bounded domain in Cn , a proof of this conjecture was proposed by M. Suzuki in [36]. In general, even if we assume that the holomorphic bisectional curvature of X is bounded from above by -1, the conjecture above seems to be open.

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2.2. Jet spaces. – We will recall here a few basic facts concerning the jet spaces associated to complex manifolds; we refer to [11], [19] for a more complete overview. Let X be an l-dimensional complex space. We denote by J k (X) the space of k-jets of holomorphic disks, described as follows. Let f and g be two germs of analytic disks (C, 0) → (X, x), we say that they define the same k jet at x if their derivatives at zero coincide up to order k, i. e. f (j) (0) = g (j) (0) for j = 0, . . . k. The equivalence classes defined by this equivalence relation is denoted by J k (X, x); as a set, J k (X) is the union of J k (X, x) for all x ∈ X. We remark that if x ∈ Xreg is a non-singular point of X, then J k (X, x) is isomorphic to Ckl , via the identification
f → f 0 (0), . . . , f (k) (0) . This map is not intrinsic, it depends on the choice of some local coordinate system needed to express the derivatives above; at a global level the projection map J k (Xreg ) → Xreg is a holomorphic fiber bundle (which is not a vector bundle in general, since the transition functions are polynomial instead of linear). If k = 1, and x ∈ Xreg is a regular point, then J 1 (X, x) is the tangent space of X at x. We also mention here that the structure of the analytic space J k (X) at a singular point of X is far more complicated. We assume that X is a subset of a complex manifold M ; then for each positive integer k we have a natural inclusion J k (X) ⊂ J k (M ) and one can see that the space J k (X) is the Zariski closure of the analytic space J k (Xreg ) in the complex manifold J k (M ) (note that this coincides with the topological closure) Next we recall the definition of the main geometric objects we will use in the analysis of the structure of the subvarieties of complex tori which are Zariski closure of some parabolic image. As before, let x ∈ Xreg be a regular point of X; we consider a coordinate system (x1 , . . . , xl ) of X centered at x. We consider the symbols dx1 , . . . , dxl , d2 x1 , . . . , d2 xl , . . . , dk x1 , . . . , dk xl and we say that the weight of the symbol dp xr is equal to p, for any r = 1, . . . , l. A jet differential of order k and degree m at x is a homogeneous polynomial of GG degree m in (dp xr )p=1,...,k,r=1,...,l ; we denote by Ek,m (X, x) the vector space of all such polynomials. We denote the set [
GG GG Ek,m Xreg := Ek,m (X, x). x∈Xreg

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It has a structure of vector bundle, whose global sections are called jet differentials GG of weight m and order k. A global section P of the bundle Ek,m (X) can be written locally as X P = aα (dx)α1 . . . (dk x)αk ; |α1 |+···+k|αk |=m

here we use the standard multi-index notation. Let f : (C, 0) → (X, x) be a k-jet at x. The group Gk of k--jets of biholomorphisms of (C, 0) acts on Jk (X), and we say that the operator P is invariant if for ϕ ∈ Gk ,

m 0 (k) P (f ◦ ϕ) , . . . , (f ◦ ϕ) = ϕ0 P f 0 , . . . , f (k) . The bundle of invariant jet differentials is denoted by Ek,m (X). We will recall next an alternative description of this bundle, which will be very useful in what follows. Along the next few lines, we indicate a compactification of the quotient Jkreg (X)/Gk following [11], where Jkreg (X) denote the space of non-constants jets. We start with the pair (X, V ), where V ⊂ TX is a subbundle of the tangent space of X. Then we define X1 := P(TX ), and the bundle V1 ⊂ TX1 is defined fiberwise by V1,(x,[v]) := {ξ ∈ TX1 (x,[v]) : dπ(ξ) ∈ Cv}, where π : X1 → X is the canonical projection and v ∈ V . It is easy to see that we have the following parallel description of V1 . Consider a non-constant disk u : (C, 0) → (X, x). We can lift it to X1 and denote the resulting germ by u1 . Then the derivative of u1 belongs to the V1 directions. In a more formal manner, we have the exact sequence 0 → TX1 /X → V1 → O X1 (−1) → 0, where O X1 (−1) is the tautological bundle on X1 , and TX1 /X is the relative tangent bundle corresponding to the fibration π. This shows that the rank of V1 is equal to the rank of V . Inductively by this procedure we get a tower of manifolds (Xk , Vk ), starting from (X, TX ) and it turns out that we have an embedding Jkreg /Gk → Xk . On each manifold Xk , we have a tautological bundle O Xk (−1), and the positivity of its dual plays an important role here. We denote
by πk : Xk → X the projection, and consider the direct image sheaf πk∗ O Xk (m) . The result is a vector bundle Ek,m (X) whose sections are precisely the invariant jet differentials considered above. The fiber of πk at a non-singular point of X is denoted by R n,k ; it is a rational manifold, and it is a compactification of the quotient Cnk \ 0/Gk . The articles [3], [11], [34] (to quote only a few) show that the existence of jet differentials are crucial in the analysis of the entire maps f : C → X. As we will see in the next sections, they play a similar role in the study of the images of the parabolic Riemann surfaces.

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3. Basics of Nevanlinna Theory for Parabolic Riemann Surfaces Let Y be a parabolic Riemann surface; as we have recalled in the introduction, this means that there exists a smooth exhaustion function σ : Y → [1, ∞[ such that the function R τ := log σ is harmonic in the complement of a compact subset of Y , and we have Y ddc τ = 1. We denote by B(r) ⊂ Y the parabolic ball of radius r, that is to say B(r) := {y ∈ Y : σ(y) ≤ r}. For almost every value r ∈ R, the sphere S(r) := ∂B(r) is a smooth curve drawn on Y . The induced length measure on S(r) is equal to dµr := dc log σ |

S(r)

.

Let v : Y → [−∞, ∞[ be a function defined on Y , such that locally near every point of Y it can be written as a difference of two subharmonic functions, i.e., ddc v is of order zero. Then we recall here the following formula, which will be very useful in what follows. Proposition 3.1. – (Jensen formula) For every r ≥ 1 large enough we have with τ = log σ, Z r Z Z Z Z dt ddc v = vdµr − vddc τ = vdµr + O(1). 1 t B(t) S(r) B(r0 ) S(r) Proof. – The arguments are standard. To start with, we remark that for each regular value r of σ the function v is integrable with respect to the measure dµr over the sphere S(r). Next, we have Z r Z Z
dt c dd v = log r − log σ ddc v t B(t) B(r) 1 Z Z r
+ r c = log dd v = vddc log+ σ σ Z Z c = vdµr − vdd τ. S(r)

B(r)

Remark 3.2. – As we can see, the Jensen formula above holds true even without the assumption that the function τ is harmonic R outside a compact set. The only difference is eventually as r → ∞, since the term B(r) vddc τ may tend to infinity. We reformulate next the notion of mean Euler characteristic in analytic terms. To this end, we first recall that the tangent bundle T Y of a non-compact parabolic surface admits a trivializing global holomorphic section v ∈ H 0 ( Y , T Y ), cf. [15] (actually, any such Riemann surface admits a submersion into C). Using the Poincaré-Hopf index theorem, we obtain the following result.

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Proposition 3.3. – Let ( Y , σ) be a parabolic Riemann surface, so that log σ is harmonic in the complement of a compact set. Then we have Z Xσ (r) = log |dσ(v)|2 dµr + O(1) S(r)

for any r ≥ 1. Proof. – Since v is a vector of type (1,0), dσ(v) = ∂σ(v). The quantity Z Z r dt ddc log |∂ log σ(v)|2 1 t B(t) is equal to the weighted Euler characteristic Xσ (r) of the domains B(r), in particular it is independent of v, up to a bounded term. This can be seen as a consequence of the Poincaré-Hopf index theorem, combined with the fact that the function ∂v log σ is holomorphic, so ddc log |∂v log σ|2 count the critical points of ∂v σ. On the other hand, by Jensen’s formula it is equal to: Z log |dσ(v)|2 dµr + O(1). S(r)

Definition 3.4. – We will need the following proximity function for the critical set: Z X+ (r) = log+ |dσ(v)|2 dµr . σ S(r)

As usual log+ := max(log, 0). When Xσ (r) = O(log r), then we also have X+ σ (r) = O(log r). We will see that for Y = C \ E with E discrete, we can choose σ, such that X+ σ (r) = Xσ (r) + O(1), see Example 4.1. We obtain next the first main theorem and the logarithmic derivative lemma of Nevanlinna theory in the parabolic setting. The results are variations on well-known techniques (see [35] and the references therein). But for the reader’s convenience we will give here a proof. Let X be a compact complex manifold, and let L → X be a line bundle on X, endowed with a smooth metric h. We make no particular assumptions concerning the curvature form Θh (L). Let s be a non-trivial section of L normalized such that supX |s| = 1, and let f : Y → X be a holomorphic map, where Y is parabolic. We define the usual characteristic function of f with respect to Θh (L) as follows Z r Z dt Tf,Θh (L) (r) := f ? Θh (L). r0 t B(t) If the form Θh (L) is positive definite, then precisely as in the classical case Y = C, the area of the image of f will be finite if and only if Tf,Θh (L) (r) = O(log r) as r → ∞. Let Z r dt Nf,s (r) := nf,s (t) t r0

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be the counting function, where nf,s (t) is the number of zeroes of s◦f in the parabolic ball of radius t (counted with multiplicities). Hence we assume implicitly that the image of f is not contained in the set (s = 0). Moreover, in our context the proximity function becomes Z 1 mf,s (r) := log dµr . |s ◦ f |h S(r) In the important case of a (meromorphic) function F : Y → P1 , one usually takes L := O (1), hence Θh (L) is the Fubini-Study metric, and s the section vanishing at infinity. The proximity function becomes Z (4) mf,∞ (r) := log+ |f |dµr , S(r)

where F := [f0 : f1 ], f = f1 /f0 . As a consequence of Jensen formula, we derive the next result. Theorem 3.5. – With the above notations, we have (5)

Tf,Θh (L) (r) = Nf,s (r) + mf,s (r) + O(1)

as r → ∞. Proof. – The argument is similar to the usual one: we apply Jensen’s formula cf. Proposition 3.1 to the function v := log |s ◦ f |2h . Recall that that the Poincaré-Lelong equation gives ddc log |s|2h = [s = 0] − Θh (L), which implies X ddc v = mj δaj − f ? (Θh (L)), j

so that by integration we obtain (5). Remark 3.6. – If the measure ddc τ does not have a compact support, then the term O(1) in the equality (5) is to be replaced by Z (6) log |s ◦ f |2h ddc τ. B(r)

We observe that, thanks to the normalization condition we impose to s, the term (6) is negative. In particular we infer that Z (7) Tf,Θh (L) (r) ≥ Nf,s (r) + log |s ◦ f |2h ddc τ B(r)

for any r ≥ r0 . We will discuss now a version of Theorem 3.5 which will be very useful in dealing with singular foliations. Let J ⊂ O X be a coherent ideal of holomorphic functions. We consider a finite covering of X with coordinate open sets (Uα )α∈Λ , such that on Uα the ideal J is generated by the holomorphic functions (gαi )i=1...Nα .

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Then we can construct a function ψ J , such that for each α ∈ Λ the difference X (8) ψ J − log( |gαi |2 ) i

is bounded on Uα . Indeed, let ρα be a partition of unity subordinated to (Uα )α∈Λ . We define the function ψ J as follows X X (9) ψ J := ρα log( |gαi |2 ) α

i

and the boundedness condition (8) is verified, since there exists a constant C > 0 such that P |gαi |2 −1 C ≤ Pi ≤C 2 i |gβi | holds on Uα ∩ Uβ , for each pair of indexes α, β. In the preceding context, the function ψ J corresponds to log |s|2h . We can assume that ψ J ≤ 0. We define a counting function and a proximity function for a holomorphic map f : Y → X with respect to the analytic set defined by J , as follows. Let (tj ) ⊂ Y be the set of solutions of the equation
exp ψ J ◦ f (y) = 0. For each r > 0 we can write ψ J ◦ f (y)|

B(r)

=

X

νj log |y − tj |2 + O (1)

σ(tj ) 0, we have that mf 0 /f,∞ (r) ≤ (1 + δ)2 (log Tf (r)) + (1 + δ) log r + X+ σ (r) + O(1) holds true for all r outside a set of finite Lebesgue measure. We also get a similar estimate for higher order derivatives.

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Proof. – Within the framework of Nevanlinna theory, this kind of results can be derived in many ways if Y is the complex plane; the proof presented here follows an argument due to Selberg in [32]. On the complex plane C ⊂ P1 we consider the coordinate w corresponding to [1 : w] in homogeneous coordinates on P1 . The form √ 1 −1 (15) Ω := dw ∧ dw |w|2 (1 + log2 |w|) 2π on C has finite volume, as one can easily check by a direct computation. In what follows, we will use the same letter to denote the expression of the meromorphic function Y → C induced by f . For each t ≥ 0, we denote by n(t, f, w) the number of zeroes of the function z → f (z) − w in the parabolic ball B(t) and we have Z Z ? n(t, f, w)Ω f Ω= (16) B(t)

C

by the change of variables formula. Next, by integrating the relation (16) above and using Theorem 3.5, we infer by Remark 3.5. the following , where C is a positive constant: Z r Z Z Z Z dt (17) f ?Ω = Nf −w,∞ (r)Ω ≤ Tf (r)Ω + CΩ. 1 t B(t) C C C This last quantity is smaller than C0 Tf (r), where C0 is a positive constant. Thus, we have Z r Z dt (18) f ? Ω ≤ C0 Tf (r). 1 t B(t) A simple algebraic computation shows the next inequality    |df (ξ)|2  |df (ξ)|2 log 1 + ≤ log 1 + |f (z)|2 |f (z)|2 (1 + log2 |f (z)|2 )|dσ(ξ)|2 + log(1 + log2 |f (z)|2 ) + log(1 + |dσ(ξ)|2 ) and therefore we get  |df (ξ)|2  log 1 + dµr |f (z)|2 S(r) Z 1 |df (ξ)|2 + dµr ≤ log 2 2 2 |dσ z (ξ)| S(r) |f (z)| (1 + log |f (z)|) Z + log(1 + log2 |f (z)|2 )dµr S(r) Z + log(1 + |dσz (ξ)|2 )dµr . Z

2mf 0 /f,∞ (r) ≤

S(r)

We have used the concavity of the log function.

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By the formula (14) we obtain Z 1 |df (ξ)|2 dµr 2 2 2 S(r) |f (z)| (1 + log |f (z)|) |dσz (ξ)| Z 1 d |df (ξ)|2 1 = dσ ∧ dc σ. r dr B(r) |f (z)|2 (1 + log2 |f (z)|) |dσz (ξ)|2 Next we show that we have |df (ξ)|2 1 dσ ∧ dc σ = f ? Ω. 2 2 2 |dσ |f (z)| (1 + log |f (z)|) z (ξ)| ∂ Indeed this is clear, since we can choose a local coordinate z such that ξ = ∂z and 2 √ −1 ∂σ c moreover we have dσ ∧ d σ = 2π ∂z dz ∧ dz (we are using here the fact that f is holomorphic). Let H be a positive, strictly increasing function defined on (0, ∞). It is immediate to check that the set of numbers s ∈ R+ such that the inequality

H 0 (s) ≤ H 1+δ (s) is not verified, is of measure. By applying this calculus lemma to the R rfinite R Lebesgue ? function H(r) := 1 dt f Ω for all r outside a set of finite measure, we obtain t B(t) Z 1+δ Z + 1 + 1 d ? f Ω ≤ log f ?Ω + O(1) log r dr B(r) r B(r) Z Z r  hd i1+δ  dt ≤ log+ rδ f ?Ω + O(1) dr 1 t B(t)  Z r dt Z (1+δ)2 ≤ δ log r + log+ + O(1) f ?Ω 1 t B(t) ≤ δ log r + (1 + δ)2 log+ (C0 Tf (r)) + O(1). The term

Z

log(1 + log2 |f (z)|2 )dµr

S(r)

is bounded, up to a constant, by log Tf (r); combined with Definition 3.4, this implies the desired inequality. Remark 3.9. – Observe that if ddc (log σ) is of finite total mass without being compactly supported, then the measures µr have uniformly bounded mass. The logarithmic derivative lemma is still valid. Indeed, the use of the concavity of log just introduces a constant to normalize µr . Hence we have just to add a term O(1) in the right hand side of the inequality. It is a simple matter to deduce the so-called second main theorem of Nevanlinna theory starting from the logarithmic derivative lemma (cf. e.g., [12]). The parabolic version of this result can be stated as follows.

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Theorem 3.10. – Let f : ( Y , σ) → P1 be a meromorphic function. We denote by NRf (r) the Nevanlinna counting function for the ramification divisor associated to f . Also, we use the classical notation δf (a) := limr

mf,a (r) Na (r) = 1 − limr . Tf (r) Tf (r)

Then for any set of distinct points (aj )1≤j≤p in P1 there exists a set Λ ⊂ R+ of finite Lebesgue measure such that NRf (r) +

p X

+ mf,aj (r) ≤ 2Tf (r) + X+ σ (r) + O log r + log Tf (r)




j=1

for all r ∈ R+ \ Λ. For the proof we refer e.g., to [12]; as we have already mentioned, it is a direct consequence of the logarithmic derivative lemma. As a consequence, if log r = o(Tf (r)), we have p X X+ (r) . δf (aj ) ≤ 2 + limr→∞ σ Tf (r) j=1 4. The Vanishing Theorem Let P be an invariant jet differential of order k and degree m. We assume that it has values in the dual of an ample line bundle, that is to say P

? ∈ H 0 (X, Ek,m TX ⊗ A−1 ),

where A is an ample line bundle on a projective manifold X. In what follows Y is always a parabolic Riemann surface. Let f : Y → X be a holomorphic map. Assume that we are given the exhaustion function σ and a vector field ξ such that we have (19)

limr→∞

X+ σ (r) =0 Tf,ω (r)

on Y . Observe that condition (19), implies that log(r) = o(Tf,ω (r)) and hence the image is of infinite area. If X+ σ (r) = O(log(r)) condition (19) means only that log r = o(Tf,ω (r)) i.e., that the area of f is infinite. As we have already recalled in the preliminaries, the operator P can be seen as section of O Xk (m) on Xk . On the other hand, the curve f admits a canonical lift to Xk as follows. One first observes that the derivative df : T Y → f ? TX induces a map f1 : Y → P(TX ). We remark that to do so we do not need any supplementary data, since df (v1 ) and df (v2 ) are proportional, provided that vj ∈ T Y ,t are tangent vectors at the same

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point. Using the notations of section two, it turns out that the curve f1 is tangent to V1 ⊂ TX1 , so that we can continue this procedure and define inductively fk : Y → Xk . We prove next the following result. Theorem 4.1. – We assume that condition (19) holds. Then the image of fk is contained in the zero set of the section of O Xk (m)⊗A−1 defined by the jet differential P ,
i.e., we have P dfk−1 (ξ)⊗m = 0. Proof. – We observe that we have dfk−1 (ξ) : Y → fk? O Xk (−1)




that is to say, the derivative of fk−1 is a section of the inverse image of the tautological ? bundle. It is in Ek,m TX ⊗ A−1 . Thus the quantity
⊗m P dfk−1 (ξ) is a section of fk? (A−1 ), where the above notation means that we are evaluating P at the point fk (t) on the m-th power of the section above at t ∈ Y . As a consequence, if ωA is the curvature form of A, we have
(20) ddc log | P dfk−1 (ξ)⊗m |2 ≥ fk? (ωA ). The missing term involves the Dirac masses at the critical points of f . We observe that the positivity of the bundle A is fully used at this point: we obtain an upper bound for the characteristic function of f . By integrating and using Jensen formula, we infer that we have Z
log | P dfk−1 (ξ)⊗m |2 dµr ≥ Tf,ωA (r) + O(1) S(r)

as r → ∞. Now we follow the arguments in [12]. There exists a finite set of rational functions uj : Xk → P1 and a positive constant C such that we have X
|d(uj ◦ fk−1 )(ξ)|2 log+ | P dfk−1 (ξ)⊗m |2 ≤ C log+ |uj ◦ fk−1 |2 j pointwise on Y . Indeed, we can use the meromorphic functions uj as local
coordinates on X, and then we can write the jet differential P as Q f, dp (log uj ◦ f ) , hence the previous inequality. We invoke next the logarithmic derivative lemma (Theorem 3.8) established in the previous section, and so we infer that we have Z
(21) log | P dfk−1 (ξ)⊗m |2 dµr ≤ C(log Tfk−1 (r) + log r + X+ σ (r)) S(r)

out of a set of finite Lebesgue measure. It is not difficult to see that the characteristic function corresponding to fk−1 is smaller than CTf (r), for some constant C. Combining the relations (20) and (21) we have
Tf (r) ≤ C log Tf (r) + log r + X+ σ (r)

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in the complement of a set
of finite Lebesgue measure. Since, by assumption, we have X+ (r) + log r = o T (r) , we get a contradiction. Therefore, if the image of fk is not f σ contained in the zero set of P , then condition (19) is not satisfied. Let E ⊂ C be a polar subset of the complex plane. In the case Y

= C \ E,

we show that we have the following version of the previous result in the context of arbitrary jet differentials. Theorem 4.2. – Let f : C \ E → X be a holomorphic curve; we assume that condition (19) is satisfied. Then P (f 0 , . . . , f (k) ) ≡ 0 for any holomorphic jet differential P of degree m and order k with values in the dual of an ample line bundle. Proof. – The argument is similar to the proof of the preceding Theorem 4.1, except that we use the pointwise inequality log+ | P (f 0 , . . . , f (k) )| ≤ C

k XX j

log |dl log(uj ◦ f )|

l=1

combined with Theorem 3.8 in order to derive a contradiction. The following statement is an immediate consequence of Theorem 4.1. Corollary 4.3. – Let X be a projective manifold whose cotangent bundle is ample. Then X does not admit any holomorphic curve f : Y → X such that condition (19) is satisfied. Proof. – There exists an ample line bundle A such that the sections of ? H 0 (X, S m TX ⊗ A−1 )

separate points on the jet space. Hence the Vanishing Theorem implies the result. We recall that in the articles [7], [40], [8] it is shown that any generic complete intersection X of sufficiently high degree and codimension in Pn satisfies the hypothesis of Corollary 4.3. To state our next result, we consider the following data. Let X be a non-singular, projective manifold and let D = Y1 + · · · + Yl be an effective divisor, such that the pair (X, D) is log-smooth (this last condition means that the hypersurfaces Yj are nonsingular, and that they have transverse intersections). In some cases (e.g., if KX +D is big, cf. logarithmic version of [13]) we have
? (22), H 0 X, Ek,m TX hDi ⊗ A−1 6= 0 ? where Ek,m TX hDi is the log version of the space of invariant jet differentials of order k and degree m. Roughly speaking, the sections of the bundle in (22) are homogeneous polynomials in dp log z1 , . . . , dp log zd , dp zd+1 , . . . , dp zn where p = 1, . . . k and z1 z2 . . . zd = 0 is a local equation of the divisor D.

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We have the following result, which is a more general version of Theorem 4.2. Theorem 4.4. – Let f : C \ E → X \ D be a non-algebraic, holomorphic map. If the map f verifies
X+ σ (r) = o Tf (r) , then for any invariant log-jet differential P

? ∈ H 0 X, Ek,m TX hDi ⊗ A−1




we have P (f 0 , . . . , f (k) ) ≡ 0. Proof. – We only have to notice that the “logarithmic derivative lemma” type argument used in the proof of the vanishing theorem is still valid in our context, despite of the fact that the jet differential has poles along D (see [29] Chapter 7, for a complete treatment). Thus, the result follows as above. It is useful at this point to have some examples. 4.1. A few examples. – At the end of this paragraph we will discuss some examples of parabolic surfaces; we will try to emphasize in particular the properties of the function Xσ (r). (1) Let E ⊂ C be a finite subset of the complex plane. Define X r log σ := log+ |z| + log+ |z − a| a∈E

for r > 0 small enough. Then clearly we have Xσ (r) = X+ σ (r) = O(log r) for Y := C\E. (2) We treat next the case of Y := C \ E, where E = (aj )j≥1 is a closed, countable set of points in C. As we will see, in this case it is natural to use the Jensen formula without assuming that the support of the measure ddc log σ is compact, see Remark 3.9. Let (rj )j≥1 be a sequence of positive real numbers, P such that the Euclidean disks Dj := D(aj , rj ) are disjoints. We can assume that j≥1 rj < ∞. As in the preceding example, we define the exhaustion function σ such that X rj log σ = log+ |z| + rj log+ . |z − aj | j≥1

We can use a smoothing of the function log+ without changing the estimates, since the chosen disks are disjoint. The difference here is that dµr is no longer a probability measure. Removing a set of r0 s of finite length, we can assume that S(r) does not S intersect |z − aj | = rj . We then have  1 ∂σ  X 1 1 rj χD(ai ,1) + χ(|z|>1) , = σ ∂z |z − aj | |z| j≥1

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which holds true on the parabolic sphere S(r). Observe that the terms in the sum are all ≥ 1, and their supports are disjoint. Hence on S(r), 1 ∂σ 1 ∂σ log+ = log + O(log r). σ ∂z σ ∂z It follows that X+ σ (r) = Xσ (r) + O(log r). Therefore, we can bound the characteristic X+ σ (r), by the counting function for (aj )j≥1 . From the interpretation of Xσ (r) as a weighted Euler characteristic, it is clear that we have Xσ (r) = N(aj )j≥1 (r) + O(log r). As a corollary of Theorem 4.4, we obtain the following statement. Corollary 4.5. – Let (X, D) be a pair as above, and let f : C → X be a non-algebraic, holomorphic map; we define E := f −1 (D). Moreover, we assume that there exists an invariant log-jet differential P with values in A−1 such that P (f 0 , . . . , f (k) ) is not identically zero. RWe denote by NE (r) the
Nevanlinna counting function associated r to E, i.e., NE (r) = 0 dt card E ∩ (σ < r) . Then there exists a constant C > 0 such t that we have NE (r) (23) lim inf > C. r Tf (r) Moreover, the constant C is independent of f . Remark 4.6. – Observe that we are not in the situation of Theorem 4.4, since the image of f can intersect the support of the divisor D. Actually the main point in the previous statement is to analyze the intersection of f with D. Proof. – Let σ denote the parabolic exhaustion associated to Y = C \ E as in Example 2 above. By the proof of the vanishing theorem we infer the existence of a constant C > 0 such that the inequality
Tf (r) ≤ C −1 log Tf (r) + O(log r) + X+ σ (r) holds as r → ∞ for any map f such that P (f 0 , . . . , f (k) ) is not identically zero. On the other hand, we have X+ σ (r) ≤ NE (r)+O(log r), so the relation (23) is verified. The fact that C is independent of f can be seen directly from the previous inequality. Remark 4.7. – It turns out that there is another important class of Riemann surfaces for which we can establish a vanishing theorem of type 4.2. Let f : D → X be a holomorphic map defined on a unit disk, which satisfies the growth condition T (f,r) = ∞. Then the vanishing Theorem 4.2 holds for f , i.e., given any suprk≥i

ωji dzj ∧ dz i ,

j,i≥k+1

where the Hermitian matrix (ωji ) is positive definite. If moreover the condition (b) is fulfilled, then the family of vector fields vα defining the foliation F can be seen as a global section VB of the bundle TX hBi ⊗ T F?

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47

and we have the following version of the metric constructed above. For each vector ξ ∈ T F ,x we define kξk2hs,B := |VB,x (ξ)|2ωX,B . As in the case discussed before, we can give the local expression of the metric on T F , as follows. Let k n X X ∂ ∂ vα = aiα zi i + aiα i ∂z ∂z i=1 i=k+1

be a logarithmic vector field trivializing the tangent bundle of the foliation on a coordinate set Uα . Then the local weight φα,B of the metric hs,B induced by the metric ωX,B is given by the expression X (32) φα,B = − log aiα ajα ωij . i,j

In particular we see that this weight is less singular than the one in the expression (31). This will be crucial in the applications. As far as the curvature current is concerned, the metric hs as well as its logarithmic variant hs,B seem useless: given the definition above. Its associated curvature is neither positive nor negative. Indeed, φα may tend to infinity along the singular set of the foliation F , and it may tend to minus infinity along the singularities of the metric ω. However, we will present a few applications of this construction in the next paragraphs. 6.3. Degree of currents associated to parabolic Riemann surfaces on the tangent bundle of foliations. – Let (X, ω) be a compact complex hermitian manifold, and let F be a holomorphic foliation on X of dimension 1. Let f : Y → X be a holomorphic map, where ( Y , σ) is a parabolic Riemann surface tangent to F , and let T [f ] := lim Tr [f ] r

be a Nevanlinna current associated to it. In this section we will derive a lower bound in arbitrary dimension for the quantity Z T [f ] ∧ c1 (T F ), X

in the same spirit as [21], [9]. Prior to this, we introduce a few useful notations. Let J F s be the coherent ideal associated to the singularities of F ; this means that locally on Uα the generators of J F s are precisely the coefficients (aα ) of the vector vα defining F , i.e., n X ∂ vα = aiα i . ∂z i=1 As we have already mentioned in Paragraph 3, there exists a function ψsing defined on X and having the property that locally on each open set Uα we have ψsing ≡ log |vα |2ω modulo a bounded function.

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P Let B = j Wj be a divisor on X, such that the pair (X, B) satisfies the requirements (a) and (b) in the preceding paragraph. Then the local generator of T F can be written in this case as (33)

vα,B =

k X

aiα zi

i=1

We denote by J we have

F s,B

n X ∂ ∂ + aiα i . i ∂z ∂z i=k+1

the coherent ideal defined by the functions (aiα ) in (33). Then J

Fs

⊂ J

F s,B

,

and the inclusion may be strict. We denote by ψsing,B the associated function. The counting function with respect to the ideal defined by F sing will be denoted Z r Z X
r dt νj log = Nf, J F s (r) = ddc ψsing ◦ f s , σ(tj ) 0 t B(t) 0 5—that the same bound dn = δn holds for Kobayashi hyperbolicity would then be a consequence of the GreenGriffiths-Lang conjecture. In the same vein, we present a construction of hyperbolic hypersurfaces of Pn+1 for all degrees d > 4n2 . The main idea is inspired from the method of ShiffmanZaidenberg [58]; by using again Wronskians, it is possible to give a direct and selfcontained argument. I wish to thank Damian Brotbek, Ya Deng, Simone Diverio, Gianluca Pacienza, Erwan Rousseau, Mihai Păun and Mikhail Zaidenberg for very stimulating discussions on these questions. These notes owe a lot to their work.

1. Hyperbolicity concepts 1.1. Kobayashi pseudodistance and pseudometric. – We first recall a few basic facts concerning the concept of hyperbolicity, according to S. Kobayashi [39, 40, 41, 42]. Let X be a complex space. Given two points p, q ∈ X, let us consider a chain of analytic disks from p to q, that is a sequence of holomorphic maps f0 , f1 , . . . , fk : D → X from the unit disk D = D(0, 1) ⊂ C to X, together with pairs of points a0 , b0 , . . . , ak , bk of D such that p = f0 (a0 ),

q = fk (bk ),

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fi (bi ) = fi+1 (ai+1 ),

i = 0, . . . , k − 1.

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Denoting this chain by α, we define its length `(α) to be (1.10 )

`(α) = dP (a1 , b1 ) + · · · + dP (ak , bk ),

where dP is the Poincaré distance on D, and the Kobayashi pseudodistance dK X on X to be (1.100 )

dK X (p, q) = inf `(α). α

A Finsler metric (resp. pseudometric) on a vector bundle E is a homogeneous positive (resp. nonnegative) function N on the total space E, that is, N (λξ) = |λ| N (ξ)

for all λ ∈ C and ξ ∈ E,

but in general N is not assumed to be subbadditive (i.e., convex) on the fibers of E. A Finsler (pseudo-)metric on E is thus nothing but a hermitian (semi-)norm on the tautological line bundle O P (E) (−1) of lines of E over the projectivized bundle Y = P (E). The Kobayashi-Royden infinitesimal pseudometric on X is the Finsler pseudometric on the tangent bundle TX defined by (1.2)  kX (ξ) = inf λ > 0 ; ∃f : D → X, f (0) = x, λf 0 (0) = ξ , x ∈ X, ξ ∈ TX,x . If Φ : X → Y is a morphism of complex spaces, by considering the compositions Φ◦f : D → Y , this definition immediately implies the monotonicity property Φ∗ kY 6 kX , i.e., (1.3)

kY (Φ∗ ξ) 6 kX (ξ) for all x ∈ X and ξ ∈ TX,x .

When X is a manifold, it follows from the work of H.L. Royden ([55], [56]) that dK X is the integrated pseudodistance associated with the pseudometric, i.e., Z (1.4) dK (p, q) = inf kX (γ 0 (t)) dt, X γ

γ

where the infimum is taken over all piecewise smooth curves joining p to q ; in the case of complex spaces, a similar formula holds, involving jets of analytic curves of arbitrary order, cf. S. Venturini [65]. When X is a non-singular projective variety, it has been shown in [24] that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal pseudometric can be computed by looking only at analytic disks that are contained in algebraic curves.

Definition 1.5. – A complex space X is said to be hyperbolic (in the sense of K Kobayashi) if dK X is actually a distance, namely if dX (p, q) > 0 for all pairs of distinct points (p, q) in X.

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1.2. Brody criterion. – In the above context, we have the following well-known result of Brody [6]. Its main interest is to relate hyperbolicity to the non-existence of entire curves. Brody reparametrization lemma 1.6. – Let ω be a hermitian metric on X and let f : D → X be a holomorphic map. For every ε > 0, there exists a radius R > (1 − ε)kf 0 (0)kω and a homographic transformation ψ of the disk D(0, R) onto (1 − ε)D such that 1 k(f ◦ ψ)0 (0)kω = 1, for every t ∈ D(0, R). k(f ◦ ψ)0 (t)kω 6 1 − |t|2 /R2 Proof. – Select t0 ∈ D such that (1 − |t|2 )kf 0 ((1 − ε)t)kω reaches its maximum for t = t0 . The reason for this choice is that (1 − |t|2 )kf 0 ((1 − ε)t)kω is the norm of the differential f 0 ((1 − ε)t) : TD → TX with respect to the Poincaré metric |dt|2 /(1 − |t|2 )2 on TD , which is conformally invariant under Aut(D). One then adjusts R and ψ so 2 that ψ(0) = (1 − ε)t0 and |ψ 0 (0)| kf 0 (ψ(0))kω = 1. As |ψ 0 (0)| = 1−ε R (1 − |t0 | ), the only possible choice for R is R = (1 − ε)(1 − |t0 |2 )kf 0 (ψ(0))kω > (1 − ε)kf 0 (0)kω . The inequality for (f ◦ψ)0 follows from the fact that the Poincaré norm is maximum at the origin, where it is equal to 1 by the choice of R. Using the Ascoli-Arzelà theorem we obtain immediately: Corollary (Brody) 1.7. – Let (X, ω) be a compact complex hermitian manifold. Given a sequence of holomorphic mappings fν : D → X such that lim kfν0 (0)kω = +∞, one can find a sequence of homographic transformations ψν : D(0, Rν ) → (1 − 1/ν)D with lim Rν = +∞, such that, after passing possibly to a subsequence, (fν ◦ ψν ) converges uniformly on every compact subset of C towards a nonconstant holomorphic map g : C → X with kg 0 (0)kω = 1 and supt∈C kg 0 (t)kω 6 1. An entire curve g : C → X such that supC kg 0 kω = M < +∞ is called a Brody curve; this concept does not depend on the choice of ω when X is compact, and one can always assume M = 1 by rescaling the parameter t. Brody criterion 1.8. – Let X be a compact complex manifold. The following properties are equivalent. (a) X is hyperbolic. (b) X does not possess any entire curve f : C → X. (c) X does not possess any Brody curve g : C → X. (d) The Kobayashi infinitesimal metric kX is uniformly bounded below, namely kX (ξ) > ckξkω ,

c > 0,

for any hermitian metric ω on X. When property (b) holds, X is said to be Brody hyperbolic.

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Proof. – (a) ⇒ (b). If X possesses an entire curve f : C → X, then by looking at arbitrary large analytic disks f : D(t0 , R) ⊂ C and rescaling them on D as t 7→ f (t0 + Rt), it is easy to see that the Kobayashi distance of any two points in f (C) is zero, so X is not hyperbolic. (b) ⇒ (c) is trivial. (c) ⇒ (d). If (d) does not hold, there exists a sequence of tangent vectors ξν ∈ TX,xν with kξν kω = 1 and kX (ξν ) → 0. By definition, this means that there exists an analytic curve fν : D → X with f (0) = xν and kfν0 (0)kω > (1 − ν1 )/kX (ξν ) → +∞. One can then produce a Brody curve g = C → X by Corollary 1.7, contradicting (c). (d) ⇒ (a). In fact (d) implies after integrating that dK X (p, q) > c dω (p, q) where dω is the geodesic distance associated with ω, so dK must be non degenerate. X As a consequence, any projective variety containing a rational curve C (i.e., a curve normalized by C ' P1C ' C ∪ {∞} or an elliptic curve (i.e., a curve normalized by a nonsingular elliptic curve C/(Z⊕Zτ )) is non-hyperbolic. An immediate consequence of the Brody criterion is the openness property of hyperbolicity for the metric topology: Proposition 1.9. – Let π : X → S be a holomorphic family of compact complex manifolds. Then the set of s ∈ S such that the fiber Xs = π −1 (s) is hyperbolic is open in the metric topology. Proof. – Let ω be an arbitrary hermitian metric on X , (Xsν )sν ∈S a sequence of nonhyperbolic fibers, and s = lim sν . By the Brody criterion, one obtains a sequence of entire maps fν : C → Xsν such that kfν0 (0)kω = 1 and kfν0 kω 6 1. Ascoli’s theorem shows that there is a subsequence of fν converging uniformly to a limit f : C → Xs , with kf 0 (0)kω = 1. Hence Xs is not hyperbolic and the collection of non-hyperbolic fibers is closed in S. 1.3. Relationship of hyperbolicity with algebraic properties. – In the case of projective algebraic varieties, Kobayashi hyperbolicity is expected to be an algebraic property. In fact, the following classical conjectures would give a necessary and sufficient algebraic characterization. Recall that a projective variety X of dimension n = dimC X is said e to be of general type if the canonical bundle K e = Λn T ∗ of some desingularization X X

e X

of X is big. When n = dimC X = 1, this is equivalent to say that X is not rational or elliptic. Some classical conjectures 1.10. – Let X be a projective variety.

(i) (Green-Griffiths-Lang conjecture) If X is of general type, there should exist a proper algebraic variety Y ( X (possibly empty ) containing all nonconstant entire curves f : C → X. (ii) Conversely, if X is Kobayashi hyperbolic and nonsingular, it is expected that KX e should should be ample. More generally, if X is singular, any desingularization X be of general type.

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(iii) (Conjectural algebraic characterization of Kobayashi hyperbolicity). A projective variety X is Kobayashi hyperbolic if and only if every positive dimensional algebraic subvariety Y ⊂ X (including X itself ) is of general type. In fact, since every analytic subspace of Kobayashi hyperbolic space is again hyperbolic by definition, it is not difficult to see by induction on dimension that 1.10 (iii) would follow formally from 1.10 (i) and (ii) [the “if” part is a consequence of 1.10 (i), and the “only if” part follows from 1.10 (ii)]. Thanks to fundamental work of Clemens [10], Ein [31, 32] and Voisin [66], it is known that every subvariety Y of a generic algebraic hypersurface X ⊂ Pn+1 of degree d > 2n + 1 is of general type for n > 2 ; Pacienza [51] has even shown that this holds for d > 2n when n > 5. The GreenGriffiths-Lang conjecture would then imply that these hypersurfaces are Kobayashi hyperbolic. Definition 1.11. – Let X be a projective algebraic manifold, and A a very ample line bundle on X. We say that X is algebraically hyperbolic if there exists ε > 0 such that every closed irreducible curve C ⊂ X has a normalization C such that its Euler characteristic satisfies −χ(C) = 2g(C) − 2 > ε degA (C), R where g(C) is the genus and degA (C) = C · A = C c1 (A). Theorem 1.12. – Every Kobayashi hyperbolic projective variety is algebraically hyperbolic. More generally, if X is a hyperbolic compact complex manifold equipped with a hermitian metric ω, there exists ε > 0 such that every closed irreducible curve C ⊂ X satisfies R 2g(C) − 2 > ε degω (C) where degω (C) = C ω. Proof ([16]). – When Γ is a nonsingular compact curve of genus at least 2, the unib → Γ is isomorphic to formization theorem implies that the universal cover ρ : Γ the unit disk D, and one then sees that the Kobayashi metric kΓ is induced by he Kobayashi metric of the disk, i.e., k2D =

idz ∧ dz . (1 − |z|2 )2

i These metrics have constant negative curvature − 2π ∂∂ log k2Γ = − π1 k2Γ , hence Z 1 k2 = −χ(Γ) = 2g(Γ) − 2 π Γ Γ

by the Gauss-Bonnet formula. Now, if X is hyperbolic and C ⊂ X is a closed analytic curve, the monotonicity formula (1.3) applied to the normalization map ν : C → X implies kC > ν ∗ kX , and we also have k2X > c2 ω for some c > 0 by 1.8 (d). Therefore Z Z Z Z 1 1 1 c2 c2 2 ∗ 2 2 2g(C) − 2 = kC > ν kX = kX > ω= degω (C). π C π C π C π C π

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It is not very difficult to check that the proof can be extended to the case of singular hyperbolic compact complex spaces (a smooth hermitian metric on X being a metric that has extensions with respect to local embeddings of X in open sets U ⊂ CN ). Proposition 1.13. – Let X → S be an algebraic family of projective algebraic manifolds, given by a projective morphism X → S. Then the set of t ∈ S such that the fiber Xt is algebraically hyperbolic is open with respect to the countable Zariski topology Proof. – After replacing S by a Zariski open subset, we may assume that the total space X itself is quasi-projective. Let ω be the Kähler metric on X obtained by pulling back the Fubini-Study metric via an embedding in a projective space. If integers d > 0, g > 0 are fixed, the set Ad,g of t ∈ S such that Xt contains an algebraic 1-cycle P P C = mj Cj with degω (C) = d and g(C) = mj g(C j ) 6 g is a closed algebraic subset of S (this follows from the existence of a relative cycle space of curves of given degree, and from the fact that the geometric genus is Zariski lower semicontinuous). Now, the set of non algebraically hyperbolic fibers is by definition \ [ Ad,g . k>0

2g−2 2. However, V is a saturated subsheaf of O (TX ), i.e., O (TX )/ V has no torsion: in fact, if the components of a section have a common divisorial component, one can always simplify this divisor and produce a new section without any such common divisorial component. Instead of defining directed manifolds by picking a linear space V , one could equivalently define them by considering saturated coherent subsheaves V ⊂ O (TX ). One could also take the dual viewpoint, looking at arbitrary quotient morphisms Ω1X → W = V ∗ (and recovering V = W ∗ = Hom O ( W , O ), as V = V ∗∗ is reflexive). We want to stress here that no assumption need be made on the Lie bracket tensor [•, •] : V × V → O (TX )/ V , i.e., we do not assume any kind of integrability for V or W . Even though we will not consider such situations here, one can even generalize the concept of directed structure to the case when X is a singular (say reduced) complex space X. In fact VX 0 should then be a holomorphic vector subbundle of TX 0 on some analytic Zariski open set X 0 ⊂ Xreg , and if U ,→ Z is an embedding of an open neighborhood U ⊂ X of a point x0 ∈ X into an open set Z ⊂ CN , we demand that the directed structure VU be a (closed and analytic) subspace of TZ , obtained as the closure of VX 0 ∩U in TZ via the obvious “inclusion 0 morphism” (X 0 ∩ U, VX 0 ∩U ) ,→ (Z, TZ ). A morphism f : (C, TC ) → (X, V ) in the category of directed varieties is the same as a holomorphic curve t 7→ f (t) that is tangent to V , i.e., f 0 (t) ∈ Vf (t) for all t. The concept of Koabayashi hyperbolicity can be extended to directed varieties as follows. Definition 2.2. – Let (X, V ) be a complex directed manifold. The Kobayashi-Royden infinitesimal metric of (X, V ) is the Finsler metric on V defined for any x ∈ X and ξ ∈ Vx by  k(X,V ) (ξ) = inf λ > 0 ; ∃f : (D, TD ) → (X, V ), f (0) = x, λf 0 (0) = ξ . We say that (X, V ) is infinitesimally hyperbolic if k(X,V ) is positive definite on every fiber Vx and satisfies a uniform lower bound k(X,V ) (ξ) > εkξkω in terms of any smooth hermitian metric ω on X, when x runs over a compact subset of X. When X is compact, the Brody criterion shows that this is equivalent to the nonexistence of nonconstant entire curves f : (C, TC ) → (X, V ), or even to the nonexistence of entire curves g : (C, TC ) → (X, V ) with sup kg 0 (t)kω = kg 0 (0)kω = 1. In this context we have the Generalized Green-Griffiths-Lang conjecture 2.3. – Let (X, V ) be a projective directed manifold where V ⊂ TX is nonsingular (i.e., a subbundle of TX ). Assume that (X, V ) is of “general type” in the sense that KV := det V ∗ is a big line bundle. Then there should exist a proper algebraic subvariety Y ( X containing the images f (C) of all entire curves f : C → X tangent to V .

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A similar statement can be made when V is singular, but then KV has to be replaced by a certain (nonnecessarily invertible) rank 1 sheaf of “locally bounded” forms of O (det V ∗ ), with respect to a smooth hermitian form ω on TX . The reader will find a more precise definition in [21].

2.2. The 1-jet functor. – The basic idea is to introduce a functorial process which e Ve ) from a given one (X, V ). The new produces a new complex directed manifold (X, e e structure (X, V ) plays the role of a space of 1-jets over X. First assume that V is non-singular. We let (2.4)

e = P (V ), X

Ve ⊂ TXe

be the projectivized bundle of lines of V , together with a subbundle Ve of TXe defined e associated with a vector v ∈ Vx \ {0}, as follows: for every point (x, [v]) ∈ X  (2.40 ) Ve (x,[v]) = ξ ∈ TX, Cv ⊂ Vx ⊂ TX,x , e (x,[v]) ; π∗ ξ ∈ Cv , e = P (V ) → X is the natural projection and π∗ : TXe → π ∗ TX is its where π : X e = P (V ) we have the tautological line bundle O Xe (−1) ⊂ π ∗ V such differential. On X that O Xe (−1)(x,[v]) = Cv. The bundle Ve is characterized by the two exact sequences (2.5)

π∗ e (−1) −→ 0, 0 −→ TX/X −→ Ve −→ OX e

(2.50 )

−→ 0, 0 −→ O Xe −→ π ∗ V ⊗ O Xe (1) −→ TX/X e

e → X. The where TX/X denotes the relative tangent bundle of the fibration π : X e first sequence is a direct consequence of the definition of Ve , whereas the second is a relative version of the Euler exact sequence describing the tangent bundle of the fibers P (Vx ). From these exact sequences we infer (2.6)

e = n + r − 1, dim X

rank Ve = rank V = r,

and by taking determinants we find det(TX/X ) = π ∗ det V ⊗ O Xe (r), thus e (2.7)

det Ve = π ∗ det V ⊗ O Xe (r − 1).

e Ve ) → (X, V ) is a morphism of complex directed manifolds. By definition, π : (X, Clearly, our construction is functorial, i.e., for every morphism of directed manifolds Φ : (X, V ) → (Y, W ), there is a commutative diagram 99K

 π π e Ve ) −→ e yΦ(Ye , W f ) −→ (X, (X, V )Φ (Y, W ), where the left vertical arrow is the meromorphic map P (V ) > P (W ) induced by the e is actually holomorphic if Φ∗ : V → Φ∗ W is injective). differential Φ∗ : V → Φ∗ W (Φ

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2.3. Lifting of curves to the 1-jet bundle. – Suppose that we are given a holomorphic curve f : DR → X parametrized by the disk DR of center 0 and radius R in the complex plane, and that f is a tangent curve of the directed manifold, i.e., f 0 (t) ∈ Vf (t) for every t ∈ DR . If f is nonconstant, there is a well defined and unique tangent line [f 0 (t)] for every t, even at stationary points, and the map (2.8)

e fe : DR → X,

t 7→ fe(t) := (f (t), [f 0 (t)])

is holomorphic (at a stationary point t0 , we just write f 0 (t) = (t − t0 )s u(t) with s ∈ N∗ and u(t0 ) 6= 0, and we define the tangent line at t0 to be [u(t0 )], hence fe(t) = (f (t), [u(t)]) near t0 ; even for t = t0 , we still denote [f 0 (t0 )] = [u(t0 )] for simplicity of notation). By definition f 0 (t) ∈ O Xe (−1)fe(t) = C u(t), hence the derivative f 0 defines a section f 0 : TDR → fe∗ O Xe (−1).

(2.9) Moreover π ◦ fe = f , therefore

π∗ fe0 (t) = f 0 (t) ∈ Cu(t) =⇒ fe0 (t) ∈ Ve (f (t),u(t)) = Ve fe(t) e Ve ). We say that fe is the canonical and we see that fe is a tangent trajectory of (X, e e e Ve ), then by lifting of f to X. Conversely, if g : DR → X is a tangent trajectory of (X, e definition of V we see that f = π ◦ g is a tangent trajectory of (X, V ) and that g = fe (unless g is contained in a vertical fiber P (Vx ), in which case f is constant). For any point x0 ∈ X, there are local coordinates (z1 , . . . , zn ) on a neighborhood Ω of x0 such that the fibers (Vz )z∈Ω can be defined by linear equations n o X X ∂ (2.10) Vz = ξ = ξj ; ξj = ajk (z)ξk for j = r + 1, . . . , n , ∂zj 16j6n

16k6r

where (ajk ) is a holomorphic (n − r) × r matrix. It follows that a vector ξ ∈ Vz is completely determined by its first r components (ξ1 , . . . , ξr ), and the affine chart ξj 6= 0 of P (V )Ω can be described by the coordinate system  ξ1 ξj−1 ξj+1 ξr  (2.11) z1 , . . . , zn ; , . . . , , ,..., . ξj ξj ξj ξj Let f ' (f1 , . . . , fn ) be the components of f in the coordinates (z1 , . . . , zn ) (we suppose here R so small that f (DR ) ⊂ Ω). It should be observed that f is uniquely determined by its initial value x and by the first r components (f1 , . . . , fr ). Indeed, as f 0 (t) ∈ Vf (t) , we can recover the other components by integrating the system of ordinary differential equations X (2.12) fj0 (t) = ajk (f (t))fk0 (t), j > r, 16k6r

on a neighborhood of 0, with initial data f (0) = x. We denote by m = m(f, t0 ) the multiplicity of f at any point t0 ∈ DR , that is, m(f, t0 ) is the smallest integer m ∈ N∗ (m) such that fj (t0 ) 6= 0 for some j. By (2.12), we can always suppose j ∈ {1, . . . , r},

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(m)

for example fr (t0 ) 6= 0. Then f 0 (t) = (t − t0 )m−1 u(t) with ur (t0 ) 6= 0, and the lifting fe is described in the coordinates of the affine chart ξr 6= 0 of P (V )Ω by  f0  f0 . (2.13) fe ' f1 , . . . , fn ; 10 , . . . , r−1 fr fr0 2.4. The Semple tower. – Let X be a complex n-dimensional manifold. Following ideas of Green-Griffiths [35], we let Jk X → X be the bundle of k-jets of germs of parametrized curves in X, that is, the set of equivalence classes of holomorphic maps f : (C, 0) → (X, x), with the equivalence relation f ∼ g if and only if all derivatives f (j) (0) = g (j) (0) coincide for 0 6 j 6 k, when computed in some local coordinate system of X near x. The projection map Jk X → X is simply f 7→ f (0). If (z1 , . . . , zn ) are local holomorphic coordinates on an open set Ω ⊂ X, the elements f of any fiber Jk Xx , x ∈ Ω, can be seen as Cn -valued maps f = (f1 , . . . , fn ) : (C, 0) → Ω ⊂ Cn , and they are completely determined by their Taylor expansion of order k at t = 0 t2 00 tk f (0) + · · · + f (k) (0) + O(tk+1 ). 2! k! In these coordinates, the fiber Jk Xx can thus be identified with the set of k-tuples of vectors (ξ1 , . . . , ξk ) = (f 0 (0), . . . , f (k) (0)) ∈ (Cn )k . It follows that Jk X is a holomorphic fiber bundle with typical fiber (Cn )k over X (however, Jk X is not a vector bundle for k > 2, because of the nonlinearity of coordinate changes. According to the philosophy of directed structures, one can also introduce the concept of jet bundle in the general situation of complex directed manifolds. If X is equipped with a holomorphic subbundle V ⊂ TX , one associates to V a k-jet bundle Jk V as follows. f (t) = x + t f 0 (0) +

Definition 2.14. – Let (X, V ) be a complex directed manifold. We define Jk V → X to be the bundle of k-jets of curves f : (C, 0) → X which are tangent to V , i.e., such that f 0 (t) ∈ Vf (t) for all t in a neighborhood of 0, together with the projection map f 7→ f (0) onto X. It is easy to check that Jk V is actually a subbundle of Jk X. In fact, by using (2.10) and (2.12), we see that the fibers Jk Vx are parametrized by
(k) (f10 (0), . . . , fr0 (0)); (f100 (0), . . . , fr00 (0)); · · · ; (f1 (0), . . . , fr(k) (0)) ∈ (Cr )k for all x ∈ Ω, hence Jk V is a locally trivial (Cr )k -subbundle of Jk X. Alternatively, connection ∇ on V such that for any germs Pwe can pick a local holomorphic P w = 16j6n wj ∂z∂ j ∈ O (TX,x ) and v = 16λ6r vλ eλ ∈ O (V )x in a local trivializing frame (e1 , . . . , er ) of VΩ we have X X ∂vλ (2.15) ∇w v(x) = wj eλ (x) + Γµjλ (x)wj vλ eµ (x). ∂zj 16j6n, 16λ6r

16j6n, 16λ,µ6r

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We can of course take the frame obtained from (2.10) by lifting the vector fields ∂/∂z1 , . . . , ∂/∂zr , and the “trivial connection” given by the zero Christof⊕k fel symbols Γ = 0. One then obtains a trivialization J k VΩ ' VΩ by considering Jk Vx 3 f 7→ (ξ1 , ξ2 , . . . , ξk ) = (∇f (0), ∇2 f (0), . . . , ∇k f (0)) ∈ Vx⊕k and computing inductively the successive derivatives ∇f (t) = f 0 (t) and ∇s f (t) via  X d ∇s−1 f eλ (f ) ∇s f = (f ∗ ∇)d/dt (∇s−1 f ) = dt λ 16λ6r   X + Γµjλ (f )fj0 ∇s−1 f eµ (f ). 16j6n, 16λ,µ6r

λ

This identification depends of course on the choice of ∇ and cannot be defined globally in general (unless we are in the rare situation where V has a global holomorphic connection. We now describe a convenient process for constructing “projectivized jet bundles,” which will later appear as natural quotients of our jet bundles Jk V (or rather, as suitable desingularized compactifications of the quotients). Such spaces have already been considered since a long time, at least in the special case X = P2 , V = TP2 (see Gherardelli [34], Semple [57]), and they have been mostly used as a tool for establishing enumerative formulas dealing with the order of contact of plane curves (see [12], [11]); the article [1] is also concerned with such generalizations of jet bundles, as well as [43] by Laksov and Thorup. One defines inductively the projectivized k-jet bundle Xk (or Semple k-jet bundle) and the associated subbundle Vk ⊂ TXk by (2.16)

(X0 , V0 ) = (X, V ),

e k−1 , Ve k−1 ). (Xk , Vk ) = (X

In other words, (Xk , Vk ) is obtained from (X, V ) by iterating k-times the lifting cone Ve ) described in § 2.B. By (2.4–2.9), we find struction (X, V ) 7→ (X, (2.17)

dim Xk = n + k(r − 1),

rank Vk = r,

together with exact sequences (πk )∗

(2.18)

0 −→ TXk /Xk−1 −→ Vk −−−−→ O Xk (−1) −→ 0,

(2.180 )

0 −→ O Xk −→ πk∗ Vk−1 ⊗ O Xk (1) −→ TXk /Xk−1 −→ 0,

where πk is the natural projection πk : Xk → Xk−1 and (πk )∗ its differential. Formula (5.4) yields (2.19)

det Vk = πk∗ det Vk−1 ⊗ O Xk (r − 1).

Every nonconstant tangent trajectory f : DR → X of (X, V ) lifts to a well defined and 0 unique tangent trajectory f[k] : DR → Xk of (Xk , Vk ). Moreover, the derivative f[k−1] gives rise to a section (2.20)

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0 ∗ f[k−1] : TDR → f[k] O Xk (−1).

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101

In coordinates, one can compute f[k] in terms of its components in the various affine charts (5.9) occurring at each step: we get inductively (2.21)

f[k] = (F1 , . . . , FN ),

 Fs0  F0 , f[k+1] = F1 , . . . , FN , s01 , . . . , r−1 Fsr Fs0r

where N = n + k(r − 1) and {s1 , . . . , sr } ⊂ {1, . . . , N }. If k > 1, {s1 , . . . , sr } contains the last r − 1 indices of {1, . . . , N } corresponding to the “vertical” components of the projection Xk → Xk−1 , and in general, sr is an index such that m(Fsr , 0) = m(f[k] , 0), that is, Fsr has the smallest vanishing order among all components Fs (sr may be vertical or not, and the choice of {s1 , . . . , sr } need not be unique). By definition, there is a canonical injection O Xk (−1) ,→ πk∗ Vk−1 , and a composition with the projection (πk−1 )∗ (analogue for order k−1 of the arrow (πk )∗ in the sequence (2.18)) yields for all k > 2 a canonical line bundle morphism (2.22)

O Xk (−1)

(πk )∗ (πk−1 )∗

,−→ πk∗ Vk−1 −−−−−−−→ πk∗ O Xk−1 (−1),

which admits precisely Dk = P (TXk−1 /Xk−2 ) ⊂ P (Vk−1 ) = Xk as its zero divisor (clearly, Dk is a hyperplane subbundle of Xk ). Hence we find (2.23)

O Xk (1)

= πk∗ O Xk−1 (1) ⊗ O (Dk ).

Now, we consider the composition of projections (2.24)

πj,k = πj+1 ◦ · · · ◦ πk−1 ◦ πk : Xk −→ Xj .

Then π0,k : Xk → X0 = X is a locally trivial holomorphic fiber bundle over X, −1 and the fibers Xk,x = π0,k (x) are k-stage towers of Pr−1 -bundles. Since we have (in both directions) morphisms (Cr , TCr ) ↔ (X, V ) of directed manifolds which are bijective on the level of bundle morphisms, the fibers are all isomorphic to a “universal” non-singular projective algebraic variety of dimension k(r − 1) which we will denote by R r,k ; it is not hard to see that R r,k is rational (as will indeed follow from the proof of Theorem 3.11 below).

Remark 2.25. – When (X, V ) is singular, one can easily extend the construction of the Semple tower by functoriality. In fact, assume that X is a closed analytic subset of some open set Z ⊂ CN , and that X 0 ⊂ X is a Zariski open subset on which VX 0 is a subbundle of TX 0 . Then we consider the injection of the nonsingular directed manifold (X 0 , V 0 ) into the absolute structure (Z, W ), W = TZ . This yields an injection (Xk0 , Vk0 ) ,→ (Zk , Wk ), and we simply define (Xk , Vk ) to be the closure of (Xk0 , Vk0 ) into (Zk , Wk ). It is not hard to see that this is indeed a closed analytic subset of the same dimension n + k(r − 1), where r = rank V 0 .

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3. Jet differentials and Green-Griffiths bundles 3.1. Green-Griffiths jet differentials. – We first introduce the concept of jet differentials in the sense of Green-Griffiths [35]. The goal is to provide an intrinsic geometric description of holomorphic differential equations that a germ of curve f : (C, 0) → X may satisfy. In the sequel, we fix a directed manifold (X, V ) and suppose implicitly that all germs of curves f are tangent to V . Let Gk be the group of germs of k-jets of biholomorphisms of (C, 0), that is, the group of germs of biholomorphic maps t 7→ ϕ(t) = a1 t + a2 t2 + · · · + ak tk ,

a1 ∈ C∗ , aj ∈ C, j > 2,

in which the composition law is taken modulo terms tj of degree j > k. Then Gk is a k-dimensional nilpotent complex Lie group, which admits a natural fiberwise right action on Jk V . The action consists of reparametrizing k-jets of maps f : (C, 0) → X by a biholomorphic change of parameter ϕ : (C, 0) → (C, 0), that is, (f, ϕ) 7→ f ◦ ϕ. There is an exact sequence of groups 1 → G0k → Gk → C∗ → 1, where Gk → C∗ is the obvious morphism ϕ 7→ ϕ0 (0), and G0k = [Gk , Gk ] is the group of k-jets of biholomorphisms tangent to the identity. Moreover, the subgroup H ' C∗ of homotheties ϕ(t) = λt is a (non-normal) subgroup of Gk , and we have a semidirect decomposition Gk = G0k n H. The corresponding action on k-jets is described in coordinates by λ · (f 0 , f 00 , . . . , f (k) ) = (λf 0 , λ2 f 00 , . . . , λk f (k) ). GG ∗ Following [35], we introduce the vector bundle Ek,m V → X whose fibers are com0 00 (k) plex valued polynomials Q(f , f , . . . , f ) on the fibers of Jk V , of weighted degree m with respect to the C∗ action defined by H, that is, such that

(3.1)

Q(λf 0 , λ2 f 00 , . . . , λk f (k) ) = λm Q(f 0 , f 00 , . . . , f (k) )

for all λ ∈ C∗ and (f 0 , f 00 , . . . , f (k) ) ∈ Jk V . Here we view (f 0 , f 00 , . . . , f (k) ) as indeterminates with components
(k) (f10 , . . . , fr0 ); (f100 , . . . , fr00 ); · · · ; (f1 , . . . , fr(k) ) ∈ (Cr )k . Notice that the concept of polynomial on the fibers of Jk V makes sense, for all coordinate changes z 7→ w = Ψ(z) on X induce polynomial transition automorphisms on the fibers of Jk V , given by a formula (3.2) (Ψ ◦ f )(j) = Ψ0 (f ) · f (j) +

s=j X

X

cj1 ···js Ψ(s) (f ) · (f (j1 ) , . . . , f (js ) )

s=2 j1 +j2 +···+js =j

with suitable integer constants cj1 ···js (this is easily checked by induction on s). GG ∗ In the case V = TX , we get the bundle of “absolute” jet differentials Ek,m TX . GG ∗ If Q ∈ Ek,m V is decomposed into multihomogeneous components of multidegree

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(`1 , `2 , . . . , `k ) in f 0 , f 00 , . . . , f (k) (the decomposition is of course coordinate dependent), these multidegrees must satisfy the relation `1 + 2`2 + · · · + k`k = m. GG ∗ The bundle Ek,m V will be called the bundle of jet differentials of order k and weighted degree m. It is clear from (3.2) that a coordinate change f 7→ Ψ ◦ f transforms every monomial (f (•) )` = (f 0 )`1 (f 00 )`2 · · · (f (k) )`k of partial weighted degree |`|s := `1 + 2`2 + · · · + s`s , 1 6 s 6 k, into a polynomial ((Ψ ◦ f )(•) )` in (f 0 , f 00 , . . . , f (k) ) which has the same partial weighted degree of order s if `s+1 = · · · = `k = 0, and a larger or equal partial degree of order s otherwise. Hence, for each s = 1, . . . , k, we GG ∗ get a well defined (i.e., coordinate invariant) decreasing filtration Fs• on Ek,m V as follows: ( ) GG ∗ Q(f 0 , f 00 , . . . , f (k) ) ∈ Ek,m V involving p GG ∗ (3.3) Fs (Ek,m V ) = , ∀p ∈ N. only monomials (f (•) )` with |`|s > p p GG ∗ GG ∗ The graded terms Grpk−1 (Ek,m V ) associated with the filtration Fk−1 (Ek,m V ) are 0 (k) precisely the homogeneous polynomials Q(f , . . . , f ) whose monomials (f • )` all have partial weighted degree |`|k−1 = p (hence their degree `k in f (k) is such that GG ∗ m − p = k`k , and Grpk−1 (Ek,m V ) = 0 unless k divides m − p). The transition automorphisms of the graded bundle are induced by coordinate changes f 7→ Ψ ◦ f , and they are described by substituting the arguments of Q(f 0 , . . . , f (k) ) according to formula (3.2), namely f (j) 7→ (Ψ ◦ f )(j) for j < k, and f (k) 7→ Ψ0 (f ) ◦ f (k) for j = k p+1 (when j = k, the other terms fall in the next stage Fk−1 of the filtration). Therefore (k) f behaves as an element of V ⊂ TX under coordinate changes. We thus find

(3.4)

GG ∗ GG k Gm−k` (Ek,m V ) = Ek−1,m−k` V ∗ ⊗ S `k V ∗ . k−1 k

GG ∗ V Combining all filtrations Fs• together, we find inductively a filtration F • on Ek,m such that the graded terms are

(3.5)

GG ∗ Gr` (Ek,m V ) = S `1 V ∗ ⊗ S `2 V ∗ ⊗ · · · ⊗ S `k V ∗ ,

` ∈ Nk ,

|`|k = m.

GG ∗ The bundles Ek,m V have other interesting properties. In fact, M GG ∗ GG ∗ Ek,• V := Ek,m V m>0

is in a natural way a bundle of graded algebras (the product is obtained simply by GG ∗ GG taking the product of polynomials). There are natural inclusions Ek,• V ⊂ Ek+1,• V∗ S GG ∗ GG ∗ of algebras, hence E∞,• V = k>0 Ek,• V is also an algebra. Moreover, the sheaf GG ∗ of holomorphic sections O (E∞,• V ) admits a canonical derivation DGG given by a collection of C-linear maps GG ∗ GG DGG : O (Ek,m V ) → O (Ek+1,m+1 V ∗ ),

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GG ∗ constructed in the following way. A holomorphic section of Ek,m V on a coordinate open set Ω ⊂ X can be seen as a differential operator on the space of germs f : (C, 0) → Ω of the form X (3.6) Q(f ) = aα1 ···αk (f ) (f 0 )α1 (f 00 )α2 · · · (f (k) )αk , |α1 |+2|α2 |+···+k|αk |=m

in which the coefficients aα1 ···αk are holomorphic functions on Ω. Then DGG Q is given by the formal derivative (DGG Q)(f )(t) = d(Q(f ))/dt with respect to the 1-dimenGG sional parameter t in f (t). For example, in dimension 2, if Q ∈ H 0 (Ω, O (E2,4 )) is the section of weighted degree 4 Q(f ) = a(f1 , f2 ) f103 f20 + b(f1 , f2 ) f1002 , GG we find that DGG Q ∈ H 0 (Ω, O (E3,5 )) is given by

(DGG Q)(f ) =

∂a ∂a ∂b (f1 , f2 ) f104 f20 + (f1 , f2 ) f103 f202 + (f1 , f2 ) f10 f1002 ∂z1 ∂z2 ∂z1 ∂b (f1 , f2 ) f20 f1002 + a(f1 , f2 ) 3f102 f100 f20 + f103 f200 ) + b(f1 , f2 ) 2f100 f1000 . + ∂z2

GG ∗ Associated with the graded algebra bundle Ek,• V , we define an analytic fiber bundle

(3.7)

GG ∗ XkGG := Proj(Ek,• V ) = (Jk V \ {0})/C∗

over X, which has weighted projective spaces P(1[r] , 2[r] , . . . , k [r] ) as fibers (these weighted projective spaces are singular for k > 1, but they only have quotient singularities, see [30] ; here Jk V \ {0} is the set of nonconstant jets of order k ; we refer e.g., to Hartshorne’s book [37] for a definition of the Proj functor). As such, it possesses a canonical sheaf O XkGG (1) such that O XkGG (m) is invertible when m is a multiple of lcm(1, 2, . . . , k). Under the natural projection πk : XkGG → X, the direct image (πk )∗ O XkGG (m) coincides with polynomials (3.8)

X

P (z ; ξ1 , . . . , ξk ) = α`

aα1 ···αk (z) ξ1α1 · · · ξkαk

∈Nr , 16`6k

of weighted degree |α1 |+2|α2 |+· · ·+k|αk | = m on J k V with holomorphic coefficients; GG ∗ in other words, we obtain precisely the sheaf of sections of the bundle Ek,m V of jet differentials of order k and degree m. Proposition 3.9. – By construction, if πk : XkGG → X is the natural projection, we have the direct image formula GG ∗ (πk )∗ O XkGG (m) = O (Ek,m V )

for all k and m.

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105

3.2. Invariant jet differentials. – In the geometric context, we are not really interested in the bundles (Jk V \ {0})/C∗ themselves, but rather on their quotients (Jk V \ {0})/Gk (would such nice complex space quotients exist!). We will see that the Semple bundle Xk constructed in § 2.D plays the role of such a quotient. First we GG ∗ introduce a canonical bundle subalgebra of Ek,• V . GG ∗ Definition 3.10. – We introduce a subbundle Ek,m V ∗ ⊂ Ek,m V , called the bundle of invariant jet differentials of order k and degree m, defined as follows: Ek,m V ∗ is the set of polynomial differential operators Q(f 0 , f 00 , . . . , f (k) ) which are invariant under arbitrary changes of parametrization, i.e., for every ϕ ∈ Gk

Q (f ◦ ϕ)0 , (f ◦ ϕ)00 , . . . , (f ◦ ϕ)(k) ) = ϕ0 (0)m Q(f 0 , f 00 , . . . , f (k) ). 0

GG ∗ Gk GG ∗ Alternatively, Ek,m V ∗ = (Ek,m V ) is the set of invariants of Ek,m V under the S L 0 ∗ ∗ GG ∗ action of Gk . Clearly, E∞,• V = k>0 m>0 Ek,m V is a subalgebra of Ek,m V GG (observe however that this algebra is not invariant under the derivation D , since e.g., fj00 = DGG fj is not an invariant polynomial).

Theorem 3.11. – Suppose that V has rank r > 2. Let π0,k : Xk −→ X be the Semple jet bundles constructed in section 2.B, and let Jk V reg be the bundle of regular k-jets of maps f : (C, 0) → X, that is, jets f such that f 0 (0) 6= 0. (i) The quotient Jk V reg /Gk has the structure of a locally trivial bundle over X, and there is a holomorphic embedding Jk V reg /Gk ,→ Xk over X, which identifies Jk V reg /Gk with Xkreg (thus Xk is a relative compactification of Jk V reg /Gk over X). (ii) The direct image sheaf (π0,k )∗ O Xk (m) ' O (Ek,m V ∗ ) can be identified with the sheaf of holomorphic sections of Ek,m V ∗ . (iii) For every m > 0, the relative base locus of the linear system | O Xk (m)| is equal to the set Xksing of singular k-jets. Moreover, O Xk (1) is relatively big over X. Proof. – (i) For f ∈ Jk V reg , the lifting fe is obtained by taking the derivative (f, [f 0 ]) without any cancelation of zeroes in f 0 , hence we get a uniquely defined (k − 1)-jet e Inductively, we get a well defined (k − j)-jet f[j] in Xj , and the fe : (C, 0) → X. value f[k] (0) is independent of the choice of the representative f for the k-jet. As the lifting process commutes with reparametrization, i.e., (f ◦ ϕ)∼ = fe ◦ ϕ and more generally (f ◦ ϕ)[k] = f[k] ◦ ϕ, we conclude that there is a well defined set-theoretic map Jk V reg /Gk → Xkreg , f mod Gk 7→ f[k] (0). This map is better understood in coordinates as follows. Fix coordinates (z1 , . . . , zn ) near a point x0 ∈ X, such that Vx0 = Vect(∂/∂z1 , . . . , ∂/∂zr ). Let f = (f1 , . . . , fn ) be a regular k-jet tangent to V . Then there exists i ∈ {1, 2, . . . , r} such that fi0 (0) 6= 0, and there is a unique reparametrization t = ϕ(τ ) such that f ◦ϕ = g = (g1 , g2 , . . . , gn )

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with gi (τ ) = τ (we just express the curve as a graph over the zi -axis, by means of a change of parameter τ = fi (t), i.e., t = ϕ(τ ) = fi−1 (τ )). Suppose i = r for the simplicity of notation. The space Xk is a k-stage tower of Pr−1 -bundles. In the corresponding inhomogeneous coordinates on these Pr−1 ’s, the point f[k] (0) is given by the collection of derivatives
(k) (k) 0 00 (g10 (0), . . . , gr−1 (0)); (g100 (0), . . . , gr−1 (0)); · · · ; (g1 (0), . . . , gr−1 (0)) . [Recall that the other components (gr+1 , . . . , gn ) can be recovered from (g1 , . . . , gr ) by integrating the differential system (5.10)]. Thus the map Jk V reg /Gk → Xk is a bijection onto Xkreg , and the fibers of these isomorphic bundles can be seen as unions of r affine charts ' (Cr−1 )k , associated with each choice of the axis zi used to describe d d the curve as a graph. The change of parameter formula dτ = f 01(t) dt expresses all r

(j)

(j)

derivatives gi (τ ) = dj gi /dτ j in terms of the derivatives fi (t) = dj fi /dtj f0 f0  0 (g10 , . . . , gr−1 ) = 10 , . . . , r−1 ; fr fr0  f 00 f 0 − f 00 f 0 f 00 f 0 − f 00 f 0  00 (3.12) (g100 , . . . , gr−1 ) = 1 r 03 r 1 , . . . , r−1 r 03 r r−1 ; fr fr .. . (k) (k) (g1 , . . . , gr−1 )

(k) 0 (k)  f (k) f 0 − f (k) f 0  fr−1 fr0 − fr fr−1 r r 1 1 = + (order < k). ,..., 0k+1 0k+1 fr fr (k)

Also, it is easy to check that fr02k−1 gi is an invariant polynomial in f 0 , f 00 , . . . , f (k) of total degree 2k − 1, i.e., a section of Ek,2k−1 . (ii) Since the bundles Xk and Ek,m V ∗ are both locally trivial over X, it is sufficient −1 to identify sections σ of O Xk (m) over a fiber Xk,x = π0,k (x) with the fiber Ek,m Vx∗ , reg at any point x ∈ X. Let f ∈ Jk Vx be a regular k-jet at x. By (6.6), the deriva0 tive f[k−1] (0) defines an element of the fiber of O Xk (−1) at f[k] (0) ∈ Xk . Hence we get a well defined complex valued operator (3.13)

0 Q(f 0 , f 00 , . . . , f (k) ) = σ(f[k] (0)) · (f[k−1] (0))m .

Clearly, Q is holomorphic on Jk Vxreg (by the holomorphicity of σ), and the Gk -invariance condition of Definition 3.10 is satisfied since f[k] (0) does not depend on reparametrization and 0 (f ◦ ϕ)0[k−1] (0) = f[k−1] (0)ϕ0 (0).

Now, Jk Vxreg is the complement of a linear subspace of codimension n in Jk Vx , hence Q extends holomorphically to all of Jk Vx ' (Cr )k by Riemann’s extension theorem (here we use the hypothesis r > 2 ; if r = 1, the situation is anyway not interesting since Xk = X for all k). Thus Q admits an everywhere convergent power series X Q(f 0 , f 00 , . . . , f (k) ) = aα1 ···αk (f 0 )α1 (f 00 )α2 · · · (f (k) )αk . α1 ,α2 ,...,αk ∈Nr

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The Gk -invariance (3.10) implies in particular that Q must be multihomogeneous in the sense of (3.1), and thus Q must be a polynomial. We conclude that Q ∈ Ek,m Vx∗ , as desired. Conversely, for all w in a neighborhood of any given point w0 ∈ Xk,x , we can find a holomorphic family of germs fw : (C, 0) → X such that (fw )[k] (0) = w and (fw )0[k−1] (0) 6= 0 (just take the projections to X of integral curves of (Xk , Vk ) integrating a nonvanishing local holomorphic section of Vk near w0 ). Then every Q ∈ Ek,m Vx∗ yields a holomorphic section σ of O Xk (m) over the fiber Xk,x by putting
−m . (3.14) σ(w) = Q(fw0 , fw00 , . . . , fw(k) )(0) · (fw )0[k−1] (0) (iii) By what we saw in (i)–(ii), every section σ of O Xk (m) over the fiber Xk,x is given by a polynomial Q ∈ Ek,m Vx∗ , and this polynomial can be expressed on the reg Zariski open chart fr0 6= 0 of Xk,x as (3.15)

b 0 , g 00 , . . . , g (k) ), Q(f 0 , f 00 , . . . , f (k) ) = fr0m Q(g

b is a polynomial and g is the reparametrization of f such that gr (τ ) = τ . where Q b is obtained from Q by substituting fr0 = 1 and fr(j) = 0 for j > 2, and In fact Q b by using the substitutions (3.12). conversely Q can be recovered easily from Q In this context, the jet differentials f 7→ f10 , . . . , f 7→ fr0 can be viewed as sections of O Xk (1) on a neighborhood of the fiber Xk,x . Since these sections vanish exactly on Xksing , the relative base locus of O Xk (m) is contained in Xksing for every m > 0. We see that O Xk (1) is big by considering the sections of O Xk (2k − 1) associated with (j) the polynomials Q(f 0 , . . . , f (k) ) = fr02k−1 gi , 1 6 i 6 r − 1, 1 6 j 6 k; indeed, these reg sections separate all points in the open chart fr0 6= 0 of Xk,x . sing Now, we check that every section σ of O Xk (m) over Xk,x must vanish on Xk,x . sing Pick an arbitrary element w ∈ Xk and a germ of curve f : (C, 0) → X such that 0 f[k] (0) = w, f[k−1] (0) 6= 0 and s = m(f, 0)  0 (such an f exists by Corollary 6.14). There are local coordinates (z1 , . . . , zn ) on X such that f (t) = (f1 (t), . . . , fn (t)) b be the polynomials associated with σ in these coordiwhere fr (t) = ts . Let Q, Q 00 α2 0 α1 nates and let (f ) (f ) · · · (f (k) )αk be a monomial occurring in Q, with αj ∈ Nr , |αj | = `j , `1 + 2`2 + · · · + k`k = m. Putting τ = ts , the curve t 7→ f (t) becomes a Puiseux expansion τ 7→ g(τ ) = (g1 (τ ), . . . , gr−1 (τ ), τ ) in which gi is a power series in τ 1/s , starting with exponents of τ at least equal to 1. The derivative g (j) (τ ) may involve negative powers of τ , but the exponent is always > 1 + 1s − j. Hence b 0 , g 00 , . . . , g (k) ) can only involve powers of τ of exponent the Puiseux expansion of Q(g 1 > − max` ((1 − s )`2 + · · · + (k − 1 − 1s )`k ). Finally fr0 (t) = sts−1 = sτ 1−1/s , thus the lowest exponent of τ in Q(f 0 , . . . , f (k) ) is at least equal to    1 1 1  1− m − max 1 − `2 + · · · + k − 1 − `k ` s s s     1 1 k − 1 > min 1 − `1 + 1 − `2 + · · · + 1 − `k , ` s s s

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where the minimum is taken over all monomials (f 0 )α1 (f 00 )α2 · · · (f (k) )αk , |αj | = `j , occurring in Q. Choosing s > k, we already find that the minimal exponent is positive, hence Q(f 0 , . . . , f (k) )(0) = 0 and σ(w) = 0 by (3.14). Theorem 3.11 (iii) shows that O Xk (1) is never relatively ample over X for k > 2. In order to overcome this difficulty, we define for every a• = (a1 , . . . , ak ) ∈ Zk a line bundle O Xk (a• ) on Xk such that (3.16)

O Xk (a• )

∗ ∗ = π1,k O X1 (a1 ) ⊗ π2,k O X2 (a2 ) ⊗ · · · ⊗ O Xk (ak ).

∗ ∗ By (6.9), we have πj,k O Xj (1) = O Xk (1) ⊗ O Xk (−πj+1,k Dj+1 − · · · − Dk ), thus by ∗ ∗ ∗ putting Dj = πj+1,k Dj+1 for 1 6 j 6 k − 1 and Dk = 0, we find an identity

(3.17)

O Xk (a• )

= O Xk (bk ) ⊗ O Xk (−b• · D∗ ),

where

k

b• = (b1 , . . . , bk ) ∈ Z , bj = a1 + · · · + aj , X ∗ b• · D∗ = bj πj+1,k Dj+1 . 16j6k−1

In particular, if b• ∈ Nk , i.e., a1 + · · · + aj > 0, we get a morphism (3.18)

O Xk (a• )

= O Xk (bk ) ⊗ O Xk (−b• · D∗ ) → O Xk (bk ).

The following result gives a sufficient condition for the relative nefness or ampleness of weighted jet bundles. Proposition 3.19. – Take a very ample line bundle A on X, and consider on Xk the line bundle ∗ Lk = O Xk (3k−1 , 3k−2 , . . . , 3, 1) ⊗ πk,0 A⊗3

k

⊗2 ∗ ∗ defined inductively by L0 = A and Lk = O Xk (1) ⊗ πk,k−1 L⊗3 is k−1 . Then Vk ⊗ Lk a nef vector bundle on Xk , which is in fact generated by its global sections, for all k > 0. Equivalently k−1

∗ k−2 ∗ L0k = O Xk (1) ⊗ πk,k−1 L⊗2 , 2 · 3k−3 , . . . , 6, 2, 1) ⊗ πk,0 A⊗2·3 k−1 = O Xk (2 · 3

is nef over Xk (and generated by sections ) for all k > 1. Let us recall that a line bundle L → X on a projective variety X is said to nef if L · C > 0 for all irreducible algebraic curves C ⊂ X, and that a vector bundle E → X is said to be nef if O P(E) (1) is nef on P(E) := P (E ∗ ) ; any vector bundle generated by global sections is nef (cf. [24] for more details). The statement concerning ∗ L0k is obtained by projectivizing the vector bundle E = Vk−1 ⊗ L⊗2 k−1 on Xk−1 , whose associated tautological line bundle is O P(E) (1) = L0k on P(E) = P (Vk−1 ) = Xk . Also one gets inductively that (3.20)

∗ Lk = O P(Vk−1 ⊗L⊗2 ) (1) ⊗ πk,k−1 Lk−1 k−1

PANORAMAS & SYNTHÈSES 56

is very ample on Xk .

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A SIMPLE PROOF OF THE KOBAYASHI CONJECTURE

Proof. – Let X ⊂ PN be the embedding provided by A, so that A = O PN (1)X . As is well known, if Q is the tautological quotient vector bundle on PN , the twisted cotangent bundle TP∗N ⊗ O PN (2) = ΛN −1 Q ∗ ∗ ⊗2 is nef; hence its quotients TX ⊗A⊗2 and V0∗ ⊗L⊗2 are nef (any tensor power 0 = V ⊗A of nef vector bundles is nef, and so is any quotient). We now proceed by induction, ∗ assuming Vk−1 ⊗ L⊗2 k−1 to be nef, k > 1. By taking the second wedge power of the central term in (6.40 ), we get an injection
0 −→ TXk /Xk−1 −→ Λ2 πk? Vk−1 ⊗ O Xk (1) .

By dualizing and twisting with O Xk−1 (2) ⊗ πk? L⊗2 k−1 , we find a surjection ? ? ⊗ O Xk (2) ⊗ πk? L⊗2 πk? Λ2 (Vk−1 ⊗ Lk−1 ) −→ TX k−1 −→ 0. k /Xk−1 ? ⊗ O Xk (2) ⊗ πk? L⊗2 By the induction hypothesis, we see that TX k−1 is nef. Next, k /Xk−1 the dual of (6.4) yields an exact sequence ? 0 −→ O Xk (1) −→ Vk? −→ TX −→ 0. k /Xk−1

As an extension of nef vector bundles is nef, the nefness of Vk∗ ⊗ L⊗2 k will follow if ⊗2 ? we check that O Xk (1) ⊗ L⊗2 and T ⊗ L are both nef. However, this follows k k Xk /Xk−1 again from the induction hypothesis if we observe that the latter implies ∗ Lk > πk,k−1 Lk−1

∗ and Lk > O Xk (1) ⊗ πk,k−1 Lk−1

in the sense that L00 > L0 if the “difference” L00 ⊗ (L0 )−1 is nef. All statements remain valid if we replace “nef” with “generated by sections” in the above arguments. ∗ Corollary 3.21. – A Q-line bundle O Xk (a• ) ⊗ πk,0 A⊗p , a• ∈ Qk , p ∈ Q, is nef (resp. ample) on Xk as soon as X aj > 3aj+1 for j = 1, 2, . . . , k − 2 and ak−1 > 2ak > 0, p > 2 aj ,

resp. aj > 3aj+1 for j = 1, 2, . . . , k − 2 and ak−1 > 2ak > 0, p > 2

X

aj .

Proof. – This follows easily by taking convex combinations of the Lj and L0j and applying Proposition 3.19 and our observation (3.20). Remark 3.22. – As GkL is a non-reductive group, it is a priori unclear whether the graded ring A n,k,r = m∈Z Ek,m V ? (taken pointwise over X) is finitely generated. This can be checked manually ([Dem07a], [Dem07b]) for n = 2 and k 6 4. Rousseau [54] also checked the case n = 3, k = 3, and then Merker [47, 48] proved the finiteness for n = 2, 3, 4, k 6 4 and n = 2, k = 5. Recently, Bérczi and Kirwan [3] made an attempt to prove the finiteness in full generality, but it appears that the general case is still unsettled.

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3.3. Fundamental vanishing theorem. – We prove here a fundamental vanishing theorem due to Siu and Yeung ([62, 63], [60]). Their original proof makes use of Nevanlinna theory, especially of the logarithmic derivative lemma, see also [17] for a more detailed account (in French). An alternative simpler proof based on the Ahlfors lemma and on algebraic properties of jet differentials can be found in [21] (cf. also [16]). Fundamental vanishing theorem 3.23. – Let (X, V ) be a projective directed manifold and A an ample divisor on X. Then P (f ; f 0 , f 00 , . . . , f (k) ) = 0 for every entire GG ∗ curve f : (C, TC ) → (X, V ) and every global section P ∈ H 0 (X, Ek,m V ⊗ O (−A)). Proof. – We first give a proof of 3.23 in the special case where f is a Brody curve, i.e., supt∈C kf 0 (t)kω < +∞ with respect to a given Hermitian metric ω on X. In fact, the proof is much simpler in that case, and thanks to the Brody criterion 1.8, this is sufficient to establish the hyperbolicity of (X, V ). After raising P to a power P s and replacing O (−A) with O (−sA), one can always assume that A is a very ample divisor. GG ∗ We interpret Ek,m V ⊗ O (−A) as the bundle of complex valued differential operators whose coefficients aα (z) vanish along A. Fix a finite open covering of X by coordinate balls B(pj , Rj ) such that the balls Bj (pj , Rj /4) still cover X. As f 0 is bounded, there exists δ > 0 such that for f (t0 ) ∈ B(pj , Rj /4) we have f (t) ∈ B(pj , Rj /2) whenever |t − t0 | < δ, uniformly for every t0 ∈ C. The Cauchy inequalities applied to the components of f in each of the balls imply that the derivatives f (j) (t) are bounded on C, and therefore, since the coefficients aα (z) of P are also uniformly bounded on each of the balls B(pj , Rj /2) we conclude that g := P (f ; f 0 , f 00 , . . . , f (k) ) is a bounded holomorphic function on C. After moving A in the linear system |A|, we may further assume that Supp A intersects f (C). Then g vanishes somewhere, hence g ≡ 0 by Liouville’s theorem, as expected. Next we consider the case where P ∈ H 0 (X, Ek,m V ∗ ⊗ O (−A)) is an invariant differential operator. We may of course assume P 6= 0. Then we get an associated ∗ non-zero section σ ∈ H 0 (Xk , O Xk (m) ⊗ πk,0 O (−A)). Thanks to Corollary 3.21, the line bundle ∗ 0 ∗ ∗ L = O Xk (a• ) ⊗ πk,0 O (pA) = O Xk (m ) ⊗ O Xk (−b• · D ) ⊗ πk,0 O (pA)

is ample on Xk for suitable b• > 0 and m0 , p > 0. Let hL be a smooth metric on L such that ωk = ΘL,hL is a Kähler metric on Xk . Then we can produce a singular hermitian metric h on O Xk (−1) by putting 0

0

kξkh = (kσ p · ξ pm+m kh−1 )1/(pm+m ) , ξ ∈ O Xk (−1), L

0

∗ O (−pA) ⊂ O (L−1 ). The and viewing σ p · ξ pm+m as an element in O Xk (−m0 ) ⊗ πk,0 ϕ metric h has a weight e that is continuous, with zeroes contained in the union of {σ = 0} and of the vertical divisor D∗ . Moreover the curvature tensor Θ OXk (1),h−1 = i 0 −1 −1 > (pm+m ) ωk . On the other hand, 2π ∂∂ log h satisfies by construction Θ O Xk (1),h the continuity of the weight of h and the compactness of Xk imply that there exists a constant C > 0 such that kdπk,k−1 (η)kh 6 Ckηkωk for all vectors η ∈ Vk (notice

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111

0 that ξ = dπk,k−1 (η) ∈ O Xk (−1)). Now, the derivative f[k−1] can be seen as a section ∗ of f[k] O Xk (−1), and we use this to define a singular hermitian metric γ(t) i dt ∧ dt on C by taking 0 (t)k2h(f[k] (t)) . γ(t) = kf[k−1]

If f[k] (C) is not contained in the divisor {σ = 0}, then γ is not identically zero and, in the sense of distributions, we find i ∗ ∗ ωk > C −1 (pm + m0 )−1 γ. Θ OXk (1),h−1 > (pm + m0 )−1 f[k] ∂∂ log γ > f[k] 2π The final inequality comes from the inequality relating h and ωk when we take 0 0 η = f[k] (t) and ξ = f[k−1] (t). However, the Ahlfors lemma shows that a hermitian metric on C with negative curvature bounded away from 0 cannot exist, thus we must have f[k] (C) ⊂ {σ = 0}. This proves our vanishing theorem in the case where P is invariant. The general case of a nonnecessarily invariant operator P will not be used here; a proof can be obtained by decomposing P into invariant parts and using an induction on m (cf. [21] for details), or alternatively by means of Nevanlinna theory arguments ([63], [60], see also [17]). Especially, we can apply the above vanishing theorem for any global invariant jet differential P ∈ H 0 (X, Ek,m V ∗ ⊗ O (−A)). In that case, P corresponds bijectively to a section (3.24)

∗ σ ∈ H 0 (Xk , O Xk (m) ⊗ πk,0 O (−A)),

and assuming P 6= 0, the vanishing theorem can be reinterpreted by stating S −1 that f[k] (C) is contained in the zero divisor Zσ ⊂ Xk . Let ∆k = 26`6k πk,` (D` ) be the union of the vertical divisors (see (2.22) and (2.23)). Then f[k] (C) cannot be contained in ∆k (as otherwise we would have f 0 (t) = 0 identically). We define the k-stage Green-Griffiths locus of (X, V ) to be the Zariski closure  \  ∗ O (−A) (3.25) GGk (X, V ) = (Xk r ∆k ) ∩ base locus of O Xk (m) ⊗ πk,0 m∈N

(trivially independent of the choice of A), and \ (3.26) GG(X, V ) = πk,0 GGk (X, V )). k∈N∗

Then Theorem 3.23 implies that f[k] (C) must be contained in GGk (X, V ) for every entire curve f : (C, TC ) → (X, V ), and also that f (C) ⊂ GG(X, V ). Corollary 3.27. – If GG(X, V ) = ∅, then (X, V ) is hyperbolic. In particular, if there exists k > 1 and a weight a• ∈ Nk such that O Xk (a• ) is ample on Xk , then (X, V ) is hyperbolic.

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It should be observed that Corollary 3.27 yields a sufficient condition for hyperbolicity, but this is not a necessary condition. In fact, if we take X = C1 × C2 to be a product of curves of genus > 2 and V = TX , it is easily checked that GG(X) = GG(X, TX ) = X. More general examples have been found by Diverio and Rousseau [28]. In a similar way, the Green-Griffiths-Lang conjecture holds for (X, V ) if Y := GG(X, V ) ( X, but this is only a sufficient condition. The following fundamental existence theorem, however, has been proved in [18], using holomorphic Morse inequalities of [15] as an essential tool. We only state the main result, as it will not be used here. Theorem 3.28. – Let (X, V ) be a projective directed manifold of general type, in the sense that the sheaf KV of locally bounded sections of O (det V ∗ ) is big. Let A be an ample Q-divisor on X such that O (det V ∗ ) ⊗ O (−A) is still ample. Then    m 1 1  0 ∗ H Xk , O Xk (m) ⊗ πk,0 O − 1 + + ··· + A 6= 0 kr 2 k for m  k  1 and m sufficiently divisible (so that the multiple of A is an integral divisor ). In particular GGk (X, V ) ( Xk for k  1.

4. Existence of hyperbolic hypersurfaces of low degree We give here a self-contained proof of the existence of hyperbolic surfaces of low degree in Pn+1 , using various techniques borrowed from the work of Toda [64], Fujimoto [33], Green [36], Nadel [50], Siu-Yeung [62], Masuda-Noguchi [45] and Shiffman-Zaidenberg [58]. The main idea is to produce ad hoc differential equations for entire curves by means of Wronskian operators. This can be seen as a variation of Nadel’s approach, that was actually based on Wronkians associated with meromorphic connections—Wronskian operators have the advantage of being much easier to handle than general jet differentials, thanks to their straightforward relationship with linear degeneracy. 4.1. General Wronskian operators. – This section follows closely the work of D. Brotbek [8]. Let U be an open set of a complex manifold X, dim X = n, and s0 , . . . , sk ∈ O X (U ) be holomorphic functions. To these functions, we can associate a Wronskian operator of order k defined by s0 (f ) s1 (f ) ··· sk (f ) D(s0 (f )) D(s1 (f )) · · · D(sk (f )) , (4.1) Wk (s0 , . . . , sk )(f ) = .. .. .. . . . Dk (s (f )) Dk (s (f )) · · · Dk (s (f )) 0 1 k

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A SIMPLE PROOF OF THE KOBAYASHI CONJECTURE

113

where f : t 7→ f (t) ∈ U ⊂ X is a germ of holomorphic curve (or a k-jet of curve), and d D = dt . For a biholomorphic change of variable ϕ of (C, 0), we find by induction on ` polynomial differential operators Q`,i of order 6 ` acting on ϕ satisfying X D` (sj (f ◦ ϕ)) = ϕ0` D` (sj (f )) ◦ ϕ + Q`,i (ϕ0 , . . . , ϕ(`) ) Di (sj (f )) ◦ ϕ. i 0, while the existence of suitable

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sections sj ∈ H 0 (X, L) can be achieved only when L is ample, so the strategy seems a priori unapplicable. It turns out that one can sometimes arrange the Wronkian operator coefficients to be divisible by a section σ∆ ∈ H 0 (X, O X (∆)) possessing a large zero divisor ∆, so that
−1 ∗ (4.7) σ∆ Wk (s0 , . . . , sk ) ∈ H 0 Xk , O Xk (k 0 ) ⊗ πk,0 (Lk+1 ⊗ O X (−∆)) , and we can then hope that Lk+1 ⊗ O X (−∆)) < 0. The strategy is to find a variety X and sections σ0 , . . . , σk ∈ H 0 (X, L) for which the associated Wronskian Wk (s0 , . . . , sk ) is highly divisible. 4.2. Hyperbolicity of certain Fermat-Waring hypersurfaces. – Let Z be a non-singular (n + 1)-dimensional projective variety, and let A be a very ample divisor on Z ; the fundamental example is of course Z = Pn+1 and A = O Pn+1 (1). Our goal is to show that a well chosen (n-dimensional) hypersurface X = {x ∈ Z ; σ(x) = 0} defined by a section σ ∈ H 0 (Z, Ad ), d  1, is Kobayashi hyperbolic. The construction explained below follows closely the ideas of Shiffman-Zaidenberg [58] and is based similarly on a use of Fermat-Waring type hypersurfaces. Our proof is however completely selfcontained. The reader can consult Brody-Green [7], Nadel [50] and Masuda-Noguchi [45] for constructions based on other techniques. Theorem 4.8. – Let Z be a non-singular (n + 1)-dimensional projective variety, A a very ample divisor on Z, and τj ∈ H 0 (Z, A), 0 6 j 6 N , sufficiently general sections. Then for N > 2n and d > N 2 , the hypersurface X = σ −1 (0) associated with P σ = 06j6N τjd ∈ H 0 (Z, Ad ) is Kobayashi hyperbolic. In particular, Theorem 4.8 provides examples of hyperbolic hypersurfaces of Pn+1 for all n > 1 and all degrees d > 4n2 . A substantially improved bound d > d(n + 3)2 /4e has been obtained recently by [38] via a deformation argument for certain unions of hyperplanes, but the methods are quite different from the techniques used here. As in [58], the main step of our proof is the following proposition due to Toda [64], Fujimoto [33] and Green [36]. Proposition 4.9. – Let gj : C → C, 0 6 j 6 N , be non-zero entire functions such that P N the curve g = [g0 : · · · : gN ] : C → P satisfies 06j6N gjd = 0. If d > N 2 , there exists a partition J1 , . . . , Jq of {0, 1, . . . , N } such that |Js | > 2, gj /gi is constant for P all i, j ∈ Js , and j∈Js gjd = 0 for all s = 1, 2, . . . , q. If g is nonconstant, we must have q > 2. Proof. – The result is true for N = 1 (with a single J1 = {0, 1}), and for higher values N > 2 we apply induction and use vanishing arguments for Wronskians. The map g = [g0 : · · · : gN ] : C → PN can be seen P as an dentire Ncurve drawn in the (smooth, irreducible) Fermat hypersurface Y = 06j6N zj of P . We set k = N − 1 and consider on Y the Wronskian operator Wk (s0 , . . . , sk ) where sj (z) = zjd ,

PANORAMAS & SYNTHÈSES 56

sj ∈ H 0 (Y, O (d)).

A SIMPLE PROOF OF THE KOBAYASHI CONJECTURE

115

Then WN −1 (s0 , . . . , sN −1 ) ∈ H 0 (Y, Ek,k0 TY∗ ⊗ O (N d)). Since D` (sj ) is divisible by zjd−k for ` 6 k, we conclude that WN −1 (s0 , . . . , sN −1 ) is Q divisible by j 2. The case q = 1 corresponds to g being constant. Proposition 4.9 follows. Proof of Theorem 4.8. – We argue by induction on n > 1. For n = 1, an easy adjunction argument shows that it is enough to take d > 4: sections of A can be used to embed the polarized surface (Z, A) in PN (e.g., with N = 5), and whenever X = σ −1 (0) is a smooth curve, we have KX = KZX ⊗ Ad and a surjective restriction morphism Ω2PN → KZ = Λ2 TZ∗ . As Ω2PN ⊗ O (3) = ΛN −2 (TPN ⊗ O (−1)) is generated by sections, one sees that KZ ⊗ A3 is also generated by sections, hence KX is ample for d > 4. Now, assume that the result is already proved for n − 1 and consider a (nonP constant) entire curve f : C → X where X = { 06j6N τjd = 0} ⊂ Z. For suitably chosen sections τj ∈ H 0 (Z, A), 0 6 j 6 N and N > dim Z = n + 1, the map τ := [τ0 : · · · : τN ] : Z → PN can be taken to be a generically finite morphism. If τj ◦ f vanishes for some j, say j = N , then f is drawn in the hypersurface X 0 P −1 d 0 of Z 0 = τN (0) associated with σ 0 = 06j6N −1 τj . We can suppose that Z is 0 smooth and, by the induction hypothesis for (n − 1, N − 1), that X is hyperbolic (notice that N − 1 > 2(n − 1) and d > (N − 1)2 ); this is a contradiction. Without loss of generality, we can thus assume that all sections gj := τj ◦ f are non-zero. P Also dsuppose that g = τ ◦ f is nonconstant. By definition of X, we have 06j6N gj = 0, and Proposition 4.9 shows that there exists a partition J = {J1 , . . . , Jq } of {0, 1, . . . , N } such that q > 2, |Js | > 2, and the ratios gj 0 /gj are

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P constant for j, j 0 ∈ Js , and j∈Js gjd = 0 for all s = 1, 2, . . . , q. Set js = min Js and wj = gj /gjs ∈ C∗ for j ∈ Js \ {js }. Then g = [g0 : · · · : gN ] = τ ◦ f is drawn in a projective linear subspace YJ,w ⊂ PN −1 of dimension q − 1 defined by the equations X (4.10) YJ,w : zj = wj zjs for j ∈ Js \ {js }, 1 + wjd = 0, 1 6 s 6 q. j∈Js \{js }

Theorem 4.8 is now a consequence of the following lemma, which forces g = τ ◦ f , and hence f , to be constant. Lemma 4.11. – For N > 2n and τj ∈ H 0 (Z, A) sufficiently general, 0 6 j 6 N , the P hypersurface X = { 06j6N τjd = 0} is smooth and the map τ = [τ0 : · · · : τN ] : Z → PN has a restriction τ : X → PN that is a finite morphism. Moreover, for all partitions J = {Js } and all choices of w = (wj ) ∈ (C∗ )N +1−q as in (4.10), the preimage τ −1 (YJ,w ) in Z is finite. P Proof. – Let (σ1 , . . . , σm ) be a basis of H 0 (Z, A). We write τj = 16`6m aj` σ` and P consider the matrix a = (aj` ) ∈ Cm(N +1) . The singular locus of X = { 06j6N τjd = 0} is described by the equations d X  X aj` σ` (x) = 0, 06j6N

16`6m

X  X

∂ ∂xs

06j6N

d ! aj` σ` (x)

= 0,

16s6n+1

16`6m

in coordinates. As the σ` ’s generate all 1-jets at every point x ∈ X, we have (n + 2) independent equations in terms of a, hence the bad locus L of points (x, a) ∈ Z × Cm(N +1) admits a fibration pr1 : L → Z whose fibers are of dimension m(N + 1) − (n + 2) in Cm(N +1) . Therefore we get dim L 6 m(N + 1) − 1 and pr2 (L) does not cover Cm(N +1) . Any matrix a taken in the complement Cm(N +1) \ pr2 (L) will produce a smooth hypersurface X. Similary, as the σ` ’s separate points of Z, the set S of triples (x1 , x2 , a) ∈ Z × Z × Cm(N +1) with x1 , x2 ∈ X, x1 6= x2 and τ (x1 ) = τ (x2 ) is such that the fibers of S → Z × Z in Cm(N +1) are described by N + 1 independent equations d X  X aj` σ` (x1 ) = 0, 06j6N

16`6m

 X

 aj` σ` (x1 )

16`6m

06j6N

=

 X 16`6m

 aj` σ` (x2 )

∈ PN .

06j6N

Therefore dim S = dim(Z × Z) + m(N + 1) − (N + 1) 6 m(N + 1) + 1 and the projection S → Cm(N +1) has a fiber of dimension at most 1 over a generic point

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a ∈ Cm(N +1) . For such a choice of a, if F = τ −1 (y) is a fiber of τ : X → PN , then S contains F × F \ ∆F , hence we must have dim F = 0, and all fibers F are finite. In order to study the finiteness of τ −1 (YJ,w ), we look at the incidence variety VJ of 4-tuples (x1 , x2 , a, w) ∈ Z 2 × Cm(N +1) × WJ such that x1 6= x2 and P τ (x1 ) = τ (x2 ) ∈ YJ,w , where WJ is the set of points w = (wj ) such that 1 + j∈Js \{js } wjd = 0, 1 6 s 6 q. This variety will detect the fibers τ −1 (YJ,w ) that contain at least two distinct points. Notice alsoP that we have only finitely many subvarieties WJ involved, and that dim WJ = (|Js | − 2) = N + 1 − 2q. The variety VJ is defined by 2(N + 1 − q) + q − 1 linear equations in the aj` : X (aj` − wj ajs ` )σ` (xi ) = 0, j ∈ Js \ {js }, 1 6 s 6 q, i = 1, 2, 16`6m

 X 16`6m

 ajs ` σ` (x1 ) 16s6q

=

 X 16`6m

 ajs ` σ` (x2 )

∈ Pq−1 .

16s6q

These equations are independent: this is again a consequence of the fact that the σ` ’s separate points of Z. The dimension of VJ is thus
dim VJ = m(N + 1) + 2(n + 1) + (N + 1 − 2q) − 2(N + 1 − q) + (q − 1) = m(N + 1) + 2n + 2 − N − q. For q > 2 and N > 2n, we have dim VJ 6 n(N + 1), therefore the projection VJ → Cm(N +1) Shas finite fibers over a Zariski open set Cm(N +1) \ SJ . Hence, for a ∈ Cm(N +1) \ SJ , we infer that all sets τ −1 (YJ,w ) are finite. (For N > 2n + 1, we could even take a outside of the projections of the incidence varieties VJ , and in that case, for a generic, the sets τ −1 (YJ,w ) have at most one point).

5. Proof of the Kobayashi conjecture on the hyperbolicity of general hypersurfaces In this section, our more ambitious goal is to give a simple proof of the Kobayashi conjecture, combining ideas of Green-Griffiths [35], Demailly [16], Brotbek [8] and Ya Deng [25, Chapter 4], in chronological order. Related ideas had been used earlier in [67] and then in [9], to establish Debarre’s conjecture on the ampleness of the cotangent bundle of generic complete intersections, when their codimension is at least equal to the dimension. 5.1. Using blow-ups of Wronskian ideal sheaves. – Let X be a projective non-singular algebraic variety and L → X a line bundle over X. We consider a linear system Σ ⊂ H 0 (X, L) producing some non-zero Wronskian sections Wk (s0 , . . . , sk ), so that dim Σ > k + 1. As the Wronskian is alternate and multilinear in the arguments sj , we get a meromorphic map Xk > P (Λk+1 Σ∗ ) by sending a k-jet γ = f[k] (0) ∈ Xk to the point of projective coordinates [Wk (ui0 , . . . , uik )(f )(0)]i0 ,...,ik , where (uj )j∈J is

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a basis of Σ and i0 , . . . , ik ∈ J are in increasing order. This assignment factorizes through the Plücker embedding into a meromorphic map Φ : Xk

>

Grk+1 (Σ)

into the Grassmannian of dimension k + 1 subspaces of Σ∗ (or codimension k + 1 subspaces of Σ, alternatively). In fact, if LU ' U ×C is a trivialization of L in a neighborhood of a point x0 = f (0) ∈ X, we can consider the map ΨU : Xk → Hom(Σ, Ck+1 ) given by
−1 πk,0 (U ) 3 f[k] 7→ s 7→ (D` (s(f ))06`6k ) , and associate either the kernel Ξ ⊂ Σ of ΨU (f[k] ), seen as a point Ξ ∈ Grk+1 (Σ), or Λk+1 Ξ⊥ ⊂ Λk+1 Σ∗ , seen as a point of P (Λk+1 Σ∗ ) (assuming that we are at a point where the rank is equal to k + 1). Let O Gr (1) be the tautological very ample line bundle on Grk+1 (Σ) (equal to the restriction of O P (Λk+1 Σ∗ ) (1)). By construction, Φ is induced by the linear system of sections ∗ Wk (ui0 , . . . , uik ) ∈ H 0 (Xk , O Xk (k 0 ) ⊗ πk,0 Lk+1 ),

and we thus get a natural isomorphism (5.1)

O Xk (k

0

∗ ) ⊗ πk,0 Lk+1 ' Φ∗ O Gr (1) on Xk \ Bk ,

where Bk ⊂ Xk is the base locus of our linear system of Wronskians. The presence of the indeterminacy set Bk may create trouble in analyzing the positivity of our line bundles, so we are going to use an appropriate blow-up to resolve the indeterminacies. For this purpose, we introduce the ideal sheaf J k,Σ ⊂ O Xk generated bk,Σ → Xk in such a way by the linear system Σ, and take a modification µk,Σ : X ∗ bk,Σ . Then Φ is resolved that µk,Σ J k,Σ = O Xbk,Σ (−Fk,Σ ) for some divisor Fk,Σ in X bk,Σ → Grk+1 (Σ), and on X bk,Σ , (5.1) becomes an everyinto a morphism Φ ◦ µk,Σ : X where defined isomorphism (5.2)

∗ µ∗k,Σ O Xk (k 0 ) ⊗ πk,0 Lk+1 ) ⊗ O Xbk,Σ (−Fk,Σ ) ' (Φ ◦ µk,Σ )∗ O Gr (1).

bk to be the normalized blow-up of J k,Σ , i.e., the normalIn fact, we can simply take X bk → Xk to be ization of the closure Γ ⊂ Xk × Grk+1 (Σ) of the graph of Φ and µk,Σ : X b the  composition of the normalization map Xk → Γ with the first projection Γ → Xk . bk by a The Hironaka desingularization theorem would possibly allow us to replace X nonsingular modification, and Fk,Σ by a simple normal crossing divisor on the desingularization; we will avoid doing so here, as we would otherwise need to show the existence of universal desingularizations when (Xt , Σt ) is a family of linear systems of k-jets of sections associated with a family of algebraic varieties. The following basic lemma was observed by Ya Deng [25, Chapter 4]. Lemma 5.3. – Locally over coordinate open sets U ⊂ X on which LU is trivial, there is a maximal “Wronskian ideal sheaf ” J X k ⊃ J k,Σ in O Xk achieved by linear systems Σ ⊂ H 0 (U, L). It is attained globally on X whenever the linear system Σ ⊂ H 0 (X, L) generates k-jets of sections of L at every point. Finally, it is “universal” in the sense

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that is does not depend on L and behaves functorially under immersions: if ψ : X → Y Y is an immersion and J X k , J k are the corresponding Wronskian ideal sheaves in O Xk , Y X O Yk , then ψk∗ J k = J k with respect to the induced immersion ψk : Xk → Yk . Proof. – The (local) existence of such a maximal ideal sheaf is merely a consequence of the strong Noetherian property of coherent ideals. As observed at the end of Section 2.D, the bundle Xk → X is a locally trivial tower of Pn−1 -bundles, with a fiber R n,k that is a rational k(n − 1)-dimensional variety; over any coordinate open set U ⊂ X equipped with local coordinates (z1 , . . . , zn ) ∈ B(0, r) ⊂ Cn , it is isomorphic to the product U × R n,k , the fiber over a point x0 ∈ U being identified with the central fiber through a translation (t 7→ f (t)) 7→ (t 7→ x0 + f (t)) of germs of curves. In this setting, J X k is generated by the functions in O Xk associated with Wronskians Xk U 3 ξ = f[k] 7→ Wk (s0 , . . . , sk )(f ) ∈ O Xk (k 0 ) R n,k ,

sj ∈ H 0 (U, O X ),

by taking local trivializations O Xk (k 0 )ξ0 ' O Xk ,ξ0 at points ξ0 ∈ Xk . In fact, it is enough to take Wronskians associated with polynomials sj ∈ C[z1 , . . . , zn ]. To see this, one can e.g., invoke Krull’s lemma for local rings, which implies J X k,ξ0 = T X `+1 `>0 ( J k,ξ0 + mξ0 ), and to observe that `-jets of Wronskians Wk (s0 , . . . , sk ) (mod `+1 mξ0 ) depend only on the (k + `)-jets of the sections sj in O X,x0 /mk+`+1 , where x0 x0 = πk,0 (ξ0 ). Therefore, polynomial sections sj or arbitrary holomorphic functions sj define the same `-jets of Wronskians for any `. Now, in the case of polynomials, it is X clear that translations (t 7→ f (t)) 7→ (t 7→ x0 + f (t)) leave J X k invariant, hence J k is the pull-back by the second projection Xk U ' U × R n,k → R n,k of its restriction −1 to any of the fibers πk,0 (x0 ) ' R n,k . As the k-jets of the sj ’s at x0 are sufficient −1 to determine the restriction of our Wronskians to πk,0 (x0 ), the first two claims of Lemma 5.3 follow. The universality property comes from the fact that LU is trivial (cf. Property 4.3 b) and that germs of sections of O X extend to germs of sections of O Y via the immersion ψ. (Notice that in this discussion, one may have to pick −1 Taylor expansions of order > k for f to reach all points of the fiber πk,0 (x0 ), the order 2k − 1 being sufficient by [Dem95, Proposition 5.11], but this fact does not play any role here). A consequence of universality is that J X k does not depend on coordinates nor on the geometry of X. The above discussion combined with Lemma 5.3 leads to the following statement. Proposition 5.4. – Assume that L generates all k-jets of sections (e.g., take L = Ap with A very ample and p > k), and let Σ ⊂ H 0 (X, L) be a linear system that also generates k-jets of sections at any point of X. Then we have a universal isomorphism ∗ µ∗k O Xk (k 0 ) ⊗ πk,0 Lk+1 ) ⊗ O Xbk,Σ (−Fk ) ' (Φ ◦ µk )∗ O Grk+1 (Σ) (1),

bk → Xk is the normalized blow-up of the (maximal ) ideal sheaf where µk : X X bk J k ⊂ O Xk associated with order k Wronskians, and Fk the universal divisor of X X resolving J k .

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5.2. Specialization to suitable hypersurfaces. – As in §4.B, let Z be a non-singular (n + 1)-dimensional projective variety polarized with a very ample divisor A. We are going to show that a sufficiently general algebraic hypersurface X = {x ∈ Z ; σ(x) = 0} defined by σ ∈ H 0 (Z, Ad ) is Kobayashi hyperbolic when d is large. Brotbek’s main idea developed in [8] is that a carefully selected hypersurface (of a more complicated type than the Fermat-Waring hypersurfaces considered in §4) may have enough Wronskian sections to directly imply the ampleness of some tautological jet line bundle—a Zariski open property. Here, we take σ be a sum of terms X (5.5) σ = aj mδj , aj ∈ H 0 (Z, Aρ ), mj ∈ H 0 (Z, Ab ), n < N 6 k, d = δb+ρ, 06j6N

where δ  1 and the mj are “monomials” of the same degree b, i.e., product of b “linear” sections τI ∈ H 0 (Z, A), and the factors aj are general enough. The integer ρ is taken in the range [k, k + b − 1], first to ensure that H 0 (Z, Aρ ) generates k-jets of sections, and second, to allow d to be an arbitrary large integer (once δ > δ0 has been chosen large enough). The monomials mj will be chosen in such a way that for suitable c ∈ N, 1 6 c 6 N , any subfamily of c terms mj shares a common factor τI ∈ H 0 (X, A). To this
end, we consider all subsets I ⊂ {0, 1, . . . , N } with card I = c ; there are B = N c+1 subsets Q of this type. For all such I, we select sections τI ∈ H 0 (Z, A) such that I τI = 0 is a simple normal crossing divisor in Z (with all of its components of multiplicity 1).
N For j = 0, 1, . . . , N given, the number of subsets I containing j is b = c−1 . We put Y (5.6) mj = τI ∈ H 0 (Z, Ab ). I3j

The first step consists in checking that we can achieve X to be smooth with these constraints. Lemma 5.7. – Assume N > c(n + 1). Then, for a generic choice of the sections aj ∈ H 0 (Z, Aρ ) and τI ∈ H 0 (Z, A), the hypersurface X = σ −1 (0) ⊂ Z defined by (5.5), (5.6) is non-singular. Moreover, under the same condition for N , the Q intersection of τI = 0 with X can be taken to be a simple normal crossing divisor in X. Proof. – As the properties considered in the Lemma are Zariski open properties in terms of the (N + B + 1)-tuple (aj , τI ), it is sufficient to prove the result for a specific ρ−1 choice of the aj ’s: we fix here aj = τ˜j τI(j) where τ˜j ∈ H 0 (X, A), 0 6 j 6 N are new Q Q sections such that τ˜j τI = 0 is a simple normal crossing divisor, and I(j) is any subset of cardinal c containing j. Let H be the hypersurface of degree d of PN +B defined in homogeneous coordinates (zj , zI ) ∈ CN +B+1 by h(z) = 0 where X Y ρ−1 h(z) = zj zI(j) zIδ , 06j6N

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and consider the morphism Φ : Z → PN +B such that Φ(x) = (˜ τj (x), τI (x)). With our choice of the aj ’s, we have σ = h ◦ Φ. Now, when the τ˜j and τI are general enough, the map Φ defines an embedding of Z into PN +B (for this, one needs N +B > 2 dim Z+1 = 2n + 3, which is the case by our assumptions). Then, by definition, X is isomorphic to the intersection of H with Φ(Z). Changing generically the τ˜j and τI ’s can be achieved by composing Φ with a generic automorphism g ∈ Aut(PN +B ) = PGLN +B+1 (C) (as GLN +B+1 (C) acts transitively on (N + B + 1)-tuples of linearly independent linear forms). As dim g ◦ Φ(Z) = dim Z = n + 1, Lemma 5.7 will follow from a standard Bertini argument if we can check that Sing(H) has codimension at least n+2 in PN +B . In fact, this condition implies Sing(H) ∩ (g ◦ Φ(Z)) = ∅ for g generic, while g ◦ Φ(Z) can be chosen transverse to Reg(H). Now, a sufficient condition for smoothness is that one of the differentials dzj , 0 6 j 6 N , appears with a non-zero factor in dh(z) (just neglect the other differentials ∗dzI in this argument). We infer from this and Q the fact that δ > 2 that Sing(H) consists of the locus defined by I3j zI = 0 for all j = 0, 1, . . . , N . It is the union of the linear subspaces zI0 = · · · = zIN = 0 for all S possible choices of subsets Ij such that Ij 3 j. Since card Ij = c, the equality Ij = {0, 1, . . . , N } implies that there are at least d(N + 1)/ce distinct subsets Ij involved in each of these linear subspaces, and the equality can be reached. Therefore codim Sing(H) = d(N + 1)/ce > n + 2 as soon as N > c(n + 1). By the same argument, we can assume that the intersection of g ◦ Φ(Z) with at least (n + 2) Q distinct hyperplanes zI = 0 is empty. In order that τI = 0 defines a normal crossing divisor at a point x ∈ X, it is sufficient to ensure that for any family G of coordinate S hyperplanes zI = 0, I ∈ G , with card G 6 n + 1, we have a “free” index j ∈ / I∈ G I such that xI 6= 0 for all I 3 j, so that dh involves a non-zero term ∗ dzj independent of the dzI , I ∈ G . If this fails, there must be at least (n + 2) hyperplanesSzI =
0 containing x, associated either with I ∈ G , or with other I’s covering { I∈ G I . The corresponding bad locus is of codimension at least (n + 2) in PN +B T and can be avoided by g ◦ Φ(Z) for a generic choice of g ∈ Aut(PN +B ). Then X ∩ I∈ G τI−1 (0) is smooth of codimension equal to card G .

5.3. Construction of highly divisible Wronskians. – To any families s, τˆ of sections s1 , . . . , sr ∈ H 0 (Z, Ak ), τˆ1 , . . . , τˆr ∈ H 0 (Z, A), and any subset J ⊂ {0, 1, . . . , N } with card J = c, we associate a Wronskian operator of order k (i.e., a (k+1)×(k+1)-determinant) (5.8)
Wk,s,ˆτ ,a,J = Wk s1 τˆ1d−k , . . . , sr τˆrd−k , (aj mδj )j∈{J , r = k +c−N, {J = N −c+1. Q Q We assume here again that the τˆj are chosen so that τˆj τI = 0 defines a simple normal crossing divisor in Z and X. Since sj τˆjd−k , aj mδj ∈ H 0 (Z, Ad ), formula (4.6) applied with L = Ad implies that (5.9)

Wk,s,ˆτ ,a,J ∈ H 0 (Z, Ek,k0 TZ∗ ⊗ A(k+1)d ).

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However, we are going to see that Wk,s,ˆτ ,a,J and its restriction Wk,s,ˆτ ,a,JX are divisible by monomials τˆα τ β of very large degree, where τˆ, resp. τ , denotes the collection of sections τˆj , resp. τI in H 0 (Z, A). In this way, we will see that we can even obtain a negative exponent of A after simplifying τˆα τ β in Wk,s,ˆτ ,a,JX . This simplification process is a generalization of techniques already considered by [59] and [50] (and later [23]), in relation with the use of meromorphic connections of low pole order. Lemma 5.10. – Assume that δ > k. Then the Wronskian operator Wk,s,ˆτ ,a,J , resp. Wk,s,ˆτ ,a,JX , is divisible by a monomial τˆα τ β , resp. τˆα τ β τJδ−k (with a multi-index Q α Q notation τˆα τ β = τˆj j τIβI ), and |α| = r(d − 2k),

α, β > 0,

|β| = (N + 1 − c)(δ − k)b.

Proof. – Wk,s,ˆτ ,a,J is obtained as a determinant whose r first columns are the derivatives D` (sj τˆjd−k ) and the last N + 1 − c columns are the D` (aj mδj ), divisible respectively by τˆjd−2k and mδ−k . As mj is of the form τ γ , |γ| = b, this implies the divisibility j of Wk,s,ˆτ ,a,J by a monomial of the form τˆα τ β , as asserted. Now, we explain why one can gain the additional factor τJδ−k dividing the restriction Wk,s,ˆτ ,a,JX . First notice that τJ does not appear as a factor in τˆα τ β , precisely because the Wronskian involves only terms aj mδj with j ∈ / J, hence these mj ’s do not contain τJ . Let us P pick j0 = min({J) ∈ {0, 1, . . . , N }. Since X is defined by 06j6N aj mδj = 0, we have identically X X aj0 mδj0 = − ai mδi − ai mδi i∈J

i∈{J\{j0 }

in restriction to X, whence (by the alternate property of Wk (•)) X
Wk,s,ˆτ ,a,JX = − Wk s1 τˆ1d−k , . . . , sr τˆrd−k , ai mδi , (aj mδj )j∈{J\{j0 } X . i∈J

However, all terms mi , i ∈ J, contain by definition the factor τJ , and the derivatives D` (•) leave us a factor mδ−k at least. Therefore, the above restricted Wronskian i Q Q is also divisible by τJδ−k , thanks to the fact that τˆj τI = 0 forms a simple normal crossing divisor in X. Corollary 5.11. – For δ > k, there exists a monomial τˆαJ τ βJ dividing Wk,s,ˆτ ,a,JX such that |αJ | + |βJ | = (k + c − N )(d − 2k) + (N + 1 − c)(δ − k)b + (δ − k) and we have ∗ fk,s,ˆτ ,a,JX := (ˆ W τ αJ τ βJ )−1 Wk,s,ˆτ ,a,JX ∈ H 0 (X, Ek,k0 TX ⊗ A−p ),

where (5.12) p = |αJ | + |βJ | − (k + 1)d = (δ − k) − (k + c − N )2k − (N + 1 − c)(kb + ρ).

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In particular, we have p > 0 for δ large enough (all other parameters being fixed or bounded ), and under this assumption, the fundamental vanishing theorem implies that all entire curves f : C → X are annihilated by these Wronskian operators. Proof. – In fact, (k + 1)d = (k + c − N )d + (N + 1 − c)d = (k + c − N )d + (N + 1 − c)(δb + ρ) and we get (5.12) by subtraction. 5.4. Control of the base locus for sufficiently general coefficients aj in σ. – The next step is to control more precisely the base locus of these Wronskians and to find conditions on N , k, c, d = bδ +P ρ ensuring that the base locus is empty for a generic choice of the sections aj in σ = aj mj . Although we will not formally use it, the next lemma is useful to realize that the base locus is related to a natural rank condition. Lemma 5.13. – Set uj := aj mδj . The base locus in Xkreg of the above Wronskians Wk,s,ˆτ ,a,JX , when s, τˆ vary, consists of jets f[k] (0) ∈ Xkreg such that the matrix (D` (uj ◦ f )(0))06`6k, j∈{J is not of maximal rank (i.e., of rank < card {J = S N + 1 − c) ; if δ > k, this includes all jets f[k] (0) such that f (0) ∈ I6=J τI−1 (0). When J also varies, the base locus of all Wk,s,ˆτ ,a,JX in the Zariski open set S Xk0 := Xkreg \ |I|=c τI−1 (0) consists of all k-jets such that rank(D` (uj ◦ f )(0))06`6k, 06j6N 6 N − c. Proof. – If δ > k and mj ◦f (0) = 0 for some j ∈ J, we have in fact D` (uj ◦f )(0) = 0 for all derivatives ` 6 k, because the exponents involved in all factors of the differentiated monomial aj mδj are at least equal to δ − k > 0. Hence the rank of the matrix cannot be maximal. Now, assume that mj ◦ f (0) 6= 0 for all j ∈ {J, i.e., [ [ (5.14) x0 := f (0) ∈ X \ m−1 τI−1 (0). j (0) = X \ j∈{J

I6=J

We take sections τˆj so that τˆj (x0 ) 6= 0, and then adjust the k-jet of the sections s1 , . . . , sr in order to generate any matrix of derivatives (D` (sj (f )ˆ τj (f )d−k )(0))06`6k, j∈{J (the fact that f 0 (0) 6= 0 is used for this!). Therefore, by expanding the determinant according to the last N + 1 − c columns, we see that the base locus is defined by the equations (5.15)

det(D` (uj (f ))(0))`∈L, j∈{J = 0,

∀L ⊂ {0, 1, . . . , k}, |L| = N + 1 − c,

equivalent to the non-maximality of the rank. The last assertion follows by a simple linear algebra argument.

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For a finer control of the base locus, we adjust the family of coefficients a = (aj )06j6N ∈ S := H 0 (Z, Aρ )⊕(N +1) P in our section σ = aj mδj ∈ H 0 (Z, Ad ), and denote by Xa = σ −1 (0) ⊂ Z the corresponding hypersurface. By Lemma 5.7, Q we know that there is a Zariski open set U ⊂ S such that Xa is smooth and τI = 0 is a simple normal crossing divisor in Xa for all a ∈ U . We consider the Semple tower Xa,k := (Xa )k of Xa , ba,k → Xa,k of the Wronskian ideal sheaf J a,k such the “universal blow-up” µa,k : X ∗ ba,k . By the that µa,k J a,k = O Xba,k (−Fa,k ) for some “Wronskian divisor” Fa,k in X universality of this construction, we can also embed Xa,k in the Semple tower Zk of Z, blow up the Wronskian ideal sheaf J Z k of Zk to get a Wronskian divisor Fk bk where µk : Z bk → Zk is the blow-up map. Then Fa,k is the restriction of Fk in Z ba,k ⊂ Zbk . Our section W fk,a,ˆτ ,s,JX is the restriction of a meromorphic section to X a defined on Z, namely
(5.17) (ˆ τ αJ τ βJ )−1 Wk,s,ˆτ ,a,J = (ˆ τ αJ τ βJ )−1 Wk s1 τˆ1d−k , ... , sr τˆrd−k , (aj mδj )j∈{J . S It induces over the Zariski open set Z 0 = Z \ I τI−1 (0) a holomorphic section
∗ (5.18) σk,s,ˆτ ,a,J ∈ H 0 Zbk0 , µ∗k ( O Zk (k 0 ) ⊗ πk,0 A−p ) ⊗ O Zbk (−Fk ) (5.16)

(notice that the relevant factors τˆj remain divisible on the whole variety Z). By construction, thanks to the divisibility property explained in Lemma 5.10, the restriction b0 = X ba,k ∩ Z b0 extends holomorphically to X ba,k , i.e., of this section to X a,k k
∗ ba,k , µ∗a,k ( O X (k 0 ) ⊗ πk,0 (5.19) σk,s,ˆτ ,a,JXba,k ∈ H 0 X A−p ) ⊗ O Xba,k (−Fa,k ) . a,k bk,a to be normal avoids any potential issue in the division (Here the fact that we took X
T −1 −1 bk,a ∩ µ process, as X πk,0 I∈ G τI−1 (0) has the expected codimension = card G for k any family G ). Lemma 5.20. – Let V be a finite dimensional vector space over C, Ψ : V p → C a nonzero alternating multilinear form, and let m, c ∈ N, c < m 6 p, r = p + c − m > 0. Then the subset T ⊂ V m of vectors (v1 , . . . , vm ) ∈ V m such that (∗) Ψ(h1 , . . . , hr , (vj )j∈{J ) = 0 for all J ⊂ {1, . . . , m}, |J| = c, and all h1 , . . . , hr ∈ V , is a closed algebraic subset of codimension > (c + 1)(r + 1). Proof. – A typical example is Ψ = det on a p-dimensional vector space V , then T consists of m-tuples of vectors of rank < p − r, and the assertion concerning the codimension is well known (we will reprove it anyway). In general, the algebraicity of T is obvious. We argue by induction on p, the result being trivial for p = 1 (the kernel of a non-zero linear form is indeed of codimension > 1). If K is the kernel of Ψ, i.e., the subspace of vectors v ∈ V such that Ψ(h1 , . . . , hp−1 , v) = 0 for all hj ∈ V , then Ψ induces an alternating multilinear form Ψ on V /K, whose kernel is equal to {0}. The proof is thus reduced to the case when Ker Ψ = {0}. Notice that we must have

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125

dim V > p, otherwise Ψ would vanish. If card {J = m − c = 1, condition (∗) implies that vj ∈ Ker Ψ = {0} for all j, hence codim T = dim V m > mp = (c + 1)(r + 1), as desired. Now, assume m − c > 2, fix vm ∈ V \ {0} and consider the non-zero alternating multilinear form on V p−1 such that Ψ0vm (w1 , . . . , wp−1 ) := Ψ(w1 , . . . , wp−1 , vm ). If (v1 , . . . , vm ) ∈ T , then (v1 , . . . , vm−1 ) belongs to the set Tv0 m associated with the new data (Ψ0vm , p − 1, m − 1, c, r). The induction hypothesis implies that codim Tv0 m > (c + 1)(r + 1), and since the projection T → V to the first factor admits the Tv0 m as its fibers, we conclude that codim T ∩ ((V \ {0}) × V m−1 ) > (c + 1)(r + 1). By permuting the arguments vj , we also conclude that codim T ∩ (V k−1 × (V \ {0}) × V m−k ) > (c + 1)(r + 1) S for all k = 1, . . . , m. The union k (V k−1 × (V \ {0}) × V m−k ) ⊂ V m leaves out only {0} ⊂ V m whose codimension is at least mp > (c + 1)(r + 1), so Lemma 5.20 follows. Proposition 5.21. – Consider in U × Zbk0 the set Γ of pairs (a, ξ) such that σk,s,ˆτ ,a,J (ξ) = 0 for all choices of s, τˆ and J ⊂ {0, 1, . . . , N } with card J = c. Then Γ is an algebraic set of dimension dim Γ 6 dim S − (c + 1)(k + c − N + 1) + n + 1 + kn. As a consequence, if (c + 1)(k + c − N + 1) > n + 1 + kn, there exists a ∈ U ⊂ S such ba,k lies over S Xa ∩ τ −1 (0). that the base locus of the family of sections σk,s,ˆτ ,a,J in X I I Proof. – The idea is similar to [Brot17, Lemma 3.8], but somewhat simpler in the b0 and the k-jet f[k] = µk (ξ) ∈ Z 0 , so present context. Let us consider a point ξ ∈ Z k k S −1 0 that x = f (0) ∈ Z = Z \ I τI (0). Let us take the τˆj such that τˆj (x) 6= 0. Then, we do not have to pay attention to the non-vanishing factors τˆαJ τ βJ , and the k-jets of sections mj and τˆjd−k are invertible near x. Let eA be a local generator of A near x and e L a local generator of the invertible sheaf L

= µ∗k O Zk (k 0 ) ⊗ O Zbk (−Fk )

near ξ ∈ Zbk0 . Let J k O Z,x = O Z,x /mk+1 Z,x be the vector space of k-jets of functions on Z at x. By definition of the Wronskian ideal and of the associated divisor Fk , we have a non-zero alternating multilinear form Ψ : (J k O Z,x )k+1 → C,

(g0 , . . . , gk ) 7→ µ∗k Wk (g0 , . . . , gk )(ξ)/e L (ξ).

The simultaneous vanishing of our sections at ξ is equivalent to the vanishing of
δ −d (5.22) Ψ s1 τˆ1d−k e−d ˆrd−k e−d A , . . . , sr τ A , (aj mj eA )j∈{J

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for all (s1 , . . . , sr ). Since A is very ample and ρ > k, the power Aρ generates k-jets at every point x ∈ Z, hence the morphisms H 0 (Z, Aρ ) → J k O Z,x ,

a 7→ amδj e−d A

and H 0 (Z, Ak ) → J k O Z,x ,

s 7→ sˆ τjd−k e−d A

are surjective. Lemma 5.20 applied with r = k + c − N and (p, m) replaced by (k + 1, N + 1) implies that the codimension of families a = (a0 , . . . , aN ) ∈ S = H 0 (Z, Aρ )⊕(N +1) for which σk,s,ˆτ ,a,J (ξ) = 0 for all choices of s, τˆ and J is at least (c+1)(k +c−N +1), i.e., the dimension is at most dim S − (c + 1)(k + c − N + 1). When we let ξ vary over Zbk0 which has dimension (n + 1) + kn and take into account the fibration (a, ξ) 7→ ξ, the dimension estimate of Proposition 5.21 follows. Under the assumption (c + 1)(k + c − N + 1) > n + 1 + kn

(5.23)

we have dim Γ < dim S, hence the image of the projection Γ → S, (a, ξ) 7→ a is a constructible algebraic subset distinct from S. This concludes the proof. Our final goal is to completely eliminate the base locus. Proposition 5.21 indicates that we have to pay attention to the intersections Xa ∩ τI−1 (0). For x ∈ Z, we let G be the family of hyperplane sections τI = 0 that contain x. We introduce the S set P = {0, 1, . . . , N } \ I∈ G I and the smooth intersection \ ZG = Z ∩ τI−1 (0), I∈ G 0

so that N + 1 := card P > N + 1 − c card G and dim Z G = n + 1 − card G . If a ∈ U is such that x ∈ Xa , we also look at the intersection \ X G ,a = Xa ∩ τI−1 (0), I∈ G

which is a smooth hypersurface of Z G . In that situation, we consider Wronskians Wk,s,ˆτ ,a,J as defined above, but we now take J ⊂ P , card J = c, {J = P \ J, r0 = k + c − N 0 . Lemma 5.24. – In the above setting, if we assume δ > k, the restriction Wk,s,ˆτ ,a,JX G ,a is still divisible by a monomial τˆαJ τ βJ such that |αJ | + |βJ | = (k + c − N 0 )(d − 2k) + (N 0 + 1 − c)(δ − k)b + (δ − k). Therefore, if p0 = |αJ | + |βJ | − (k + 1)d = (δ − k) − (k + c − N 0 )2k − (N 0 + 1 − c)(kb + ρ) as in (5.12), we obtain again holomorphic sections 0 ∗ fk,s,ˆτ ,a,JX := (ˆ W τ αJ τ βJ )−1 Wk,s,ˆτ ,a,JX G ,a ∈ H 0 (X G ,a , Ek,k0 TX ⊗ A−p ), G ,a


0 −1 ∗ σk,s,ˆτ ,a,Jπ−1 (X G ,a ) ∈ H 0 πk,0 (X G ,a ), µ∗a,k ( O Xa,k (k 0 ) ⊗ πk,0 A−p ) ⊗ O Xba,k (−Fa,k ) . k,0

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Proof. – The arguments are similar to those employed in the proof of Lemma 5.10. Let f[k] ∈ Xa,k be a k-jet such that f (0) ∈ X G ,a (the k-jet need not be entirely contained in X G ,a ). Putting j0 = min({J), we observe that we have on X G ,a an identity X X X aj0 mδj0 = − ai mδi = − ai mδi − ai mδi i∈P \{j0 }

Q

i∈J

P \(J∪{j0 })

S

because mi = I3i τI = 0 on X G ,a when i ∈ {P = I∈ G I (one of the factors τI is such that I ∈ G , hence τI = 0). If we compose with a germ t 7→ f (t) such that f (0) ∈ X G ,a (even though f does not necessarily lie entirely in X G ,a ), we get X X ai mδi (f (t)) − aj0 mδj0 (f (t)) = − ai mδi (f (t)) + O(tk+1 ) i∈J

P \(J∪{j0 })

as soon as δ > k. Hence we have an equality for all derivatives D` (•), ` 6 k at t = 0, and Wk,s,ˆτ ,a,JX G ,a (f[k] ) X
=− Wk s1 τˆ1d−k , . . . , sr0 τˆrd−k , ai mδi , (aj mδj )j∈P \(J∪{j0 }) X 0

G ,a

(f[k] ).

i∈J

Then, again, τJδ−k is a new additional common factor of all terms in the sum, and we conclude as in Lemma 5.10 and Corollary 5.11. Now, we analyze the base locus of these new sections on [ −1 −1 −1 b µ−1 a,k πk,0 (X G ,a ) ⊂ µk πk,0 (Z G ) ⊂ Zk . a∈U

As x runs in Z G and N 0 < N , Lemma 5.20 shows that (5.23) can be replaced by the less demanding condition (5.230 )

−1 (c + 1)(k + c − N 0 + 1) > n + 1 − card G + kn = dim µ−1 k πk,0 (Z G ).

A proof entirely similar to that of Proposition 5.21 shows that for a generic choice −1 b G ,a,k projects onto S of a ∈ U , the base locus of these sections on X I∈{ G X G ,a ∩τI (0). Arguing inductively on card G , the base locus can be shrinked step by step down to empty set (but it is in fact sufficient to stop when X G ,a ∩ τI−1 (0) reaches dimension 0). 5.5. Nefness and ampleness of appropriate tautological line bundles. – At this point, we have produced a smooth family X S → U ⊂ S of particular hypersurfaces in Z, namely Xa = {σa (z) = 0}, a ∈ U , for which a certain “tautological” line bundle has an empty base locus for sufficiently general coefficients: Corollary 5.25. – Under condition (5.23) and the hypothesis p > 0 in (5.12), the following properties hold.

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(a) The line bundle La

∗ := µ∗a,k ( O Xa,k (k 0 ) ⊗ πk,0 A−1 ) ⊗ O Xba,k (−Fa,k )

ba,k for general a ∈ U 0 , where U 0 ⊂ U is a dense Zariski open set. is nef on X P (b) Let ∆a = 26`6k λ` Da,` be a positive rational combination of vertical divisors of the Semple tower and q ∈ N, q  1, an integer such that 0

La

∗ := O Xa,k (1) ⊗ O a,k (−∆a ) ⊗ πk,0 Aq

is ample on Xa,k . Then the Q-line bundle L a,ε,η

∗ := µ∗a,k ( O Xa,k (k 0 ) ⊗ O Xa,k (−ε∆a ) ⊗ πk,0 A−1+qε ) ⊗ O Xba,k (−(1 + εη)Fa,k )

ba,k for a ∈ U 0 , for some q ∈ N and ε, η ∈ Q>0 arbitrarily small. is ample on X Proof. – (a) This would be obvious if we had global sections generating L a on the ba,k , but our sections are only defined on a stratification of X ba,k . In any case, whole of X ba,k is an irreducible curve, we take a maximal family G such that C ⊂ X G ,a,k . if C ⊂ X Then, by what we have seen, for a ∈ U general enough, we can find global sections b G ,a,k such that C is not contained in their base locus. Hence L a · C > 0 of L a on X and L a is nef for a in a dense Zariski open set U 0 ⊂ U . (b) The existence of ∆a and q follows from Proposition 3.19 and Corollary 3.21, which even provide universal values for λ` and q. After taking the blow up µa,k : ba,k → Xa,k (cf. (4.8)), we infer that X 0

L a,η

0

:= µ∗a,k L a ⊗ O Xba,k (−ηFa,k )
∗ = µ∗a,k O Xa,k (1) ⊗ O Xa,k (−∆a ) ⊗ πk,0 Aq ⊗ O Xba,k (−ηFa,k )

is ample for η > 0 small. The result now follows by taking a combination L a,ε,η

1−ε/k0

= La

0

⊗ ( L a,η )ε .



Corollary 5.26. – Let X → Ω be the universal family of hypersurfaces Xσ = {σ(z) = 0}, σ ∈ Ω, where Ω ⊂ P (H 0 (Z, Ad )) is the dense Zariski open set over which the family bσ,k of Xσ,k , let us consider the line bundle is smooth. On the “Wronskian blow-up” X L σ,ε,η

∗ := µ∗σ,k ( O Xσ,k (k 0 ) ⊗ O Xσ,k (−ε∆σ ) ⊗ πk,0 A−1+qε ) ⊗ O Xbσ,k (−(1 + εη)Fσ,k ) 0

associated with the same choice of constants as in Cor. 5.25. Then L σ,ε,η is ample bσ,k for σ in a dense Zariski open set Ω0 ⊂ Ω. on X Proof. – By 5.25 (b), we can find σ0 ∈ H 0 (Z, Ad ) such that Xσ0 = σ0−1 (0) is smooth m bσ ,k (m ∈ N∗ ). As ampleness is a Zariski open and L σ0 ,ε,η is an ample line bundle on X 0 m condition, we infer that L σ,ε,η remains ample for a general section σ ∈ H 0 (Z, Ad ), i.e., for [σ] in some Zariski open set Ω0 ⊂ Ω. Since µσ,k (Fσ,k ) is contained in the vertical divisor of Xσ,k , we conclude by Corollary 3.27 that Xσ is Kobayashi hyperbolic for [σ] ∈ Ω.

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5.6. Final conclusion and computation of degree bounds. – At this point, we fix our integer parameters to meet all conditions that have been found. We must have N > c(n + 1) by Lemma 5.7, and for such a large value of N , condition (5.23) can hold only when c > n, so we take c = n and N = n(n + 1). Inequality (5.23) then requires k large enough, k = n3 + n2 + 1 being the smallest possible value. We find    2  N n +n (n2 + n) · · · (n2 + 2) b= = =n . c−1 n−1 n! We have n2 + k = n2 (1 + k/n2 ) < n2 exp(k/n2 ) and by Stirling’s formula, √ n! > 2πn (n/e)n , hence 1

b
k. Then δ + 1 > (d − k + 1)/b and formula (5.12) yields p = (δ − k) − (n3 + 1)2k − (n2 + 2n + 1)(kb + ρ) > (d − k + 1)/b − 1 − (2n3 + 3)k − (n2 + 2n + 1)(kb + k + b − 1), therefore p > 0 is achieved as soon as
d > dn = k + b 1 + (2n3 + 3)k + (n2 + 2n + 1)(kb + k + b − 1) , where 3

2

k = n + n + 1,

 2  n +n b= . n−1

The dominant term in dn is k(n2 + 2n + 1)b2 ∼ e2n+1 n2n+2 /2π. By means of more precise numerical calculations and of Stirling’s asymptotic expansion for n!, one can check in fact that dn 6 b(n + 4) (en)2n+1 /2πc for n > 4 (which is also an equivalent and a close approximation as n → +∞), while d1 = 61, d2 = 6,685, d3 = 2,825,761. We can now state the main result of this section. Theorem 5.27. – Let Z be a projective (n + 1)-dimensional manifold and A a very ample line bundle on Z. Then, for a general section σ ∈ H 0 (Z, Ad ) and d > dn , the hypersurface Xσ = σ −1 (0) is Kobayashi hyperbolic. The bound dn for the degree can be taken to be dn = b(n + 4) (en)2n+1 /2πc

for n > 4,

and for n 6 3, one can take d1 = 4, d2 = 6,685, d3 = 2,825,761. For n = 1, we have already seen in § 4.B that d1 = 4 works (rather than the insane value d1 = 61). A simpler (and less refined) choice is d˜n = b 31 (en)2n+2 c, which is valid for all n. These bounds are only slightly weaker than the ones found by Ya Deng in his PhD thesis [25, 26], namely d˜n = O(n2n+6 ).

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5.7. Further comments. – 5.28. Our bound dn is rather large, but just as in Ya Deng’s effective approach of Brotbek’s theorem [26], the bound holds for a property that looks substantially stronger than hyperbolicity, namely the ampleness of the pull-back of some (twisted) jet bundle µ∗k O Xbk (a• ) ⊗ O Xbk (−Fk0 ). It is certainly desirable to look for more general jet differentials than Wronskians, and to relax the positivity demands on tautological line bundles to ensure hyperbolicity (see e.g., [20]). However, the required calculations appear to be much more involved. 5.29. After this chapter was written, Riedl and Yang [53] proved the important and somewhat surprising result that the lower bound estimates dGG (n) and dKob (n), respectively for the Green-Griffiths-Lang and Kobayashi conjectures for general hypersurfaces in Pn+1 , can be related by dKob (n) := dGG (2n − 2). This should be understood in the sense that a solution of the generic (2n − 2)-dimensional GreenGriffiths conjecture for d > dGG (2n − 2) implies a solution of the n-dimensional Kobayashi conjecture for the same lower bound. We refer to [53] for the precise statement, which requires an extra assumption on the algebraic dependence of the GreenGriffiths locus with respect to a variation of coefficients in the defining polynomials. In combination with [27], this gives a completely new proof of the Kobayashi conjecture, and the order 1 bound dGG (n) = O(exp(n1+ε )) of [19] implies a similar bound dKob (n) = O(exp(n1+ε )) for the Kobayashi conjecture—just a little bit weaker than what our direct proof gave (Theorem 5.26). In [49], Merker and Ta were able to im√ prove the Green-Griffiths bound to dGG (n) = o( n log n)n , using a strengthening of Darondeau’s estimates [13, 14], along with very delicate calculations. The Riedl-Yang result then implies dKob (n) = O((n log n)n+1 ), which was the best bound known when [49] appeared. 5.30. In the unpublished preprint [22], we introduced an alternative strategy for the proof of the Kobayashi conjecture which appears to be still incomplete at this point. We nevertheless hope that a refined version could one day lead to linear bounds such as dKob (n) = 2n+1. The rough idea was to establish a k-jet analogue of Claire Voisin’s proof [66] of the Clemens conjecture. Unfortunately, Lemma 5.1.18 as stated in [22] is incorrect—the assertion concerning the ∆ divisor introduced there simply does not hold. It is however conceivable that a weaker statement holds, in the form of a control of the degree of the divisor ∆, and in a way that would still be sufficient to imply similar consequences for the generic positivity of tautological jet bundles. 5.31. In [2], G. Bérczi stated a positivity conjecture for Thom polynomials of Morin singularities (see also [5]), and showed that it would imply a polynomial bound dn = 2 n9 + 1 for the generic hyperbolicity of hypersurfaces. 5.32. In September 2019, Bérczi and Kirwan [4] introduced new deep ideas in nonreductive geometric invariant theory that actually lead to polynomials bounds for the Kobayashi conjecture. Their technique is based on the use of alternative compactifications for the jet spaces.

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, “Application of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy”, J. Symbolic Comput. 45 (2010), p. 986–1074. √ [49] J. Merker & T.-A. Ta – “Degrees d > n log n)n and d > (n log n)n in the Conjectures of Green-Griffiths and of Kobayashi”, preprint arXiv:1901.04042. [50] A. M. Nadel – “Hyperbolic surfaces in P3 ”, Duke Math. J. 58 (1989), p. 749–771. [51] G. Pacienza – “Subvarieties of general type on a general projective hypersurface”, Trans. Amer. Math. Soc. 356 (2004), p. 2649–2661. [52] M. Păun – “Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity”, Math. Ann. 340 (2008), p. 875–892. [53] E. Riedl & D. Yang – “Applications of a Ggrassmannian technique in hypersurfaces”, preprint arXiv:1806.02364. [54] E. Rousseau – “Étude des jets de Demailly-Semple en dimension 3”, Ann. Inst. Fourier 56 (2006), p. 397–421. [55] H. L. Royden – “Remarks on the Kobayashi metric”, in Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Springer Lecture Notes in Math., vol. 185, 1971, p. 125–137. [56]

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Jean-Pierre Demailly, Université Grenoble Alpes, Institut Fourier Laboratoire de Mathématiques, CNRS UMR 5582, 100 rue des Maths 38610 Gières, France E-mail : [email protected]

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Panoramas & Synthèses 56, 2021, p. 135–167

MCQUILLAN’S APPROACH TO THE GREEN-GRIFFITHS CONJECTURE FOR SURFACES by Carlo Gasbarri

Abstract. – After reviewing the proof by Bogomolov of the fact that, on smooth projective surfaces with c21 > c2 , curves of bounded geometric genus form a bounded family, we explain the main steps of the proof, given by McQuillan, of the GreenGriffiths conjectures for these surfaces. Viewed from afar the two proofs follow the same strategy, but the second requires a much deeper analysis of the tools involved. In order to describe McQuillan’s proof we explain the construction of the Ahlfors currents associated to entire curves in a variety and we show how these can be used to produce a substitute for intersection numbers. A proof of the tautological inequality in both the standard case and the logarithmic case is given. We explain how the hypotheses allow us to suppose that the involved entire curve (which a posteriori should not exist) is a leaf of a foliation. In order to simplify some technical points of the proof we impose some restrictions on the singularities of this foliation (the general case requires a much more involved analysis but the main ideas of the proof are already visible under this restriction). In the last section we give a very brief description of a possible strategy (proposed by McQuillan) for the proof of the general case of the conjecture, together with an explanation of the main difficulties that must be overcome.

1. Introduction Let X be a smooth projective variety. If X is hyperbolic, then there is no, non constant analytic map f : C → X. It is also known that hyperbolicity implies algebraic hyperbolicity. Thus if X is hyperbolic and L is an ample line bundle on it, we can find a constant A for which the following holds: for every smooth projective curve C of genus g and morphism f : C → X we have deg(f ∗ (L)) ≤ A(2g − 2). For a smooth projective surface of general type we cannot expect that an inequality like the one above holds. This is, for instance, due to the fact that one can find rational or elliptic curves on it. But probably the following conjecture holds:

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Conjecture 1.1. – Let X be a smooth projective surface of general type. Let L an ample line bundle on it. Then there exist constants A and B for which the following holds: for every smooth projective curve C of genus g and morphism f : C → X we have deg(f ∗ (L)) ≤ Ag + B.

(1.1)

This conjecture is very deep and, in particular, it implies that on a surface of general type there are only finitely many rational or elliptic curves. More specifically the conjecture implies the following: Conjecture 1.2. – Let X be a smooth surface of general type. Then there exists a proper Zariski closed set Z ⊂ X for which the following holds: If C is a rational or an elliptic curve and f : C → X is a non constant map, then f factorize through Z. The Green-Griffiths conjecture is a generalization of this last conjecture: Conjecture 1.3 (Green-Gritffiths Conjecture). – Let X be a smooth surface of general type. Then there exists a proper Zariski closed set Z ⊂ X for which the following holds: every non constant analytic map f : C → X factorizes through Z. If, in the possibility that Conjecture 1.2 is false, we may weaken the Conjecture 1.3 by requiring only that the closed set Z depends only on f (but it should be one dimensional). In this chapter we will discuss some advances on these conjectures and a strategy, due to McQuillan, to attack Conjecture 1.3 in general. The reader can refer also to the paper [7] for another description of the proof.

2. Curves of bounded genus on surfaces with big cotangent bundle Definition 2.1. – Let X be a smooth projective variety and E be a vector bundle over it. We will say that E is ample, big, nef, . . . , if the tautological bundle O (1) is ample, big, nef, . . . on the projective bundle P(E) respectively. We start this section by showing how to prove algebraically a strong version of Conjecture 1.1 on varieties with ample cotangent bundle. Remark that if a variety X has ample cotangent bundle, then his canonical line bundle KX is ample too. Theorem 2.2. – Let X be a smooth projective variety with ample cotangent bundle. Then there exists a constant A with the following property: for every smooth curve C of genus g and every non constant map f : C → X we have (2.1)

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deg(f ∗ (KX )) ≤ A(2g − 2).

MCQUILLAN’S APPROACH TO THE GREEN-GRIFFITHS CONJECTURE FOR SURFACES 137

Proof. – Consider the structure morphism p : P(Ω1X ) → X. Since, by hypothesis, O (1) is ample on P(Ω1X ), there is an integer N for which the line bundle −1 MN := O (N ) ⊗ p∗ (KX ) is ample on it. Let C be a smooth projective curve and f : C → X be a non constant map. The natural map f ∗ (Ω1X ) → Ω1C gives, by functoriality, a map f 0 : C → P(Ω1X ) such that f = p ◦ f 0 . By construction f 0∗ ( O (1)) ,→ Ω1C . Thus deg(f 0∗ ( O (1)) ≤ 2g − 2. Since MN is ample on P(Ω1X )) we have that deg(f 0∗ (MN )) ≥ 0. Consequently deg(f ∗ (KX )) ≤ N (2g − 2). We observe that, in particular, we obtain that such a variety do not contain rational or elliptic curves. Bogomolov Theorem 2.3 [1] generalize Theorem 2.2 to surfaces whose cotangent bundle is big. Theorem 2.3 (Bogomolov). – Let X be a smooth projective surface with big cotangent bundle. Then there exist constants A1 and A2 with the following property: for every smooth curve C of genus g and every non constant map f : C → X we have (2.2)

deg(f ∗ (KX )) ≤ A1 (2g − 2) + A2 .

Observe that a sufficient condition for the cotangent bundle to be big is that c1 (X)2 > c2 (X) (exercise). Let’s remark an interesting corollary: Corollary 2.4. – Let X be a surface with big cotangent bundle. Then X contains only finitely many rational or elliptic curves. Proof. – Indeed, curves of bounded genus in such a surface are a bounded family and surfaces of general type are not covered by rational or elliptic curves. We now prove Theorem 2.3. Proof. – As before we fix an ample line bundle L on X. Consider the natural morphism p : P(Ω1X ) → X. If M is a line bundle on P(Ω1X ), we denote by Bs(M ) the base locus of it; it is a Zariski closed set which coincides with P(Ω1X ) if and only if H 0 (X, M ) = {0}. For every positive integers n and m, consider the closed set Bn,m := Bs(( O (m) ⊗ T p∗ (L−1 ))⊗n ) ⊂ P(Ω1X ). Let B = n,m Bn,m . Since O (1) is big, we have that B 6= P(Ω1X ). By Noetherianity we may suppose that there exists n0 and m0 such that B = Bn0 ,m0 . Let C be a smooth projective curve of genus g and f : C → X be a non constant map. Suppose that the lift f 0 : C → P(Ω1X ) of f do not factor through B. This implies that there is a global section s ∈ H 0 (P(Ω1X ); ( O (n0 ) ⊗ p∗ (L− 1))m0 ) which do not vanish identically along f 0 (C). Consequently deg(f 0( O (n0 )) ⊗ f ∗ (L−1 )) ≥ 0. Thus (2.3)

deg(f ∗ (L)) ≤ n0 (2g − 2).

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Thus, for these curves it suffices to take A2 bigger then n0 . We must now deal with the case when the morphism f 0 factor through B. Consider an irreducible component of B. By abuse of notation we will denote it again by B. If p(B) is of dimension at most one the conclusion of the theorem easily follows: f (C) can be contained in a finite list of curves inside X, the genus and the degree of which can be absorbed by the constant A2 . Suppose that p : B → X is dominant. In this case, since B 6= P(Ω1X ), the dimension of B is two. We will now show that the image of C in B is leaf of a natural foliation on it. Lemma 2.5. – there exists a smooth projective surface e → B and a natural algebraic foliation F on B e with B e be a lift of f 0 . Then, either f1 (C) belongs to f1 : C → B

e a birational morphism B, the following property: Let a finite list or f1 (C) is leaf

of the foliation F . Before we start the proof of the lemma, we recall some basic definitions of foliations on surfaces. 2.1. Standard fact about algebraic foliations on surfaces Definition 2.6. – Let Y be a smooth algebraic surface. An algebraic foliation F is a sub sheaf N de Ω1Y which is locally free of rank one such that the quotient Ω1Y /N is torsion free. We recall the following facts: a) The line bundle N is usually called the conormal sheaf of the foliation. The line bundle det(Ω1Y /N ) is usually called the canonical sheaf of F and denoted by K F . b) There is a zero dimensional subschema Z ⊂ Y (in general non reduced) which is called the singular locus of the foliation and an exact sequence (2.4)

0 −→ N −→ Ω1Y −→ IZ ⊗ K F −→ 0,

where IZ is the ideal sheaf of Z. Points in the support of Z are called singular points of the foliation. Points outside Z are called regular, or smooth, points for the foliation. c) Consequently we have KY = N ⊗ K F . d) Let M be a Riemann surface (not necessarily compact nor algebraic). A morphism ι : M → Y is said to be a leaf of the foliation if: d.1) There is a discrete set of points P ⊂ M such that ι| : M \ P → Y is a M \P local embedding; d.2) the natural map ι∗ (N ) → ι∗ (Ω1Y ) → Ω1M is the zero map. e) If z ∈ Y is a regular point for the foliation, then there is a unique leaf of the foliation passing through z. f) Denote by ∆ the one dimensional unit disk. If z ∈ Y is a regular point for the foliation, then there is an analytic neighborhood z ∈ U ⊂ Y isomorphic to ∆ × ∆

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MCQUILLAN’S APPROACH TO THE GREEN-GRIFFITHS CONJECTURE FOR SURFACES 139

with coordinates (z1 , z2 ) and the restriction of the exact sequence 2.4 to U is the exact sequence (2.5)

0 −→ O U dz1 −→ O U dz1 ⊕ O U dz2 −→ O U dz2 −→ 0.

Consequently the leaves of the foliation passing through U are given by the equations z1 = c (c ∈ ∆). Point (f) above explains a bit the geometry of a foliation on the open set of regular points: we can cover the regular locus of the foliation by open sets such that the restriction of the foliation to each of them is just a product. On the other side, near the singular locus of the foliation, the structure of the leaves may be much more complicated. g) Suppose that Y1 is a smooth projective surface and p : Y1 → Y is a dominant morphism. We have an inclusion p∗ (N ) → p∗ (Ω1Y ) → Ω1Y1 . The saturation N1 of N inside Ω1Y1 is then a foliation on Y1 . It will be called the pull back of the foliation F to Y1 via p and denoted by p∗ ( F ). One should be aware that, in general, the conormal sheaf of p∗ ( F ) is not p∗ (N ) and, in general, Kp∗ ( F ) 6= p∗ (K F ). h) A leaf M of the foliation is said to be algebraic if the Zariski closure of its image in Y is an algebraic curve. i) Suppose that S is a smooth algebraic curve and f : Y → S is a dominant morphism. The natural exact sequence (2.6)

0 −→ f ∗ (Ω1S ) −→ Ω1Y −→ Ω1Y /S −→ 0

induces a foliation F f on Y . The leaves of this foliation are the fibers of f thus they are all compact. Observe that in general f ∗ (Ω1S ) is not the conormal sheaf of F f . The conormal sheaf of F f will be the saturation of f ∗ (Ω1S ) in Ω1Y . j) A foliation F is said to be a fibration if there is a a birational morphism p : Y1 → Y and a dominant morphism f : Y1 → S where S is a smooth projective curve such that p∗ ( F ) = F f . All the leaves of a fibration are algebraic. The following theorem characterizes foliations with “many” algebraic leaves, cf. [4]: Theorem 2.7 (Jouanoulou). – Let F be a foliation on a smooth projective surface. Then F has infinitely many algebraic leaves if and only if it is a fibration (and thus all the leaves are algebraic). For a detailed reference on foliations on algebraic surfaces, cf. [2]. Let’s prove Lemma 2.5: Proof. – Over P(Ω1X ) we have the tautological exact sequence (2.7)

0 −→ Ω1P(Ω1

X )/X

(1) −→ p∗ (Ω1X ) −→ O (1) −→ 0.

Observe that if f : C → X is a morphism and f 0 : C → P(Ω1X ) is the natural lift, then the natural map f 0∗ (Ω1P(Ω1 )/X (1)) → f ∗ (Ω1X ) → Ω1C is the zero map. X

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Let B1 be a desingularization of B. denote by p : B1 → B and by h : B1 → X the natural projections. By construction we have an inclusion p ∗ (Ω1P(Ω1 /X (1))| ) → X

B

h∗ (Ω1X ) → Ω1B1 . Let N ⊂ Ω1B1 be the saturation of p ∗ (Ω1P(Ω1 /X (1))| ). Taking a B X e of B1 if necessary, we may suppose that N is locally free of rank one and blow up B e thus it defines a foliation F on B. Suppose now that f 0 (C) intersects the smooth locus of B. Thus we can lift f 0 to e By construction, and the observation above, the natural a morphism f1 : C → B. ∗ 1 map f1 (N ) → ΩC is the zero map. Thus f1 (C) is a leaf of the foliation F . Since there are only finitely many curves contained in the singular locus of B the conclusion of the Lemma follows. Now the conclusion of Theorem 2.3 easily follows from Jouanoulou Theorem 2.7: If f (C) factors through B then either it belongs to a finite list (thus his degree and genus may be absorbed by the constant A2 ) or it is leaf of an algebraic foliation F e over B. If the foliation is a fibration then all the leaves (up to finitely many) are algebraically equivalent. in particular they will be of fixed genus and degree. Thus an inequality as the one proposed by the theorem holds, up to rise the constant A1 if necessary If the foliation is not a fibration then there will be only finitely many algebraic leaves and their degree and genus may be absorbed in the constant A2 .

3. Entire curves on surfaces with big cotangent bundle In this section, the core of the chapter, we will se how a strategy which is essentially similar to the proof of Bogomolov theorem, may prove Green-Griffiths Conjecture 1.3 for surfaces with big cotangent bundle. The proof, even if it follows the same philosophy, is much more involved and requires stronger technical tools. In the meanwhile we will see that the proof will also provide a very easy proof of Theorem 2.2. The first thing we remark in the proof of Theorem 2.3 is that there is a extended use of intersection theory. Since we deal with analytic maps of C inside a variety we cannot use it but Nevanlinna theory can provide a substitute of it. 3.1. Review of Nevanlinna Theory. – Let X be a smooth projective variety and ω a positive (1, 1) form on it. Let f : C → X an analytic map. For every positive real number r we define the characteristic function Z r Z dt (3.1) Tf (r) := f ∗ (ω) t 0 |z| Cγ · [γ ∆

trans

].

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Benoît Claudon, Université de Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France • E-mail : [email protected] Url : https://perso.univ-rennes1.fr/benoit.claudon Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany and Freiburg Institute for Advanced Studies (FRIAS), Freiburg im Breisgau, Germany and University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France • E-mail : [email protected] Url : https://cplx.vm.uni-freiburg.de Behrouz Taji, School of Mathematics and Statistics F07, The University of Sydney, NSW 2006 Australia • E-mail : [email protected] Url : https://www.maths.usyd.edu.au/u/behrouzt

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Panoramas & Synthèses 56, 2021, p. 209–252

KOBAYASHI HYPERBOLICITY, NEGATIVITY OF THE CURVATURE AND POSITIVITY OF THE CANONICAL BUNDLE by Simone Diverio

Abstract. – We give a detailed account of a recent breakthrough by Wu and Yau, generalized shortly afterwards by Tosatti and Yang (and also by Diverio and Trapani). The breakthrough sits at the crossroad of complex differential geometry and Kobayashi hyperbolicity. More specifically, an old conjecture of Kobayashi, stated at the very beginning of the theory, predicts that a complex projective (or more generally compact Kähler) Kobayashi hyperbolic manifold should have ample canonical bundle. On the one hand it is also known since the beginning of the theory that a compact complex manifold with a Hermitian metric whose holomorphic sectional curvature is negative is Kobayashi hyperbolic. On the other hand a compact Kähler manifold with ample canonical bundle is known—by the celebrated work of Aubin and Yau—to admit a Kähler metric with (constant) negative Ricci curvature. Wu and Yau’s theorem states that if a smooth projective manifold admits a Kähler metric with negative holomorphic sectional curvature, then it also admits a possibly different Kähler metric whose Ricci curvature is negative. The result can be therefore seen as a weak confirmation of Kobayashi’s conjecture above, since it gives the same conclusion but with the stronger hypothesis about the holomorphic sectional curvature. Beside a careful, fully detailed presentation of the proof of the Wu-Yau theorem, we take the opportunity to give some basic background material on complex differential geometry and several results, positive and negative, about the link between curvature and Kobayashi hyperbolicity. Some natural open questions are also discussed. The proof of the Wu-Yau theorem presented here closely follows the original main ideas by Wu and Yau, but the conclusion of the proof is simplified somewhat by using the pluripotential approach of Diverio and Trapani.

1. Introduction Let X be a compact complex manifold. An entire curve traced in X is by definition a non constant holomorphic map f : C → X. By Brody’s criterion X is Kobayashi hyperbolic if and only if X does not admit any entire curve.

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At the very beginning of the theory, in the early 70s, very few examples of (higher dimensional) compact complex manifolds where known: mainly compact quotients of bounded domains in Cn . Such quotients admits the Bergman metric, whose class lies by construction in the opposite of the first Chern class of X. Thus, for them, the canonical bundle is positive and therefore ample. In particular such quotients are projective. We can guess that this lack of knowledge of examples on the one hand, and the positivity property of the canonical bundle of the known hyperbolic compact complex manifolds on the other hand, led S. Kobayashi to conjecture the following. Conjecture (Kobayashi ’70). – Let X be a compact Kähler (or projective) manifold which is Kobayashi hyperbolic. Then, KX is ample. In the same vein, Kobayashi also asked in [23] whether a compact hyperbolic complex manifold has always infinite fundamental group. While the answer to this latter question is nowadays known to be negative (we know plenty of examples of simply connected compact complex manifold, mainly given by smooth projective general complete intersections of high degree), the former question is still widely open (and believed to be true). Observe that, at least in the projective case, we now know since Mori’s breakthrough [29] that being hyperbolic implies the nefness of the canonical bundle, due to the absence of rational curves. Thus, the canonical class is at least in the closure of the ample cone. What we want is then to show that hyperbolicity pushes the canonical class a little bit further into the ample cone. Beside the class of smooth compact quotients of bounded domain, another remarkable class of compact hyperbolic manifolds—known since the beginning of the theory—is given by compact Hermitian manifolds whose holomorphic sectional curvature is negative. Even if this is for sure an important class where to test conjectures on hyperbolic manifolds, it is somehow surprising that until very recently the Kobayashi conjecture was not known even for this class. The principal aim of this chapter is to present in full detail a proof of the following statement due to Wu and Yau, which settles Kobayashi’s conjecture for negatively curved hyperbolic projective manifolds. Theorem 1.1 ([37]). – Let X be a smooth projective manifold, and suppose that X carries a Kähler metric ω whose holomorphic sectional curvature is everywhere negative. Then, X posseses a (possibly different) Kähler metric ω 0 whose Ricci curvature is everywhere negative. In particular, KX is ample. Observe that, since the holomorphic sectional curvature decreases when the metric is restricted to smooth submanifolds, as a direct consequence one obtains that every smooth submanifold of a compact Kähler manifold with negative holomorphic sectional curvature has ample canonical bundle. This observation goes in the direction of the celebrated Lang conjecture which predicts the following.

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Conjecture 1.2 ([26]). – Let X be a smooth projective complex manifold. Then, X is Kobayashi hyperbolic if and only if X as well as all of its subvarieties are of general type.

Thus, since a projective manifold with ample canonical bundle is of general type, Wu and Yau’s theorem is also a confirmation in the negatively curved case, as long as only smooth subvarieties are concerned, of (one direction of) Lang’s conjecture. It is therefore of primordial importance to extend their result in the singular case. The exact statement one should try to prove is the following. Let X be an irreducible projective variety and suppose to be able to embed X into a projective manifold Y supporting a Kähler metric whose holomorphic sectional curvature is negative, at ˜ → X, with least locally around X. Thus, there should exists a modification (1) µ : X ˜ X smooth, such that the canonical bundle KX˜ is big. Addendum. – The Wu-Yau-Tosatti-Yang theorem (as well as its generalization by Diverio and Trapani) has been very recently proved by H. Guenancia in [16] also for singular subvarieties as mentioned here above. His very general result stems from a highly non trivial generalization of ideas explained in this chapter, involving also deep results from the Minimal Model Program. Its most general version can be stated as follows. Theorem 1.3 (Guenancia [16, Theorem B]). – Let P (X, D) be a pair consisting of a projective manifold X and a reduced divisor D = i∈I Di with simple normal crossings. Let ω be a Kähler metric on X ◦ := X \ D such that there exists κ0 > 0 satisfying ∀(x, v) ∈ X ◦ × TX,x \ {0},

HSCω (x, [v]) < −κ0 .

Then, the pair (X, D) is of log general type, that is, KX + D is big. If additionally ω is assumed to be bounded near D, then KX is itself big. This theorem thus provides a full confirmation of Lang’s conjecture for compact Kähler manifolds with negative holomorphic sectional curvature. Finally, Lang’s conjecture has been settled very recently also in the particular case of compact free quotients of bounded domains (and in a slightly more general context, indeed) in [3].

˜ → X is a proper surjective holomorphic map, such that there By definition, a modification µ : X ˜ \ µ−1 (S) → X \ S is a exists a proper analytic subset S ⊂ X with the property that µ| ˜ −1 :X (1)

X\µ

(S)

biholomorphism.

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1.1. Organization of the chapter. – Beside the introduction, this chapter is made up of five sections. Section 1 is devoted to build the proper background in complex differential geometry in order to go into the proof of Wu-Yau’s theorem, as well as the basic notions of complex hyperbolicity. In particular we summarize the different kinds of curvature in the Riemannian setting as well as in the Hermitian setting, putting in evidence the relations of between these notions in the Kähler case. Moreover, we explain if and how the sign of a particular notion of curvature propagates to others. We also take the opportunity to recall how the negativity of the holomorphic sectional curvature gives the Kobayashi hyperbolicity of a compact Hermitian manifold. Section 2 has a birational geometric flavor, and we try to motivate in this framework Kobayashi’s conjecture as well as Wu-Yau’s theorem and its generalizations using standard tools and conjectures. Namely, assuming the abundance conjecture and using the Iitaka fibration, we try to make clear how compact projective manifolds with trivial real first Chern class enter naturally into the picture and how to rule them out using the negativity (or even the quasi-negativity) of the holomorphic sectional curvature. In Section 3 we present in full details an algebraic criterion due to J.-P. Demailly which a compact Hermitian manifold must satisfy in order to have negative holomorphic sectional curvature. As a consequence, we construct (still following Demailly) an example of smooth projective surface which is hyperbolic, has ample canonical bundle, but nevertheless does not admit any negatively curved Hermitian metric. This shows that Wu-Yau’s theorem is unfortunately a confirmation of Kobayashi’s conjecture only in a particular (although important) case. Section 4 is the heart of the chapter and we present therein a complete, detailed proof of Wu-Yau’s theorem. The proof is divided into several steps, in order to make the strategy more insightful. We tried to really work out every computation and estimate, perhaps even paying the price to be slightly redundant, to keep the chapter fully self-contained. Finally, in Section 5 we present a couple of generalizations of Wu-Yau’s theorem, namely the Kähler case due to V. Tosatti and X. Yang, and the (from a certain point of view, sharp) quasi-negative case due to S. Trapani and the author. Acknowledgements. – The author wishes to express his gratitude to the anonymous referee for having read with great attention the present chapter, and for the uncountable valuables suggestions which really improved the exposition.

2. Complex differential geometric background and hyperbolicity The material in this section is somehow standard, but we take the opportunity here to fix notations and explain some remarkable facts which are not necessarily in everybody’s background. We refer to [8, 21, 40] for an excellent and more systematic treatment of the subject.

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Let X be a complex manifold of complex dimension n, and let h be a Hermitian metric on its tangent space TX , which is considered as a complex vector bundle endowed with the standard complex structure J inherited from the holomorphic coordinates on X. Then, the real part g of h = g − iω defines a Riemannian metric on the underlying real manifold, while its imaginary part ω defines a 2-form on X. Now, one can consider both the Riemannian or the Hermitian theory on X. On R the one hand we have the existence of a unique connection ∇ on TX —the Levi-Civita connection—which is both compatible with the metric g and without torsion. Here the superscript R is put on TX to emphasize that we are looking at the real underlying manifold. We call the square of this connection R = ∇2 , the Riemannian curvature R R of (TX , g). It is a 2-form with values in the endomorphisms of TX . On the other hand, we can complexify TX and decompose it as a direct sum of the eigenbundles for the complexified complex structure J ⊗ IdC relatives to the eigenvalues ±i: 1,0 0,1 C TX = TX ⊗ C ' TX ⊕ TX . We have a natural vector bundle isomorphism 1,0 R ξ : TX → TX 1 v 7→ (v − iJv), 2 which is moreover C-linear: ξ ◦ J = iξ. There is a natural way to define a Hermitian C , as follows. We first consider the C-bilinear extension g C of g, and then metric on TX ˜ made up using complex conjugation in T C : its sesquilinear form h X

˜ •) := g C (•, ¯•). h(•, Such a Hermitian metric realizes the direct sum decomposition above as an orthogonal decomposition. The complexification of ω, which we still call ω by an abuse of notation, is then a real positive (1, 1)-form. These three notions, namely a Hermitian metric 1,0 on TX , a Hermitian metric on TX , and a real positive (1, 1)-form are essentially the same, since there is a canonical way to pass from one to the other. 1,0 Now, we know that there exists a unique connection D on TX which is both ˜ compatible with h and the complex structure: the Chern connection. We call the 1,0 ˜ , h). It is a (1, 1)-form square of this connection Θ = D2 , the Chern curvature of (TX 1,0 ˜ with values in the anti-Hermitian endomorphisms of (TX , h).

A basic question is then: can we compare these two theories via ξ? The answer is classical and surprisingly simple. The Riemannian theory and the Hermitian one are the same if and only if the metric h is Kähler, i.e., if and only if dω = 0. In other words, the metric is Kähler if and only if D = ξ ◦ ∇ ◦ ξ −1 , and of course, in this case, Θ = ξ ◦ R ◦ ξ −1 (see e.g., [21, §4.A]).

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2.1. Notions of curvature in Riemannian and Hermitian geometry and their correlation in the Kähler case. – We now give a brief overview of the different notions of curvature in the setting respectively of Riemannian geometry and Hermitian geometry. Then, we shall compare them in the case of Kähler metrics, with particular attention to the “propagation” of signs. 2.1.1. The Riemannian case. – Let (M, g) be a Riemannian manifold, ∇ its LeviCivita connection and R its Riemannian curvature. To this data it is attached the classical notion of sectional curvature K g of g. It is a function which assigns to each 2-plane π = Span(v, w) in TM the real number K g (π)

=−

hR(v, w) · v, wig . ||v||2g ||w||2g − |hv, wig |2

One can verify that this function completely determines the Riemannian curvature tensor. One usually also considers other “easier” tensors obtained by performing some type of contractions on R, for instance the Ricci curvature rg and the scalar curvature sg . The former is a symmetric 2-tensor defined by
rg (u, v) = trTM w 7→ R(w, u) · v . The latter is the real function on M obtained by taking the trace of the Ricci curvature with respect to g: sg = trg rg . One can straightforwardly show that, up to a positive factor which depends only on the dimension of M , the Ricci curvature can be obtained as an average of sectional curvatures and the scalar curvature as an average of Ricci curvatures. In particular the sign of the sectional curvature “dominates” the sign of the Ricci curvature which, in turn, “dominates” the sign of the scalar curvature. 2.1.2. The Hermitian case. – We now look at the complex case. We start more generally with the notion of Griffiths curvature for a holomorphic Hermitian vector bundle (E, h) over a complex manifold X. In this situation, we also have a unique connection both compatible with the metric and the holomorphic structure on E, whose curvature we still denote by Θ. It is a (1, 1)-form with values in the anti-Hermitian endomorphisms of (E, h). The 1,0 Griffiths curvature assigns to each pair (v, ζ) ∈ TX,x × Ex , x ∈ X, the real number given by θE,h (v, ζ) = hΘ(v, v¯) · ζ, ζih . It has the remarkable property (which is a special case of what is called more generally Griffiths formulae) that it decreases when passing to holomorphic subbundles. Suppose that S ⊆ E is a holomorphic subbundle of E and endow it with the restriction 1,0 metric h| . Then, given x ∈ X, ζ ∈ Sx ⊆ Ex , and v ∈ TX,x , we always have S θS,h| (v, ζ) ≤ θE,h (v, ζ). S

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1,0 Now, we look more closely at the special case where E = TX , and the Hermitian met˜ ric is h as above. In this case, the Griffiths curvature is nothing but (up to normalization) what is classically called holomorphic bisectional curvature. To be more precise, 1,0 given a point x ∈ X and two non-zero holomorphic tangent vectors v, w ∈ TX,x \ {0} we define the holomorphic bisectional curvature in the directions given by v, w as

HBCh (x, [v], [w]) =

θT 1,0 ,h˜ (v, w) X

||v||2h˜ ||w||2h˜

.

˜ and ω in By a slight abuse of notation, we may possibly confuse and interchange h, h what follows. The holomorphic sectional curvature is defined to be the restriction of the holomorphic bisectional curvature to the diagonal: HSCh (x, [v]) = HBCh (x, [v], [v]) =

θT 1,0 ,h˜ (v, v) X

||v||4h˜

.

In the spirit of the Riemannian case, we can construct a closed real (1, 1)-form, the Chern-Ricci form, by taking the trace with respect to the endomorphism part of the Chern curvature. We also normalize it in such a way that its cohomology class coincides with the first Chern class of the manifold, namely: i Ricω = tr 1,0 Θ. 2π TX The new feature here is that the Chern-Ricci tensor is a 2-form always belonging to a fixed cohomology class, the first Chern class of X, independently of the choice of the metric. This is because, in general, the trace of the curvature of a vector bundle is the curvature of the induced connection on the determinant bundle, which in this case is the dual of the canonical bundle of X. By taking again the trace, but this time with respect to ω, we get what is called the Chern scalar curvature. Thus, it is by definition the unique real function scalω : X → R such that Ricω ∧

ω n−1 ωn = scalω . (n − 1)! n!

2.1.3. Negativity of the holomorphic sectional curvature and hyperbolicity. – Before going further and explore—when the metric is Kähler—the links among the different notions of curvature we introduced in the Riemannian and Hermitian setting, we would like to explain here how the negativity of the holomorphic sectional curvature implies the Kobayashi hyperbolicity of the manifold. We want to do it here before entering the Kähler world, since this is a purely Hermitian fact. Before proceeding further, let us recall the very basic definitions and notions about Kobayashi hyperbolicity (for an exhaustive treatment we refer to [25, 24]). Let X be a complex space. We call a holomorphic disk in X a holomorphic map from the complex unit disk ∆ to X. Given two points p, q ∈ X, consider a chain of holomorphic disks from p to q, that is a chain of points p = p0 , p1 , . . . , pk = q of X,

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pairs of point a1 , b1 , . . . , ak , bk of ∆ and holomorphic maps f1 , . . . , fk : ∆ → X such that fi (ai ) = pi−1 , fi (bi ) = pi , i = 1, . . . , k. Denoting this chain by α, define its length `(α) by `(α) = ρ(a1 , b1 ) + · · · + ρ(ak , bk ) and a pseudodistance dX on X by dX (p, q) = inf `(α). α

This is the Kobayashi pseudodistance of X. Definition 2.1. – The complex space X is said to be Kobayashi hyperbolic if the pseudodistance dX is actually a distance. For ∆ the complex unit disk, it is easy to see using the usual Schwarz-Pick lemma (2) in one direction and the identity transformation in the other that dX = ρ. Then ∆ is hyperbolic. The entire complex plane is not hyperbolic: indeed the Kobayashi pseudodistance is identically zero. To see this from the very definition, take any two point z1 , z2 ∈ C and consider a sequence of holomorphic disks fj : ∆ → C ζ 7→ z1 + jζ(z2 − z1 ). It is important to remark here that the non hyperbolicity of the complex plane is connected to the possibility of taking larger and larger disks in C. It is immediate to check that the Kobayashi pseudodistance has the fundamental property of being contracted by holomorphic maps: given two complex spaces X and Y and a holomorphic map f : X → Y one has for every pair of point x, y in X dY (f (x), f (y)) ≤ dX (x, y). In particular, a Kobayashi hyperbolic complex space cannot contain any complex subspace which is not Kobayashi hyperbolic. Remark 2.2. – In other words, begin Kobayashi hyperbolic is an hereditary property for complex subspaces, analogously to what happens for the negativity of holomorphic sectional curvature, as prescribed by Griffiths’ formulae. The distance decreasing property together with the fact that the Kobayashi pseudodistance is identically zero on C, implies immediately. Proposition 2.3. – If X is a hyperbolic complex space, then every holomorphic map f : C → X is constant. (2)

We recall here that the usual Schwarz-Pick lemma says that for f : ∆ → ∆ a holomorphic map, one has the following inequality: |f 0 (ζ)| 1 ≤ . 1 − |f (ζ)|2 1 − |ζ|2 This means exactly that holomorphic maps contract the Poincar metric.

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Let us now come at the infinitesimal analogue of the Kobayashi pseudodistance introduced above. For simplicity, we shall suppose that X is a smooth complex manifold but most of the things would work on an arbitrary singular complex space. So, fix an arbitrary holomorphic tangent vector v ∈ TX,x0 , x0 ∈ X: we want to give it an intrinsic length. Thus, define kX (v) = inf{λ > 0 | ∃f : ∆ → X, f (0) = x0 , λf 0 (0) = v}, where f : ∆ → X is holomorphic. Even with this infinitesimal form, it is straightforward to check that holomorphic maps between complex manifolds contract it and that in the case of the complex unit disk, it agrees with the Poincar metric. Definition 2.4. – Let X be a complex manifold and ω an arbitrary Hermitian metric on X. We say that X is infinitesimally Kobayashi hyperbolic if kX is positive definite on each fiber and satisfies a uniform lower bound kX (v) ≥ ε||v||

ω

when v ∈ TX,x and x ∈ X describes a compact subset of X. The Kobayashi pseudodistance is the integrated form of the corresponding infinitesimal pseudometric (this is due to Royden). Theorem 2.5 ([24, (3.5.31) Theorem]). – Let X be a complex manifold. Then Z dX (p, q) = inf kX (γ 0 (t)) dt, γ

γ

where the infimum is taken over all piecewise smooth curves joining p to q. In particular, if X is infinitesimally hyperbolic, then it is hyperbolic. Next theorem, which is due to Brody, is the simplest and most useful criterion for hyperbolicity. It gives a converse of Proposition 2.3 in the case where the target X is compact. Fix any Hermitian metric ω on the compact complex manifold X; we say that a holomorphic map f : C → X is an entire curve if it is non constant and that it is a Brody curve if it is an entire curve with bounded derivative with respect to ω (or, of course, any other Hermitian metric). Theorem 2.6 ([4], see [24, (3.6.3) Theorem] for a more modern account) Let X be a compact complex manifold. If X is not (infinitesimally) hyperbolic then there exists a Brody curve in X. A first direct consequence of this theorem is that in the compact case, hyperbolicity and infinitesimal hyperbolicity are equivalent, since if X is not infinitesimally hyperbolic then there exists an entire curve in X and then two distinct points on this curve will have zero distance with respect to dX . For more information on the localization of such a curve, we refer the reader to the remarkable results of [14], which are also described in J. Duval’s chapter of the present monograph.

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The absence of entire holomorphic curves in a given complex manifold is often referred to as Brody hyperbolicity. Thus, in the compact case, Brody hyperbolicity and Kobayashi hyperbolicity do coincide. We now come back to the fact that negativity of the holomorphic sectional curvature implies the Kobayashi hyperbolicity of the manifold. For our purposes, it is sufficient to deal with the smooth compact case even if more general statements can be established. Theorem 2.7 ([25, Theorem 4.1]). – Let (X, ω) be a compact Hermitian manifold such that we have HSCω < 0. Then, X is Kobayashi hyperbolic. In the next section we will see that the negativity of the holomorphic sectional curvature is a sufficient but not necessary condition for Kobayashi hyperbolicity. The following proof is somehow slightly different (more formally than substantially) from the several others that can be found in literature. Let us highlight in particular that Brody’s theorem is not really needed for the proof and can be for instance replaced by the Ahlfors-Schwarz lemma. Proof. – By Theorem 2.6, it is sufficient to show that every holomorphic map f : C → X whose derivative is ω-bounded is constant. So let f be such a map and consider the function F : C → R ∪ {−∞} t 7→ log ||f 0 (t)||2ω , which is clearly upper semi-continuous and bounded from above. Suppose by contradiction that F is not identically −∞, which corresponds to the fact that f is not constant. Then, of course, the locus where log ||f 0 (t)||2ω is −∞ is a discrete set. We now check that log ||f 0 (t)||2ω is a subharmonic function on the whole C, which is moreover strictly subharmonic over {f 0 6= 0}. Since any bounded subharmonic function on C is constant [22, Proposition 2.7.3], this gives a contradiction, because a constant function cannot be strictly subharmonic somewhere. First of all we show the subharmonicity of log ||f 0 ||2ω , by showing that for all positive real numbers ε the smooth functions defined on the whole complex plane ¯ ε ≥ 0. For, since ψε = log(||f 0 ||2ω + ε) are subharmonic, i.e., i∂ ∂ψ log ||f 0 ||2ω = lim ψε ε→0

pointwise, and the sequence {ψε } is decreasing, then the subharmonicity of log ||f 0 ||2ω follows from [22, Theorem 2.6.1, (ii)]. So, fix ε > 0 and consider a point t0 ∈ C. Call x0 = f (t0 ) ∈ X and choose holomorphic coordinates (z1 , . . . , zn ) for X centered at x0 so that x0 corresponds to z = 0, and write f = (f1 , . . . , fn ) for f in these coordinates. Moreover, chose a

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normal coordinate frame {e1 , . . . , en } for (TX , ω) at x0 [8, (12.10) Proposition]. With this choice we have that n X

hel (z), em (z)iω = δlm −

cjklm zj z¯k + O(|z|3 ),

j,k=1

where the cjklm ’s are the coefficients of the Chern curvature of ω. Observe that, since the metric is not supposed to be Kähler, we can a priori not chose holomorphic coordinates around x0 such that the el ’s can be taken simply to be ∂/∂zl ; nevertheless, by a constant change of coordinates, we can suppose that el (x0 ) equals ∂/∂zl (x0 ), at least at x0 . Now, of course there exist holomorphic functions ϕj , j = 1, . . . , n, defined on a neighborhood of t0 such that n X

f 0 (t) =

ϕj (t) ej (t),

j=1

so that around t0 we have ||f 0 (t)||2ω = |ϕ(t)|2 −

n X

cjklm fj (t)fk (t)ϕl (t)ϕm (t) + O(|f |3 ).

j,k,l,m=1

Moreover, we have fj0 (t0 ) = ϕj (t0 ) for all j, since the el ’s and the ∂/∂zl ’s agree at t0 . Remark that, since X is compact, there exists a positive constant κ such that HSCω < −κ. This condition reads in our coordinates n X

cjklm vj v¯k vl v¯m < −κ|v|4 ,

∀v = (v1 , . . . , vn ) ∈ Cn .

j,k,l,m=1

¯ 0 ||2 and ∂ ∂||f ¯ 0 ||2 at t0 , since Now, we have to compute ∂||f 0 ||2ω , ∂||f ω ω ¯ ε= ∂ ∂ψ

−1 1 ¯ 0 ||2 + ¯ 0 ||2 . ∂||f 0 ||2ω ∧ ∂||f ∂ ∂||f ω ω 2 0 + ε) ||f ||2ω + ε

(||f 0 ||2ω

We find ∂||f 0 ||2ω |

= hϕ0 (t0 ), f 0 (t0 )i dt,

¯ 0 ||2 | ∂||f ω

= hf 0 (t0 ), ϕ0 (t0 )i dt¯,  n X = i |ϕ0 (t0 )|2 −

t0

t0

¯ 0 ||2 | i∂ ∂||f ω

t0

 0 (t ) dt ∧ dt¯ cjklm fj0 (t0 )fk0 (t0 )fl0 (t0 )fm 0

j,k,l,m=1 0


> i |ϕ (t0 )| + κ|f 0 (t0 )|4 dt ∧ dt¯, 2

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where the brackets just mean the standard Hermitian product in Cn . Putting all this together we obtain   0 0 2 0 2 0 4 ¯ ε | > i −|hf (t0 ), ϕ (t0 )i| + |ϕ (t0 )| + κ|f (t0 )| dt ∧ dt¯ i∂ ∂ψ t0 (|f 0 (t0 )|2 + ε)2 |f 0 (t0 )|2 + ε   |ϕ0 (t0 )|2 + κ|f 0 (t0 )|4 −|f 0 (t0 )|2 |ϕ0 (t0 )|2 + dt ∧ dt¯ ≥i (|f 0 (t0 )|2 + ε)2 |f 0 (t0 )|2 + ε
κ|f 0 (t0 )|6 + ε |ϕ0 (t0 )|2 + κ|f 0 (t0 )|4 dt ∧ dt¯ ≥ 0, =i (|f 0 (t0 )|2 + ε)2 where we used Cauchy-Schwarz for the second inequality, and so ψε is subharmonic at each point. To conclude, observe that the very same computation with ε = 0 makes sense away from points where f 0 = 0, and give moreover strict positivity for i∂ ∂¯ log ||f 0 ||2ω at these points. Which means that, away from {f 0 = 0}, log ||f 0 ||2ω is strictly subharmonic, as desired. 2.1.4. The Kähler case. – Suppose now that (X, ω) is a Kähler manifold, with ω = −=h. Let (X, g = 0, otherwise some power of the canonical bundle would be a pull-back of a (ample) line bundle over point, and thus would be trivial! If κ(X) = dim X, then X would be birational to a projective variety, i.e., would be a Moishezon manifold. By Moishezon’s theorem, a compact Kähler Moishezon manifold is projective. Moreover, X contains no rational curves and is of general type, and we conclude as before that KX must be ample. Next, suppose by contradiction that 1 ≤ κ(X) ≤ dim X − 1 so that if we call F the general fiber of φ, we have that F is a smooth compact Kähler manifold of positive dimension and different from X itself. Now, on the one hand, the short exact

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sequence of the fibration shows that KF ' KX | and therefore it follows that c1 (F ) F must be zero in real cohomology. On the other hand, the classical Griffiths’ formulae for curvature of holomorphic vector bundles imply that the holomorphic sectional curvature decreases when passing to submanifolds, that is for every x ∈ F ⊂ X HSCω| (x, [v]) ≤ HSCω (x, [v]), F

where v ∈ TF,x and, in the right hand side, v is seen as a tangent vector to X. The quasi-negativity of the holomorphic sectional curvature implies, since F is a general fiber, that there exists a tangent vector to F along which the holomorphic sectional curvature of ω | is strictly negative. Thus, Proposition 3.7 implies that F F cannot have trivial first real Chern class, which is absurd. As a consequence, me may indeed hope to extend Wu-Yau-Tosatti-Yang theorem to the optimal, quasi-negative case. This is precisely the main contribution of the paper [13]. Theorem 3.8 ([13, Theorem 1.2]). – Let (X, ω) be a connected compact Kähler manifold. Suppose that the holomorphic sectional curvature of ω is quasi-negative. Then, KX is ample. In particular, X is projective. We shall spend some words on this result as well as the Kähler case in the last section. Remark 3.9. – To finish this section with, unfortunately, we must confess that we are not aware of any example of a compact Kähler manifold with a Kähler metric whose holomorphic sectional curvature is quasi-negative but which does not posses any Kähler metric with strictly negative holomorphic sectional curvature. In other word, is Theorem 3.8 a true generalization of Wu-Yau-Tosatti-Yang result? We believe so. Then, such an example, if any, would be urgently needed!

4. An example by J.-P. Demailly In this section we would like to explain, following [7, §8], how one can construct examples of compact hyperbolic projective manifolds which nevertheless do not admit any Hermitian metric of negative holomorphic sectional curvature. Such examples can be generalized to higher order analogues—namely “k-jet curvature”—of holomorphic sectional curvature: this will be mentioned at the end of the section, and related conjectures that come out from this picture will be discussed at the end of the chapter. The first observation is the following algebraic criterium for the nonexistence of a metric with negative holomorphic sectional curvature. Let X be a complex manifold, C be a compact Riemann surface, and F : C → X be a non-constant holomorphic map. Let mp ∈ N be the multiplicity at p ∈ C of F . Clearly, mp = 1 except possibly at finitely many points of C, and mp ≥ 2 if and only if F is not an immersion at p.

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Theorem 4.1 (Demailly [7, Special case of Theorem 8.1]). – Consider (X, ω) a compact Hermitian manifold and let F : C → X be a non constant holomorphic map from a compact Riemann surface C of genus g = g(C) to X. Suppose that HSCω ≤ −κ for some κ ≥ 0. Then, X κ 2g − 2 ≥ degω C + (mp − 1), 2π p∈C R where degω C = C F ∗ ω > 0 is the degree of C with respect to ω. We shall use this theorem especially in the case where C is the normalization of a singular curve in X and F the normalization map. Observe that, in particular, we recover the well-known fact that on a compact Hermitian manifold with negative holomorphic sectional curvature there are no rational nor elliptic curves (even singular), and that there are no rational (possibly singular) curves on a compact Hermitian manifold with non positive holomorphic sectional curvature. The main ingredients of the proof are two: first, as already observed, curvature decreases one passing to subbundles (even if one needs some adjustments here, since at the very beginning one merely gets a locally free subsheaf and not an actual subbundle), and second, the holomorphic sectional curvature for a Riemann surface is nothing else than the gaussian curvature so that in the compact case its total integral gives the opposite of its canonical degree, by the Gauss-Bonnet Theorem and the Hurwitz Formula. Proof. – The differential F 0 of F gives us a map F 0 : TC → F ∗ TX . This map is injective at the level of sheaves, but not necessarily at the level of vector bundle, since F 0 may vanish at some point. Taking into account these vanishing points counted with multiplicities, we obtain the following injection of vector bundles F 0 : TC ⊗ O C (D) → F ∗ TX , P where we defined the effective divisor D to be p∈C (mp − 1) p. Thus, via F 0 , we realized TC ⊗ O C (D) as a subbundle of the Hermitian vector bundle (F ∗ TX , F ∗ ω), and—as such—we can endow it with the induced metric h = F ∗ ω | (observe TC ⊗ O C (D) ∗ the we consider F ω not as a pull-back of differential forms, but as a pull-back of Hermitian metrics). Now, a local holomorphic frame for TC ⊗ O C (D) around a point p ∈ C is ∂ given by η(t) = 1/tmp −1 ∂t , where t is a holomorphic coordinate centered at p. 0 ∗ Call ξ(t) = F (η(t)) ∈ (F TX )t = TX,F (t) , so that ξ is a local holomorphic frame for TC ⊗ O C (D) when seen as a subbundle of F ∗ TX . We have, for the Griffiths curvature of (TC ⊗ O C (D), h), hΘ(TC ⊗ O C (D), h)(∂/∂t, ∂/∂ t¯) · ξ, ξih = Θ(TC ⊗ O C (D), h)(∂/∂t, ∂/∂ t¯) ||ξ||2h . | {z } =||ξ||2ω

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By the classical Griffiths’ formulae, we have the following decreasing property for the Griffiths curvatures: hΘ(TC ⊗ O C (D), h)(∂/∂t, ∂/∂ t¯) · ξ, ξih ≤ hΘ(F ∗ TX , F ∗ ω)(∂/∂t, ∂/∂ t¯) · ξ, ξiF ∗ ω = hF ∗ Θ(TX , ω)(∂/∂t, ∂/∂ t¯) · ξ, ξiF ∗ ω = hΘ(TX , ω)(F 0 (∂/∂t), F 0 (∂/∂t) ) · ξ, ξiω = |tmp −1 |2 hΘ(TX , ω)(ξ, ξ¯ ) · ξ, ξiω ≤ −κ|tmp −1 |2 ||ξ||4 , ω

where the last inequality holds since hΘ(TX , ω)(ξ, ξ¯ ) · ξ, ξiω = ||ξ||4ω HSCω (ξ). Therefore, we obtain Θ(TC ⊗ O C (D), h)(∂/∂t, ∂/∂ t¯) ≤ −κ|tmp −1 |2 ||ξ||2ω = iκ (F ∗ ω)(∂/∂t, ∂/∂ t¯), where by F ∗ ω here we mean the pull-back at the level of differential forms. Summing up, we have obtained that i Θ(TC ⊗ O C (D), h) ≤ −κ F ∗ ω, as real (1, 1)-forms. But then, Z Z i κ κ Θ(TC ⊗ O C (D), h) ≤ − F ∗ω = − degω C, 2π C 2π C 2π and Z C

X i Θ(TC ⊗ O C (D), h) = deg(TC ⊗ O C (D)) = 2 − 2g + (mp − 1), 2π p∈C

since deg(TC ) = 2 − 2g by Hurwitz’s formula. The statement follows. Following Demailly, we shall now exhibit a smooth projective surface which is Kobayashi hyperbolic, has an ample canonical bundle, but which cannot admit any Hermitian metric with negative holomorphic sectional curvature. It will be constructed as a fibration of Kobayashi hyperbolic curves onto a Kobayashi hyperbolic curve, with at least one “very” singular fiber, which will violate the above criterion. Proposition 4.2 (Cf. [7, 8.2. Theorem]). – There exists a smooth projective surface S which is hyperbolic (and hence with ample canonical bundle KS ) but does not carry any Hermitian metric with negative holomorphic sectional curvature. Moreover, given any two smooth compact hyperbolic Riemann surfaces Γ, Γ0 , such a surface can be obtained as a fibration S → Γ, with hyperbolic fibers, in which (at least) one of the fibers is singular and has Γ0 as its normalization. Proof. – Take any compact hyperbolic Riemann surface Γ0 , and let g = g(Γ0 ) ≥ 2 be its genus. Now, we modify it into a singular compact Riemann surface Γ00 of the same genus, whose normalization is Γ0 . In order to do so, consider a pair of positive relatively prime integers (a, b), with a < b, and the associated affine plane curve C in C2 given by the equation y a − xb = 0,

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which has a monomial singularity of type (a, b) at 0 ∈ C2 . Its normalization is given by C 3 t 7→ (ta , tb ) ∈ C2 . Choose integers n, m such that na + mb = 1. Then, the restriction of the rational function on C2 defined by (x, y) 7→ xn y m to C gives a holomorphic coordinate on it minus the singular point (this is actually the inverse map of the normalization map outside the singularity). In particular, the set of points (x, y) ∈ C such that 0 < |x| < 1 is biholomorphic to the punctured unit disk. Now, take a point x0 ∈ Γ0 and choose a holomorphic coordinate centered at x0 such that we can select a neighborhood of x0 whose image is the unit disk via this coordinate. Finally, remove the point x0 in order to obtain a holomorphic coordinate chart whose image is the punctured unit disk. By identifying with the punctured unit disk constructed above, we replace this neighborhood of x0 with the set of point (x, y) ∈ C such that |x| < 1, thus creating the desired singularity at x0 . Call the resulting curve Γ00 . By construction, the normalization of Γ00 is exactly Γ0 , and Γ00 has a single singular point, whose singularity type is plane and monomial of type (a, b) (for an excellent and very elementary discussion around this subject we refer the reader to [28, Chapter III, Section 2]). Next, we embed Γ00 in some large projective space, and then we project it to P2 , in such a way that the singular point is left untouched and outside it we create at most a finite number of nodes (i.e., plane monomial singularity of type (2, 2)). Call the resulting projective plane curve C0 , whose normalization is of course again Γ0 . Observe that the normalization map ν : Γ0 → C0 is an immersion outside the (single) preimage of the first singular point we created. On the other hand, at this point it has multiplicity a. In order to obtain the desired surface S, we select then a so that a − 1 > 2g − 2, i.e., a ≥ 2g. Such a surface S then does contain a curve which violates the criterium given in Theorem 4.1, and we are done. Take a (reduced) homogeneous polynomial equation P0 (z0 , z1 , z2 ) = 0 for C0 in P2 . Then, we necessarily have d = deg P0 ≥ 4, since otherwise C0 would be normalized by a rational or an elliptic curve. Next, complete P0 into a basis {P0 , P1 , . . . , PN } of the space H 0 (P2 , O (d)) of homogeneous polynomials of degree d in three variables, and consider the corresponding universal family

U

=



N X
[z0 : z1 : z2 ], [α0 : · · · : αN ] ∈ P2 × PN | αj Pj (z) = 0 ⊂ P2 × PN , j=0

of curves of degree d in P2 , together with the projection π : U → PN . Our starting curve C0 is then the fiber U[1:0:···:0] over the point [1 : 0 : · · · : 0] ∈ PN . Now, we embed the first curve Γ into PN (this is of course possible since N ≥ 3) in such a way that [1 : 0 : · · · : 0] ∈ Γ. The desired fibration S → Γ will be obtained as the pull-back

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family S = U ×P N Γ

/ U

  Γ

 / PN .

Of course, we have to select carefully the embedding of Γ into PN , so that S will be non singular, and in such a way that we have a good control of the singular fibers out of U[1:0:···:0] . In order to do so, the first observation is that—as it is well-known—the locus Z in PN which corresponds to singular curve is an algebraic hypersurface and, moreover, the locus Z 0 ⊂ Z which corresponds to curves which have not only one node in their singularity set is of codimension 2 in PN . In particular, by possibly moving Γ with a generic projective automorphism of PN leaving fixed [1 : 0 : · · · : 0], we can suppose that Γ ∩ Z 0 = {[1 : 0 : · · · : 0]}, so that all the fibers of S, except from C0 , are either smooth, or with a single node. If such an S were non singular, we would be done. Indeed, by Plcker’s formula, the smooth fibers have genus (d − 1)(d − 2)/2 ≥ 3, U[1:0:···:0] has genus g ≥ 2 by construction, and the other singular fibers have genus (d − 1)(d − 2)/2 − 1 ≥ 2, since they have only one node. Therefore, S is a fibration onto a hyperbolic Riemann surface with all hyperbolic fibers and is then hyperbolic (and hence with ample canonical bundle), with a fiber which contradicts Theorem 4.1. So we are left to checking the smoothness of S, knowing that we can possibly use again generic automorphisms of PN leaving fixed [1 : 0 : · · · : 0] to move Γ. Thus, since Γ is embedded in PN , we can think of S as included in U , and since U is smooth, Bertini’s theorem immediately implies that S can be chosen non singular outside U[1:0:···:0] . Now, what about points along U[1:0:···:0] ? Fix such a point ([z0 : z1 : z2 ], [1 : 0 : · · · : 0]) ∈ U[1:0:···:0] , and suppose, just to fix ideas, that z0 6= 0. Take the corresponding affine coordinates, say ((z, w), (a1 , . . . , aN )) around this point, set pj (z, w) = Pj (1, z, w) to be the dehomogenization of the Pj ’s, and let f1 (a), . . . , fr (a) be affine equations of the curve Γ. Then, we have
to check the rank of the following Jacobian matrix at the point (z, w), (0, . . . , 0) , the affine PN equation for U being p0 (z, w) + j=1 aj pj (z, w) = 0:  ∂p0  ∂p0 ∂z (z, w) ∂w (z, w) p1 (z, w) · · · pN (z, w)   ∂f1 ∂f1 0 0 · · · ∂a (0)   ∂a1 (0) N  . .. .. .. ..     . . . . 0

0

∂fr ∂a1 (0)

···

∂fr ∂aN

(0)

Observe that the lower right block has rank equal to N − 1 being Γ smooth. Call this block A and let v = (v1 , . . . , vn ) ∈ CN be a generator for the kernel of this block, which is thus a nonzero tangent vector to Γ at 0. In order to get rank N +2−2 = N for the entire Jacobian matrix we have only to worry about (the finitely many) singular

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∂p0 0 points of C0 = U[1:0:···:0] , since at regular points either ∂p ∂z or ∂w is non zero. If (z, w) is a singular point for U[1:0:···:0] , then the condition for S to be smooth around
⊥ this point is given by (p1 , . . . , pN )t 6∈ Im(At ) = ker A , that is N X

vj pj (z, w) 6= 0.

j=1

This can be of course achieved by possibly moving again Γ with a generic projective automorphism of PN leaving fixed [1 : 0 : · · · : 0], since only the tangent line of Γ at 0 is concerned in the required condition.

5. The Wu-Yau theorem In this section we go into the details of the proof of Wu-Yau’s theorem on the positivity of the canonical class for projective manifolds endowed with a Kähler metric of negative holomorphic sectional curvature. We will present a proof which follows, for the first part, almost verbatim the original proof of Wu and Yau. On the other hand, the conclusion will be achieved with an approach which is more pluripotential in flavor, taken from [13]. Finally, we shall discuss at the end of this section several generalizations of this result (including the Kähler case, and weaker notions of negativity). The proof is achieved in essentially three steps, after a reduction as follows. As we have seen, the negativity of the curvature (or even its non-positivity) implies the non existence of rational curves on X. Then, by Mori’s Cone Theorem, we deduce that the canonical bundle of X is nef. But then, it is sufficient to prove that c1 (KX )n > 0, which in this case means that the canonical bundle is big. Indeed, if KX is big and there are no rational curves on X one can conclude the ampleness of the canonical bundle via the following standard lemma. Lemma 5.1 (Exercise 8, page 219 of [6]). – Let X be a smooth projective variety of general type which contains no rational curves. Then, KX is ample. Here is a proof, for the sake of completeness. Proof. – Since there are no rational curves on X, Mori’s theorem implies as above that KX is nef. Since KX is big and nef, the Base Point Free theorem tells us that KX is semi-ample. If KX were not ample, then the morphism defined by (some multiple of) KX would be birational but not an isomorphism. In particular, there would exist an irreducible curve C ⊂ X contracted by this morphism. Therefore, KX · C = 0. Now, take any ample divisor H. For any ε > 0 rational and small enough, KX −εH remains big and thus some large positive multiple, say m(KX −εH),

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of KX −εH is linearly equivalent to an effective divisor D. Set ∆ = ε0 D, where ε0 > 0 is a rational number. We have: (KX + ∆) · C = ε0 D · C = ε0 m(KX − εH) · C = −εε0 m H · C < 0. Finally, if ε0 is small enough, then (X, ∆) is a klt pair. Thus, the (logarithmic version of the) Cone Theorem would give the existence of an extremal ray generated by the class of a rational curve in X, contradiction. 5.1. Description of the main steps of the proof.– Keeping in mind that what we have to show is that c1 (KX )n > 0, we illustrate now the steps of the proof. Step 1: Solving an approximate Kähler-Einstein equation. – Let ω be our fixed Kähler metric (with negative holomorphic sectional curvature, but we shall not use this hypothesis for the moment). Claim 5.2. – For each ε > 0 there exists a unique smooth function uε : X → R such that i ¯ ωε := εω − Ricω + ∂ ∂u ε 2π is a positive (1, 1)-form (hence Kähler, belonging to the cohomology class c1 (KX ) + ε[ω]) satisfying the Monge-Ampre equation ωεn = euε ω n . In particular, Ric(ωε ) = −ωε + εω, whence the terminology “approximate Kähler-Einstein”, and we have the following uniform upper bound: sup uε ≤ C, X

where the constant C depends only on ω and n = dim X. Observe finally, that in particular, Ric(ωε ) ≥ −ωε . Step 2: A laplacian estimate involving the holomorphic sectional curvature. – This step is somehow a refinement of the laplacian estimate needed in order to achieve the classical C 2 -estimates to solve the complex Monge-Ampre equation on compact Kähler manifolds. In the classical setting an upper bound for the holomorphic bisectional curvature is used. Here we shall employ a lemma due to Royden in order to use only the weaker information given by the bound on the holomorphic sectional curvature, as in the hypotheses. The crucial part of this step is the following.

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Claim 5.3. – Suppose −κ < 0 is an upper bound for the holomorphic sectional curvature of ω. Suppose moreover that ω 0 is another Kähler metric on X whose Ricci curvature is comparable with ω and ω 0 as follows: Ric(ω 0 ) ≥ −λω 0 + µω, where λ, µ are non negative constants. Define the smooth function S : X → R>0 to be the trace of ω with respect to ω 0 , i.e.,
n−1 ω0 ∧ω
n S := trω0 ω = n . ω0 Then, the following differential inequality holds:   κ(n + 1) µ (1) − ∆ω0 log S ≥ + 2π S − 2πλ. 2n n Observing that Z

∆ω0 log S ω 0


n

= 0,

X

we shall use the inequality above with ω 0 = ωε , λ = 1, and µ = 0, in the following integral form: Z Z (n + 1)κ Sε ωεn ≤ 2π ωεn , (2) 2n X X where we added the subscript ε to S in order to emphasize the dependence of S from ε. Step 3: Proof of the key inequality. – One wants to show that c1 (KX )n > 0. Since ωε = − Ric(ωε ) + εω, we have that n   X
n
n−j n j−1 ωεn = − Ric(ωε ) + ε ε − Ric(ωε ) ∧ ωj . j j=1 But then, Z

Z n   X
n
n−j n j−1 = − Ric(ωε ) + ε ε − Ric(ωε ) ∧ ωj . j X X X j=1   On the other hand, since the cohomology class − Ric(ωε ) = c1 (KX ) is independent from ε, the integrals Z
n−j − Ric(ωε ) ∧ ω j = c1 (KX )n−j · [ω]j ωεn

Z

X

are purely cohomological, so that Z Z
n n ωε = − Ric(ωε ) +O(ε), X {z } |X =c1 (KX )n

and thus

Z

n

c1 (KX ) = lim

ε→0+

ωεn .

X

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What we want is therefore to show the positivity of such a limit. Claim 5.4. – The limit

Z lim+

ε→0

ωεn

X

is strictly positive. This is what we call the key inequality. During the proof of the main result, this will be the only step where what we present here differs from Wu-Yau’s original approach. 5.2. Proof of the steps.– We now proceed with the proof of the various claims stated above. Proof of Claim 5.2. – The first observation is that, since KX is nef, for each ε > 0 the cohomology class c1 (KX ) + ε [ω] = −c1 (X) + ε [ω] is a Kähler class. This implies, ¯ thanks to the ∂ ∂-lemma, that there exists a smooth real function fε on X, unique up to an additive constant, such that i ¯ ωfε := ε ω − Ricω + ∂ ∂fε 2π is a Kähler form on X. Now we use the following theorem, in order to obtain an approximate KählerEinstein metric on X. We give here Yau’s original general statement, which is the key ingredient to get his celebrated solution of the Calabi conjecture. Theorem 5.5 (Yau [39]). – Let (X, ω0 ) be a compact Kähler manifold, and F : X × Rt → R smooth function such that ∂F/∂t ≥ 0. Suppose that there exists smooth function ψ : X → R such that Z Z eF (x,ψ(x)) ω0n = ω0n . X

X

Then, there exists a unique (up to a constant if F does not actually depend on ψ) smooth function ϕ : X → R, such that ( i ¯ > 0, ω0 + 2π ∂ ∂ϕ
i ¯ n = eF (x,ϕ(x)) ω n . ω0 + ∂ ∂ϕ 0

2π

From this statement one can derive easily both the existence of a Kähler metric in a fixed Kähler class with prescribed volume form (or, equivalently, Ricci tensor), and the existence of Kähler-Einstein metrics on compact Kähler manifold with negative (resp. zero) real first Chern class. Now, we fix ε > 0, and define a smooth real function αε on X implicitly by ωfnε = e−αε ω n . We then apply the theorem above with the following data: ω = ωfε ,

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Then, there exists a unique smooth real function vε such that  n i ¯ ωfε + = evε +αε +fε ωfnε ∂ ∂vε 2π = evε +fε ω n , and i ¯ ∂ ∂vε > 0 2π on X. Now, define uε to be the sum fε + vε , so that it holds ωε := ωfε +

ωεn = euε ω n . Thus, we get for the Ricci curvature of ωε Ricωε = −

i ¯ ∂ ∂ log ωεn 2π =Ric

z }| ω { i ¯ i ¯ i ¯ n = − ∂ ∂vε − ∂ ∂fε − ∂ ∂ log ω 2π | 2π {z2π } =ε ω−ωfε

= ε ω − ωε . In particular, Ricωε ≥ −ωε . Now, we use the maximum principle in order to obtain the desired uniform upper bound for uε . To do so, pick a point x0 ∈ X such that supX uε = uε (x0 ). Then, at this point we have that the complex Hessian of uε is negative semi-definite, i.e., ¯ ε (x0 ) ≤ 0. Thus, i ∂ ∂u i ¯
ωε (x0 ) = ε ω − Ricω + ∂ ∂uε (x0 )
2π ≤ ε ω − Ricω (x0 )
≤ ε0 ω − Ricω (x0 ), if ε0 > ε. Therefore, esupX uε = euε (x0 )
i ¯ ε n (x0 ) ∂ ∂u ε ω − Ricω + 2π = ω n (x0 )
n ε0 ω − Ricω (x0 ) ≤ =: eC , ω n (x0 ) so that sup uε ≤ C,

∀ε < ε0 .

X

This complete the proof of Claim 5.2.

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Proof of Claim 5.3. – Let x0 ∈ X be a fixed point. Chose holomorphic normal coordinates (z1 , . . . , zn ) with respect to ω, centered at x0 . Without loss of generality, by a constant ω-unitary change of variables, we may also suppose that ω 0 is diagonalized with respect to ω at x0 . Thus we write ω=i

n X

ωlm dzl ∧ d¯ zm ,

ωlm (z) = δlm −

l,m=1

n X

cjklm zj z¯k + O(|z 3 |),

j,k=1

where the cjklm ’s are the coefficients of the Chern curvature tensor of (X, ω) at x0 , and n X 0 0 ω0 = i ωlm dzl ∧ d¯ zm , ωlm (z) = λl δlm + O(|z|), l,m=1

where the λj ’s are the eigenvalues at x0 of ω 0 with respect to ω. In particular, λj > 0, j = 1, . . . , n. Next, call ρ0jk the coefficients of the Ricci curvature of ω 0 , so that Ricω0 =

n i X 0 ρjk dzj ∧ d¯ zk , 2π j,k=1

where ρ0jk =

n X

c0jkll .

l=1

With these notations, the starting point is the following Lemma 5.6 (See [38, pag. 371]). – The following differential equality holds: 0 n n n ∂ω /∂zj 2 X X X ρ0ll cjjll al (3) − ∆ω0 S(x0 ) = + − , (λl )2 λj (λl )2 λa λj λl l=1

j,l,a=1

j,l=1

where the right hand side is intended to be computed at x0 . Proof. – A straightforward computation, using the adjugate matrix method to obtain the inverse, shows that n X S= Ω0ml ωlm , l,m=1

(Ω0lm )

0 where we define to be the inverse matrix of (ωlm ). We want to compute ∆ω0 S at x0 . We have, by the basic commutation relations in Kähler geometry,

¯ = −iΛω0 ∂ ∂S, ¯ ∆ω0 S = ∂¯∗ ∂S and thus, since acting with Λω0 on real (1, 1)-forms amounts to takeing the trace with respect to ω 0 , n X ∂2S ¯ =− ∆ω0 S = − trω0 i∂ ∂S Ω0kj . ∂zj ∂ z¯k j,k=1

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Now, n X ∂2 ∂2S = Ω0ml ωlm ∂zj ∂ z¯k ∂zj ∂ z¯k l,m=1

=

n X

ωlm

l,m=1

∂ 2 Ω0ml ∂ 2 ωlm ∂Ω0ml ∂ωlm ∂ωlm ∂Ω0ml . + Ω0ml + + ∂zj ∂ z¯k ∂zj ∂ z¯k ∂zj ∂ z¯k ∂zj ∂ z¯k {z } | =O(|z|)

At the end of the day, thanks to the choice of geodesic coordinates, the terms with only one derivative involved are O(|z|)’s and will disappear. Therefore, we only have to understand the summands with two derivatives of Ω0ml , and express them in terms 0 0 ’s. In order to do this, call H = (ωlm ), so that H −1 = (Ω0lm ) and observe of the ωlm that (4)

0 ≡ ∂(HH −1 ) = ∂HH −1 + H∂H −1 ,

(5)

−1 −1 ¯ ¯ ¯ −1 , 0 ≡ ∂(HH ) = ∂HH + H ∂H

and −1 −1 ¯ ¯ ¯ ∧ ∂H −1 + ∂H ∧ ∂H ¯ −1 + H∂ ∂H ¯ −1 . 0 ≡ ∂ ∂(HH ) = ∂ ∂HH − ∂H

We obtain therefore the matrix identity −1 −1 ¯ −1 = −H −1 ∂ ∂HH ¯ ¯ ∧ H −1 ∂HH −1 + H −1 ∂H ∧ H −1 ∂HH ¯ ∂ ∂H − H −1 ∂H ,

which gives us the following expression for the second derivatives of Ω0ml : n 0 X ∂ 2 ωab ∂ 2 Ω0ml =− Ω0ma Ω0 ∂zj ∂ z¯k ∂zj ∂ z¯k bl a,b=1

+

n X a,b,p,q=1

Ω0mp

0 0 ∂ωpq ∂ωpq ∂ω 0 ∂ω 0 Ω0qa ab Ω0bl + Ω0ma ab Ω0bp Ω0 . ∂ z¯k ∂zj ∂zj ∂ z¯k ql

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Thus, we get the following expression for the ω 0 -Laplacian:   n X ∂ 2 ωlm ∂ 2 Ω0ml 0 0 ∆ω 0 S = − + Ωml + Ljklm Ωkj ωlm ∂zj ∂ z¯k ∂zj ∂ z¯k j,k,l,m=1

=−

n X

Ω0kj Ω0ml

∂ 2 Ω0ml ∂ 2 ωlm + Ω0kj ωlm + Ω0kj Ljklm ∂zj ∂ z¯k ∂zj ∂ z¯k

Ω0kj Ω0ml

∂ 2 ωlm + Ω0kj Ljklm ∂zj ∂ z¯k

j,k,l,m=1

=−

n X j,k,l,m=1

(6) +

n X

ωlm Ω0kj Ω0ma Ω0bl

j,k,l,m,a,b=1 n X

−

0 ∂ 2 ωab ∂zj ∂ z¯k

ωlm Ω0kj Ω0mp Ω0qa Ω0bl

0 ∂ω 0 ∂ωab pq ∂zj ∂ z¯k

ωlm Ω0kj Ω0ma Ω0bp Ω0ql

0 ∂ω 0 ∂ωab pq . ∂zj ∂ z¯k

j,k,l,m,a,b,p,q=1 n X

−

j,k,l,m,a,b,p,q=1

0 Now, still denoting by H the matrix (ωlm ), we recall the well-known formula to determine in local coordinates the Chern curvature of ω 0 , namely
¯ −1 ∂ H ¯ Θ(TX , ω 0 ) 'loc ∂¯ H

¯ H ¯ −1 ∧ ∂ H ¯ +H ¯ −1 ∂∂ ¯ = ∂¯H ¯ H, ¯ −1 ∂¯H ¯H ¯ −1 ∧ ∂ H ¯ −1 + H ¯ −1 ∂∂ ¯ = −H where the last equality is obtained by using formula (5). So, if we write in these coordinates ∗  X ∂ ∂ 0 0 ⊗ , Θ(TX , ω ) = cjklm dzj ∧ d¯ zk ⊗ ∂zl ∂ z¯m j,k,l,m

we obtain the following expression for the coefficients of the Chern curvature tensor: (7)

c0jkal = −

n X b=1

Ω0bl

n 0 0 X ∂ 2 ωab ∂ω 0 ∂ωqp + Ω0pl Ω0bq ab . ∂zj ∂ z¯k ∂zj ∂ z¯k b,p,q=1

We can now use the above identity (7) to replace in the right hand side of formula (6) the summand n 0 X ∂ 2 ωab Ω0bl ∂zj ∂ z¯k b=1

with −c0jkal +

n X b,p,q=1

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0 ∂ω 0 ∂ωab qp . ∂zj ∂ z¯k

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With this substitution, we obtain ∆ω 0 S = −

n X j,k,l,m=1

+

∂ 2 ωlm + Ω0kj Ljklm ∂zj ∂ z¯k   n 0 ∂ω 0 X qp 0 0 0 0 0 ∂ωab ωlm Ωkj Ωma −cjkal + Ωpl Ωbq ∂zj ∂ z¯k

Ω0kj Ω0ml

n X j,k,l,m,a=1

b,p,q=1

n X

−

ωlm Ω0kj Ω0mp Ω0qa Ω0bl

0 ∂ω 0 ∂ωab pq ∂zj ∂ z¯k

ωlm Ω0kj Ω0ma Ω0bp Ω0ql

0 ∂ω 0 ∂ωab pq ∂zj ∂ z¯k

j,k,l,m,a,b,p,q=1 n X

−

(8)

j,k,l,m,a,b,p,q=1

=−

n X

Ω0kj Ω0ml

j,k,l,m=1

−

n X

∂ 2 ωlm + Ω0kj Ljklm ∂zj ∂ z¯k

ωlm Ω0kj Ω0ma c0jkal

j,k,l,m,a=1 n X

−

ωlm Ω0kj Ω0mp Ω0qa Ω0bl

j,k,l,m,a,b,p,q=1

0 ∂ω 0 ∂ωab pq . ∂zj ∂ z¯k

Now, since (∂/∂z1 , . . . , ∂/∂zn ) is merely ω 0 -orthogonal but not necessarily ω 0 -unitary at x0 , the Kähler symmetries of the coefficients c0jklm ’s at x0 read p p p p c0jklm λl λm = c0lmjk λj λk . In particular, c0jjll λl = c0lljj λj . We are now in a good position to conclude the proof of the lemma. Indeed, evaluating (8) at the point x0 with our initial choice of coordinates gives ∆ω0 S(x0 ) =

n n n 0 0 X X X c0jjll cjjll 1 ∂ωal ∂ωla − − 2 λj λl λj λl λj (λl ) λa ∂zj ∂ z¯j j,l=1 j,l=1 | {z } j,l,a=1 c0

= (λlljj )2

(9)

l

n n n 0 X X cjjll X ρ0ll |∂ωal /∂zj (x0 )|2 = − − . 2 λj λl (λl ) λj (λl )2 λa j,l=1

l=1

j,l,a=1

Our next task will be to estimate the three summands appearing on the right hand side of the differential equality of the above lemma. We begin with the term involving the Ricci curvature of ω 0 . Recall that the we are supposing that Ric(ω 0 ) ≥ −λω 0 + µω.

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Lemma 5.7. – At the point x0 ∈ X, we have   n X µ 2 ρ0ll ≥ 2π −λS + S . (λl )2 n l=1

Proof. – The hypothesis on the Ricci curvature of ω 0 , when red at the point x0 with our choice of coordinates, gives ρ0ll ≥ 2π(−λλl + µ). Thus, we get n n n X X X 1 ρ0ll 1 ≥ −2πλ . + 2πµ (λl )2 λl (λl )2 l=1

l=1

l=1

Now, since the λl ’s are the eigenvalues of ω 0 with respect to ω, the eigenvalues of omega Pn with respect to ω 0 are 1/λl , l = 1, . . . , n, and therefore S = l=1 1/λl . Moreover, by the standard inequality between 1-norm and 2-norm of vectors in Rn , we have
2 Pn Pn 2 l=1 1/(λl ) ≥ 1/n l=1 1/λl . We finally obtain   n X µ 2 ρ0ll ≥ 2π −λS + S . (λl )2 n l=1

Now, we treat the term with the first order derivatives of the metric ω 0 . In doing this, we have to keep in mind that, at the end of the day, we want to estimate ∆ω0 log S. This Laplacian is given in coordinates by ∆ω0 log S = −

n X

∂ 2 log S ∂zj ∂ z¯k | {z }

Ω0kj

j,k=1

 ∂ = ∂z

=

j

1 ∂S S ∂z ¯k

=− S12

∂S ∂S ∂zj ∂ z ¯k

1 +S

∂2 S ∂zj ∂ z ¯k

n 1 1 X 0 ∂S ∂S Ωkj ∆ω 0 S + 2 . S S ∂zj ∂ z¯k j,k=1

Once computed at x0 , we have (10)

∆

ω0

2 n 1 1 X 1 ∂S . 0 ∆ω S(x0 ) + (x ) log S(x0 ) = 0 S(x0 ) S(x0 )2 j=1 λj ∂zj

What we want to do in the lemma below is then to try to express these first order derivatives in terms of first order derivatives of S. Lemma 5.8. – At the point x0 ∈ X, we have 0 2 n n ∂ω /∂zj 2 X 1 X 1 ∂S al ≥ (x ) 0 . 2 λj (λl ) λa S(x0 ) j=1 λj ∂zj j,l,a=1

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Proof. – Since the sum we are dealing with is made up of non negative terms, we have by plain minoration 0 2 n n ∂ω /∂zj 2 X X ∂ωll0 /∂zj al . ≥ λj (λl )2 λa λj (λl )3 j,l,a=1

j,l=1

Now, let us compute ∂S/∂zj at x0 . We have n ∂ X 0 ∂S (x0 ) = Ωml ωlm | x0 toΘ ∂zj ∂zj l,m=1

n n X X ∂ωlm ∂Ω0ll ∂Ω0ml ωlm + Ω0ml x0 = (x0 ). = | ∂zj ∂zj toΘ ∂zj l=1

l,m=1

Now, we use the identity (4) to replace −

n X

Ω0la

a,b=1

∂Ω0ll /∂zj (x0 )

with

0 1 ∂ωll0 ∂ωab Ω0bl | x0 = − (x0 ). toΘ ∂zj (λl )2 ∂zj

Thus, we obtain n

X 1 ∂ω 0 ∂S ll (x0 ) = − (x0 ). ∂zj (λl )2 ∂zj l=1

Now, inspired by (10), we compute 2 n n X X 1 1 ∂S (x0 ) = λ ∂z λ j j=1 j j=1 j

2 n X 1 ∂ω 0 ll (x ) 0 (λl )2 ∂zj l=1 2 n n X 1 X 1 ∂ωll0 /∂zj (x0 ) = λ (λl )1/2 (λl )3/2 j=1 j l=1 2 n n n X 1 X 1 X ∂ωll0 /∂zj (x0 ) ≤ λ λ (λl )3 j=1 j k=1 k l=1 2 n X ∂ωll0 /∂zj (x0 ) = S(x0 ) , λj (λl )3 l,j=1

where the inequality is given by Cauchy-Schwarz. The lemma follows. Finally, we estimate the term involving the curvature of ω, using the hypothesis on the negativity of the holomorphic sectional curvature. Lemma 5.9. – At the point x0 ∈ X, we have n X cjjll κ(n + 1) 2 ≤− S . λj λl 2n

j,l=1

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Classically this term has been bounded in terms of a uniform bound on the holomorphic bisectional curvature of ω. In order to prove this lemma, we need to be able to transform an information on the sum of holomorphic bisectional curvature type terms into an estimate using holomorphic sectional curvature only. Next proposition in Hermitian linear algebra is the key point to do that. It is due to Royden. Proposition 5.10 (Royden [32]). – Let ξ1 , . . . , ξν be mutually orthogonal (but not necessarily unitary) non-zero vectors of a Hermitian vector space (V, h). Suppose that Θ(ξ, η, ζ, ω) is a symmetric “bi-Hermitian” form, i.e., Θ is sesquilinear in the first two and last two variables and has the same pointwise properties as those of the (contraction with the metric of the) Chern curvature of a Kähler metric. Suppose also that there exists a real constant K such that for all ξ ∈ V one has Θ(ξ, ξ, ξ, ξ) ≤ K ||ξ||4h . Then, X α,β

1 Θ(ξα , ξα , ξβ , ξβ ) ≤ K 2

X

||ξα ||2h

!

2 +

α

X

||ξα ||4h

.

α

Moreover, if K ≤ 0, then X

Θ(ξα , ξα , ξβ , ξβ ) ≤

α,β

X 2 ν+1 K ||ξα ||2h . 2ν α

We shall use this proposition with

 Θ(TX , ω)(•, ¯•) · •, • ω as the symmetric “bi-Hermitian” form on TX,x0 in the statement. In terms of holomorphic bisectional curvature it can be rephrased as follows, when ν = n = dim X. Suppose that a Kähler metric ω has negative holomorphic sectional curvature at the point x0 , bounded above by a negative constant −κ. Then, if ξ1 , . . . , ξn is a ω-orthogonal basis for TX,x0 we have 2  n n X κ(n + 1) X 2 2 2 ||ξα ||ω ||ξβ ||ω HBCω (ξα , ξβ ) ≤ − ||ξα ||ω . 2n α=1 α,β=1

Here is the proof. Proof. – Realize Z4 as the group of 4th roots of unity and set, for a vector A = (1 , . . . , ν ) ∈ Zν4 , X ξA = α ξα . Then, by orthogonality,

||ξA ||2h

=

α 2 α ||ξα ||h ,

P

and thus by hypothesis X 2 4 2 Θ(ξA , ξA , ξA , ξA ) ≤ K ||ξA ||h = K ||ξα ||h . α

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Now, we take the sum over all possible A ∈ Zν4 and get X 2 1 X Θ(ξA , ξA , ξA , ξA ) K ||ξα ||2h ≥ ν 4 α A 1 X X = ν α ¯β γ ¯δ Θ(ξα , ξβ , ξγ , ξδ ) 4 A α,β,γ,δ

1 X X α γ Θ(ξα , ξβ , ξγ , ξδ ). = ν 4 β δ α,β,γ,δ A

Now, fix a 4-tuple (α, β, γ, δ). We claim that only the terms with α = β and γ = δ or α = δ and β = γ can survive after summing over all A. Thus, we are left only with the following terms 1 X X α γ Θ(ξα , ξβ , ξγ , ξδ ) 4ν β δ α,β,γ,δ A X X = Θ(ξα , ξα , ξα , ξα ) + Θ(ξα , ξα , ξγ , ξγ ) + Θ(ξα , ξγ , ξγ , ξα ). α

α6=γ

The claim is straightforwardly verified, since for all the other terms, for each A ∈ Zν4 0 0

 

one can find an A0 = (01 , . . . , 0ν ) ∈ Zν4 such that αβ γδ = − α0 γ0 . β δ P Now, by symmetry of Θ, adding α Θ(ξα , ξα , ξα , ξα ) to both side and using the upper bound as in the hypotheses, we get ! X 2 X X 2 Θ(ξα , ξα , ξγ , ξγ ) ≤ K ||ξα ||2h + ||ξα ||4h . α,γ

α

α

To end the proof, observe that applying the Cauchy-Schwarz inequality in Rν to the vectors (||ξ1 ||2h , . . . , ||ξν ||2h ) and (1, . . . , 1), we have X 2 X 2 ||ξα ||h ||ξα ||4h , ≤ν α

α

so that if K ≤ 0, then K

X

||ξα ||4h ≤

α

K ν

X

||ξα ||2h

2 ,

α

and thus X

Θ(ξα , ξα , ξγ , ξγ ) ≤

α,γ

X 2 ν+1 K ||ξα ||2h , 2ν α

as desired. We are now ready to give a

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Proof of Lemma 5.9. – Set 1 ∂ ξj := p , λj ∂zj

j = 1, . . . n,

so that ξ1 , . . . , ξn is a ω-orthogonal basis for TX,x0 . Now, it suffices to observe that

 cjjll = Θ(TX , ω)(ξj , ξ¯j ) · ξl , ξl ω λj λl = ||ξj ||2ω ||ξl ||2ω HBCω (ξj , ξl ). Now take the sum over all j, l = 1, . . . , n, to obtain  n 2 n X κ(n + 1) X 1 κ(n + 1) 2 cjjll ≤− =− S . λj λl 2n λ 2n α=1 j j,l=1

Now, to conclude the proof of Claim 5.3, i.e., to show inequality (1), we just put together the three estimates of the above lemmata, and plug them into formula (10). We get: 2 n 1 1 X 1 ∂S −∆ω0 log S(x0 ) = − ∆ω0 S(x0 ) − (x ) 0 2 S(x0 ) S(x0 ) j=1 λj ∂zj   0 n n n ∂ω /∂zj (x0 ) 2 X X 1 X ρ0ll c jjll  al = + − S(x0 ) (λl )2 λj (λl )2 λa λj λl l=1 j,l,a=1 j,l=1 n 2 1 X 1 ∂S − (x0 ) 2 S(x0 ) j=1 λj ∂zj    1 µ ≥ 2π −λS(x0 ) + S(x0 )2 S(x0 ) n  n 2 κ(n + 1) 1 X 1 ∂S + S(x0 )2  (x0 ) + S(x0 ) j=1 λj ∂zj 2n 2 n 1 X 1 ∂S − (x0 ) S(x0 )2 j=1 λj ∂zj   κ(n + 1) µ = + 2π S(x0 ) − 2πλ, 2n n as desired. Proof of Claim 5.4. – We want to show that the limit Z Z lim+ ωεn = lim+ euε ω n ε→0

X

ε→0

X

is strictly positive. The first observation is that the functions uε are all ω 0 -plurisubharmonic for some fixed Kähler form ω 0 and ε > 0 small enough. For, let ` > 0 be

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such that `ω − Ricω is positive and call ω 0 = `ω − Ricω . Thus, for all 0 < ε < `, one has ¯ ε < `ω − Ricω +i∂ ∂u ¯ ε = ω 0 + i∂ ∂u ¯ ε. 0 < εω − Ricω +i∂ ∂u Therefore, since the uε ’s are all uniformly bounded from above, by [15, Proposition 2.6], either {uε } converges uniformly to −∞ on X or it is relatively compact in L1 (X). Suppose for a moment that we are in the second case. Then, there exists a subsequence {uεk } converging in L1 (X) and moreover the limit coincides a.e. with a uniquely determined ω 0 -plurisubharmonic function u. Up to passing to a further subsequence, we can also suppose that uεk converges pointwise a.e. to u. But then, euεk → eu pointwise a.e. on X. On the other hand, by Claim 5.2, we have euεk ≤ eC so that, by dominated convergence, we also have L1 (X)-convergence and Z Z lim euεk ω n = eu ω n > 0. k→∞

X

X

The upshot is that what we need to prove is that {uε } does not converge uniformly to −∞ on X. We shall thus provide a lower bound for supX uε , as follows. Recall that we defined the smooth positive function Sε on X by Sε n ω ∧ ωεn−1 = ω . n ε Now, set Tε = log Sε . In other words, Tε is the logarithm of the trace of ω with respect to ωε . Lemma 5.11. – The function Tε satisfies the following inequality: uε Tε > − . n Proof. – Let 0 < λ1 ≤ · · · ≤ λn be the eigenvalues of ωε with respect to ω, so that 0 < 1/λn ≤ · · · ≤ 1/λ1 are the eigenvalues of ω with respect to ωε . Then, 1 1 1 eTε = trωε ω = + ··· + > . λ1 λn λ1 Thus, e−Tε < λ1 so that e−nTε < (λ1 )n ≤ λ1 · · · λn . But, euε ω n = ωεn = λ1 · · · λn ω n , and so we get e−nTε < euε , or, in other words, uε Tε > − . n As announced, we now use the integral inequality (2). We write it as follows: Z Z (n + 1)κ eTε +uε ω n ≤ 2π euε ω n . 2n X X Next, if we define Cε := inf X e−uε /n , by Lemma 5.11 we have that eTε > Cε , and thus Z Z (n + 1)κ uε n (11) Cε e ω ≤ 2π eu ε ω n . 2n X X But then, we obtain that (n + 1)κ Cε ≤ 2π, 2n

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i.e., inf e−uε /n ≤ X

4πn , (n + 1)κ

so that sup uε ≥ −n log X

4πn (n + 1)κ

is the desired lower bound. 6. The Kähler case, and the quasi-negative holomorphic sectional curvature case As largely revealed in advance, the Wu-Yau theorem also holds for compact Kähler manifolds under the same curvature assumptions. This is due to Tosatti and Yang, on the lines of Wu-Yau’s proof. One can also relax, still in the Kähler case, the negativity hypothesis on the curvature to quasi-negativity, as explained in Section 3. Let us briefly explain how to obtain such results. 6.1. The Kähler case. – In order to adapt the proof explained in the preceding section to the compact Kähler case, the main point is to show that a compact Kähler manifold endowed with a Kähler metric whose holomorphic sectional curvature is negative has nef canonical bundle. Indeed, this is obtained in the projective case by Mori’s theorem, which is unknown for compact Kähler manifolds. The nefness of the canonical bundle still holds more generally under the (more natural) assumption of non positivity of the holomorphic sectional curvature. Theorem 6.1 ([33, Theorem 1.1]). – Let (X, ω) be a compact Kähler manifold with non positive holomorphic sectional curvature. Then the canonical bundle KX is nef. Sketch of the proof. – Suppose by contradiction that KX is not nef, that is −c1 (X) is not in the closure of the Kähler cone. Since t[ω] − c1 (X) becomes a Kähler class for t  0, one can select the time ε0 when t[ω]−c1 (X) cuts the boundary of the Kähler cone, that is ε0 [ω] − c1 (X) is a nef but not Kähler class. Thus, for every ε > 0 one ¯ gets a Kähler class (ε + ε0 )[ω] − c1 (X). Therefore, by ∂ ∂-lemma and Yau’s solution of Calabi’s conjecture, for every ε > 0 we obtain a Kähler metric ωε ∈ (ε+ε0 )[ω]−c1 (X) such that i ¯ ωε = (ε + ε0 )ω − Ricω + ∂ ∂u ε 2π and ωεn = euε ω n . For this metric, we have Ricωε = −ωε + (ε + ε0 )ω. Now, precisely as in Claim 5.2 and 5.3, we get a uniform upper bound on the one hand for sup euε ≤ C, X

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and, on the other hand, for  −∆ωε log Sε ≥

 µ κ(n + 1) + 2π Sε − 2πλ, 2n n

where Sε = trωε ω, and here κ = 0, λ = 1, and µ = ε + ε0 . By the maximum principle we obtain n sup trωε ω ≤ , ε + ε0 X which is uniformly bounded as ε approaches to zero, since ∆ωε log Sε is non negative at a point where Sε (and hence log Sε ) achieves a (local) maximum. Now, the elementary inequality
n−1 ωεn 1 trω ωε ≤ , trωε ω (n − 1)! ωn |{z} =euε

enables us to conclude that also trω ωε is uniformly bounded from above, so that we have B −1 ω ≤ ωε ≤ B ω, for some positive constant B. Beside this control for ωε , it is also possible to work out uniform C k -estimates for all k ≥ 0—as explained in [33, pp. 577–578]—and these together with the Ascoli-Arzel Theorem and a diagonal argument allow to obtain the existence of a sequence εk → 0 such that ωεk converges smoothly to a Kähler metric ω0 which of course satisfies [ω0 ] = ε0 [ω] − c1 (X). But this is a contradiction, since we were supposing that ε0 [ω] − c1 (X) is a nef but not Kähler class, and hence it cannot contain any Kähler metric. Once the nefness of the canonical class is known also in the Kähler setting the proof proceeds in the same way of the projective case, with a small further argument at the end. Indeed, after proving Claim 5.4, we know that c1 (KX )n > 0. Then, by [9, Theorem 0.5], we deduce that KX is big. In particular, carrying a big line bundle, X is Moishezon. Since X is Kähler and Moishezon, by Moishezon’s theorem X is projective. We have thus finally landed in the projective world, where we can apply Lemma 5.1 and conclude the proof of the ampleness. 6.2. The Kähler quasi-negative case. – Relaxing further the hypotheses, we finally come to the case of a compact Kähler manifold supporting a Kähler metric whose holomorphic sectional curvature is quasi-negative. Thus, the first problem to be faced is that we do not dispose anymore of a negative uniform upper bound for the holomorphic sectional curvature. Such a bound is here replaced by the continuous function on X: κ: X → R x 7→ −

max

v∈TX,x \{0}

HSCω (x, [v]).

The quasi-negativity of the holomorphic sectional curvature of Theorem 3.8 translates in κ ≥ 0 and κ(x0 ) > 0 for some x0 ∈ X.

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As we saw, for every ε > 0 we have the following crucial inequality which makes the holomorphic sectional curvature enter into the picture:   n+1 ε Tε (x) (12) ∆ωε Tε (x) ≥ κ(x) + e − 1. 2n n The inequality being of pointwise nature, it still holds with using the continuous function κ. Set M (x) = n+1 2n κ(x). By plain minoration of the right hand side, we obtain that the Tε ’s satisfy the following differential inequality: ∆ωε Tε (x) ≥ M (x) eTε (x) − 1.

(13)

For each ε > 0, as before, integrate (13) over X using the volume form associated to ωε , to get Z Z
n 0= ∆ωε Tε ωε ≥ M eTε − 1 ωεn . X

X

We obtain therefore the following integral inequality: Z Z Tε uε n Me e ω ≤ euε ω n , X

X

and setting vε = uε − supX uε one has Z Z M e Tε e v ε ω n ≤ e vε ω n . X

X

Next, if we define Cε := inf X e−uε /n , we have that eTε > Cε , and Z Z vε n (14) Cε Me ω ≤ e vε ω n . X

X

Moreover, recall that we are assuming by contradiction that Cε → +∞ as ε → 0+ . Now, the same reasoning made during the proof of Claim 5.4 tells us that there exists a subsequence {vεk } of {vε } converging in L1 (X) and moreover the limit coincides a.e. with a uniquely determined ω 0 -plurisubharmonic function v. Indeed, the case where {vε } converges uniformly to −∞ is not possible here since the supremum of the vε ’s is fixed and equal to 0. Again, up to pass to a further subsequence, we can also suppose that vεk converges pointwise a.e. to v. But then, evεk → ev pointwise a.e. on X. On the other hand, we have evεk ≤ 1 so that, by dominated convergence, we also have L1 (X)-convergence and therefore Z Z v εk n lim e ω = ev ω n > 0, k→∞

and

Z lim

k→∞

X

X

M e v εk ω n =

X

Z

M ev ω n > 0,

X

since M is non negative and strictly positive in at least one point, while the set of points where v = −∞ has zero measure. Plugging this information into inequality (14) we obtain the desired contradiction since the left hand side blows up while the right hand side converges to some fixed positive number.

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References [1] A. Beauville – “Variétés kähleriennes dont la première classe de Chern est nulle”, J. Differential Geom. 18 (1983), p. 755–782. [2] M. Berger – “Sur les variétés d’Einstein compactes”, in Comptes Rendus de la IIIe Réunion du Groupement des Mathématiciens d’Expression Latine (Namur, 1965), Librairie Universitaire, Louvain, 1966, p. 35–55. [3] S. Boucksom & S. Diverio – “A note on Lang’s conjecture for quotients of bounded domains”, Épijournal Géom. Algébrique 5, Art. 5. [4] R. Brody – “Compact manifolds and hyperbolicity”, Trans. Amer. Math. Soc. 235 (1978), p. 213–219. [5] G. Cho & Y. Yuan – “Comparison of invariant metrics on the symmetrize bidisc”, preprint arXiv:2004.04637. [6] O. Debarre – Higher-dimensional algebraic geometry, Universitext, Springer, 2001. [7] J.-P. Demailly – “Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., 1997, p. 285–360. [8]

, “Complex analytic and differential geometry”, https://www-fourier. ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2012.

[9] J.-P. Demailly & M. Păun – “Numerical characterization of the Kähler cone of a compact Kähler manifold”, Ann. of Math. 159 (2004), p. 1247–1274. [10] S. Diverio – “Segre forms and Kobayashi-Lübke inequality”, Math. Z. 283 (2016), p. 1033–1047. [11] S. Diverio & A. Ferretti – “On a conjecture of Oguiso about rational curves on Calabi-Yau threefolds”, Comment. Math. Helv. 89 (2014), p. 157–172. [12] S. Diverio, C. Fontanari & D. Martinelli – “Rational curves on fibered CalabiYau manifolds”, Doc. Math. 24 (2019), p. 663–675. [13] S. Diverio & S. Trapani – “Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle”, J. Differential Geom. 111 (2019), p. 303–314. [14] J. Duval – “Sur le lemme de Brody”, Invent. math. 173 (2008), p. 305–314. [15] V. Guedj & A. Zeriahi – “Intrinsic capacities on compact Kähler manifolds”, J. Geom. Anal. 15 (2005), p. 607–639. [16] H. Guenancia – “Quasi-projective manifolds with negative holomorphic sectional curvature”, to appear in Duke Math. J. [17] D. R. Heath-Brown & P. M. H. Wilson – “Calabi-Yau threefolds with ρgt; 13”, Math. Ann. 294 (1992), p. 49–57. [18] G. Heier, S. S. Y. Lu & B. Wong – “Kähler manifolds of semi-negative holomorphic sectional curvature”, J. Differential Geom. 104 (2016), p. 419–441. [19] G. Heier, S. S. Y. Lu, B. Wong & F. Zheng – “Reduction of manifolds with semi-negative holomorphic sectional curvature”, Math. Ann. 372 (2018), p. 951–962. [20] N. Hitchin – “On the curvature of rational surfaces”, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), 1975, p. 65–80.

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[21] D. Huybrechts – Complex geometry, Universitext, Springer, 2005. [22] M. Klimek – Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press Univ. Press, 1991. [23] S. Kobayashi – “Some problems on intrinsic distances and measures”, in Proceedings of the C. Carathéodory International Symposium (Athens, 1973), 1974, p. 306–317. [24] [25]

, Hyperbolic complex spaces, Grundl. math. Wiss., vol. 318, Springer, 1998. , Hyperbolic manifolds and holomorphic mappings, second ed., World Scientific Publishing Co. Pte. Ltd., 2005.

[26] S. Lang – “Hyperbolic and Diophantine analysis”, Bull. Amer. Math. Soc. (N.S.) 14 (1986), p. 159–205. [27] R. Lazarsfeld – Positivity in algebraic geometry. I, Ergebn. Math. Grenzg., vol. 48, Springer, 2004. [28] R. Miranda – Algebraic curves and Riemann surfaces, Graduate Studies in Math., vol. 5, Amer. Math. Soc., 1995. [29] S. Mori – “Projective manifolds with ample tangent bundles”, Ann. of Math. 110 (1979), p. 593–606. [30] K. Oguiso – “On algebraic fiber space structures on a Calabi-Yau 3-fold”, Internat. J. Math. 4 (1993), p. 439–465. [31] T. Peternell – “Calabi-Yau manifolds and a conjecture of Kobayashi”, Math. Z. 207 (1991), p. 305–318. [32] H. L. Royden – “The Ahlfors-Schwarz lemma in several complex variables”, Comment. Math. Helv. 55 (1980), p. 547–558. [33] V. Tosatti & X. Yang – “An extension of a theorem of Wu-Yau”, J. Differential Geom. 107 (2017), p. 573–579. [34] M. Verbitsky – “Ergodic complex structures on hyperkähler manifolds”, Acta Math. 215 (2015), p. 161–182. [35] P. M. H. Wilson – “Calabi-Yau manifolds with large Picard number”, Invent. math. 98 (1989), p. 139–155. [36] P.-M. Wong, D. Wu & S.-T. Yau – “Picard number, holomorphic sectional curvature, and ampleness”, Proc. Amer. Math. Soc. 140 (2012), p. 621–626. [37] D. Wu & S.-T. Yau – “Negative holomorphic curvature and positive canonical bundle”, Invent. math. 204 (2016), p. 595–604. [38] D. Wu, S.-T. Yau & F. Zheng – “A degenerate Monge-Ampère equation and the boundary classes of Kähler cones”, Math. Res. Lett. 16 (2009), p. 365–374. [39] S. T. Yau – “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I”, Comm. Pure Appl. Math. 31 (1978), p. 339–411. [40] F. Zheng – Complex differential geometry, AMS/IP Studies in Advanced Math., vol. 18, Amer. Math. Soc.; International Press, 2000.

Simone Diverio, Dipartimento di Matematica “Guido Castelnuovo” SAPIENZA Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma • E-mail : [email protected]

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Panoramas & Synthèses 56, 2021, p. 253–318

SOME ARITHMETIC ASPECTS OF HYPERBOLICITY by Pietro Corvaja

Abstract. – We give a survey of the study of integral points and some aspects of Diophantine approximation on algebraic varieties, and we treat arithmetic analogues of the notion of hyperbolicity for algebraic varieties. According to a conjecture by Lang and Vojta, those (quasi projective) algebraic varieties, defined over number fields, whose complex points form a hyperbolic manifold (in the complex analytic sense) should admit only degenerate sets of integral or rational points. In dimension one, after the work of Siegel and Faltings, it is known that the analytic and arithmetic notions of hyperbolicity are equivalent. We show, mainly focusing on the two-dimensional case, that many apparently unrelated Diophantine problems can be reduced to questions about the distribution of integral points on certain algebraic surfaces. Significant examples are the following. The theorem of Darmon and Granville on the generalized Fermat equation xp + y q = z r is proved here in a slightly simplified way and its connection with the hyperbolicity of the triple of exponents (p, q, r) is developed in detail. A conjecture about the denominators of rational points on elliptic curves is linked to Vojta’s conjecture, and a weaker version is unconditionally established. A main tool in the proofs of finiteness or degeneracy results for integral points on varieties is provided by Diophantine approximation. The theory of Diophantine approximation is also linked to questions of hyperbolicity, and in particular a new “gap principle” for rational points on elliptic curves is proved and its formulation is shown to be directly linked to a hyperbolicity condition.

1. Introduction 1.1. Introducing the problems. – Our main concern will be the following problem: To find geometric properties for an algebraic variety X defined over a number field κ which ensure that for every number field K ⊃ κ the set X(K) of K-rational points of X is not Zariski-dense.

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This property can be considered to be the arithmetic analogue of a weak-form of hyperbolicity, namely: there exists no entire curve f : C → X(C) with Zariski-dense image. An analogue question arises naturally concerning integral points. The investigation on these problems led to considering two other different issues, namely Diophantine approximation and gap principles. Diophantine approximation refers, at first instance, to the theory of approximating algebraic numbers by rationals. More generally, one can fix one or more ‘targets’ on an algebraic variety in which rational points (over a fixed number field) are dense, in some archimedean or p-adic topology, and look at how fast these targets can be approached by a sequence of rational points. Usually the targets are hypersurfaces on the given algebraic variety, so they are themselves points if the variety is a curve. In any case, they are supposed to be defined over the field of algebraic numbers. The so called gap principles arise when a sequence of rational points converges ‘rapidly’ to any point, possibly a transcendental one; we dispose of a gap principle if we can deduce, from the rapidity of its convergence, that the approximating sequence is ‘sparse’. In the case the ambient algebraic variety X is a curve, we have a rather satisfactory solution to all the above issues, due mainly to works of K. Roth, C.-L. Siegel, L. Mordell, A. Weil and G. Faltings. In each case, a hyperbolicity condition on the variety or on the sequence of approximants implies a finiteness or a sparseness result. More precisely, for a smooth algebraic curve C , of genus g with d points at infinity (in a smooth completion), we define its Euler characteristic χ to be the number χ = 2g − 2 + d. If d = 0, i.e., the curve is projective, then χ = 2g − 2 coincides with the degree of the canonical bundle. We say that a curve is hyperbolic if χ > 0, parabolic if χ = 0 and of elliptic type (1) if χ < 0. Hence the hyperbolicity condition reads (1.1)

χ := 2g − 2 + d > 0

(Hyperbolicity).

Let us review the mentioned arithmetic results, by starting from the problem of density. Recalling that on an irreducible curve the Zariski-dense sets are just the infinite ones, we are interested in describing those algebraic curves which can contain infinitely many rational or integral points. In the case of integral points, a theorem proved by Siegel in 1929 (see [62] and [69]) reads: Theorem (Siegel’s Theorem). – Let X ⊂ AN be an affine irreducible curve, defined over a number field κ. If the curve contains infinitely many points with coordinates in the ring of algebraic integers of κ then X is a rational curve and it has at most two points at infinity. (1)

By elliptic curve we mean something different, namely a parabolic complete curve.

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Vice-versa, if a curve is rational (i.e., of genus zero with at least one rational point) and has one or two smooth points at infinity, then a suitable model of it contains infinitely many integral points, as we now show. First, if it has exactly one point at infinity, a normalization of it is isomorphic to the affine line. On a suitable integral model (i.e., after changing coordinates) it will clearly have infinitely many integral points. Note that the coordinate-change is unnecessary if we replace the ring of integers with a suitable ring of S-integers (defined below). If a rational curve has two points at infinity, then after possibly a quadratic field extension a normalization of it becomes isomorphic to the variety Gm = A1 {0} (defined e.g., as a closed subset in the plane by the equation xy = 1) and again it has infinitely many integral points, at least after enlarging the ring of integers so to acquire infinitely many units. In view of these considerations, Siegel’s theorem can be considered to be a bestpossible result. For rational points, Faltings theorem, proved in 1983, states that: Theorem (Faltings’ Theorem). – Let X be an irreducible algebraic curve defined over a number field κ. If the genus of X is ≥ 2, then its set of κ-rational points is finite. As for Siegel’s Theorem, the above statement is essentially optimal, since, as we shall see, every algebraic curve of genus ≤ 1 contains infinitely many rational points, after suitably enlarging the ground number field. Let us now consider briefly the two other issues, starting from Diophantine approximation. It is well known that every real irrational number α admits infinitely many rational approximations p/q, where p, q are coprime integers, q > 0, such that α − p < 1 . q q2 A proof of this fact is obtained via Dirichlet’s box principle (see Chapter I of [58]); an explicit sequence of rational approximations is provided by the continued fraction development of α. The celebrated Theorem of Roth (see §2.3) asserts that for every real number δ > 2 and every real algebraic number α, the inequality α − p < 1 (1.2) q qδ admits only finitely many solutions p/q ∈ Q (where p, q are coprime integers, q > 0). Note that the approximants p/q ∈ Q (and the target α) are points on the line P1 , whose Euler characteristic χ equals −2. Hence the finiteness result of Roth requires (1.3)

χ + δ > 0,

which is the analogue of the hyperbolicity condition (1.1). We shall see (Theorem 2.21) that when approximating an algebraic point on an elliptic curve with rational ones, the analogue of Roth’s theorem holds with any exponent δ > 0; this is in accordance with the fact that the Euler characteristic of an

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elliptic curve is zero, so the inequality (1.3) holds in that case whenever δ is strictly positive. If the limit point of the sequence of rational approximations is transcendental, the conclusion of Roth’s Theorem does not hold; in fact, for every δ one can construct a real number α such that the inequality (1.2) admits infinitely many rational solutions. However, we dispose in that situation of a gap principle (Theorem 2.27), asserting that if the sequence of approximations p1 /q1 , p2 /q2 , . . . is ordered by increasing denominators, then log qn+1 lim inf ≥ δ − 1, n→∞ log qn which is a non-trivial result whenever δ > 2 (i.e., when χ + δ > 0). The analogue for elliptic curves provides, mutatis mutandis, the bound 1 + δ for the above limit, which is non trivial for every δ > 0, i.e., again when χ + δ > 0. 1.2. Integrality over algebraic varieties. – We shall formulate in a unified way the two problems (and the general results in dimension one) for the integral and for the rational points, by giving a suitable definition of what we mean by an integral point. Definition 1.1. – Let κ be a number field, S a finite set of places of κ containing the archimedean ones. The ring of S-integers of κ, denoted by O S , is defined as the set OS

= {x ∈ κ : |x|ν ≤ 1 for all v 6∈ S}.

Its group of units, called the group of S-units, is then ×

OS

= {x ∈ κ : |x|ν = 1 for all v 6∈ S}.

Definition 1.2. – Let X be a quasi projective irreducible variety, defined over a number ˜ a completion of X in a projective space PN . Then we can field κ. Let us denote by X ˜ ˜ We say that a rational write X = X D, where D is a proper closed subvariety of X. point p ∈ X(κ) is S-integral with respect to D if for no place outside S p reduces to a point of D. We note that in the above definition no mention of integral models appears: in fact, we assume that our variety is already embedded in a projective space PN , which is canonically provided with an integral model; this canonical integral model implicitely appears via the notion of reduction modulo a prime. We also note that whenever the variety X is affine, and embedded into the affine space AN , the integral points with respect to the divisor at infinity of X exactly cor˜ is projective, respond to the points of X having all their coordinates in O S . If X = X then D = ∅ and the set of S-integral points coincides with the full set of κ-rational points. An alternative definition of integrality, making use of Weil functions, will appear later. We now give some examples of integrality of rational points on quasi-projective algebraic varieties.

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˜ = P1 — Let X = A1 be the affine line, embedded into the projective line X ˜ X consists of the single by the map t 7→ (t : 1) so that the complement X point D = {(1 : 0)}, also called the point at infinity. Letting κ = Q, we can write a rational point on the line as t = a/b, where a, b ∈ Z are coprime integers, b 6= 0. Then t corresponds to the projective point (a : b), which reduces to (1 : 0) modulo the primes dividing b. It is integral if and only if there are no such primes, which amounts to b = ±1, i.e., t ∈ Z. — Let X = Gm = P1 X( O S ) = O ∗S .

{0, ∞}. For the same reason as in the previous example,

— Consider the quasi-projective surface X = A2 {(0, 0)}. It can be embedded into P2 in the usual way: (x, y) 7→ (x : y : 1) = (x : y : z), so that X = P2 D, with D consisting of the line z = 0 plus the single point (0 : 0 : 1). The set X(Z) consists of pairs (x, y) ∈ Z2 with gcd(x, y) = 1. Note that by changing ˜ e.g., replacing P2 by the plane blown up at the point the compactification X, (0 : 0 : 1), we can view X as the complement of a hypersurface in a projective surface. ˜ = P1 × P1 be the product of two lines; let D be its diagonal — Let X and X = P21 D. Each Q-rational point of P1 × P1 can be written as P = ((a : b), (c : d)) where a, b (resp. c, d) are coprime integers. The condition of integrality with respect to the diagonal
is equivalent to the quantity ad − bc being a unit, i.e., ad − bc = det ac db = ±1. Since (a, b) (resp. (c, d)) are defined up to constant, i.e., up to multiplying both of them by −1, we can normalize so that the determinant is positive and the set X(Z) is in natural bijection with P SL2 (Z) = SL2 (Z)/{±I}. — Let f (x, y), g(x, y) ∈ O S [x, y] be polynomials. Suppose that the affine curves of equations f (x, y) = 0 and g(x, y) = 0 intersect transversally at every point of intersection. Letting P1 , . . . , Pk ∈ A2 ⊂ P2 be the set of the intersection ˜ to be the projective plane blown up at these points of the two curves, define X intersection points. Let now D be the union of the pull-back of the line at infinity with the strict transform of the zero divisor of the polynomial g(x, y) and put ˜ D. Then X( O S ) is in natural bijection with the set of pairs (x, y) ∈ O 2S X=X such that g(x, y) divides f (x, y) in the ring O S . In other words, it represents the set of S-integral solutions to the equation z · g(x, y) = f (x, y). — This example will be treated in detail in §5. Let 1 < p ≤ q ≤ r be three natural numbers, S be the quasi-projective surface defined in A3 by the equation xp + y q = z r with the origin removed. The integral points in S correspond to the integral solutions (x, y, z) ∈ Z3 to the defining equation xp + y q = z r such that (x, y, z) 6≡ (0, 0, 0) (mod p) for every prime p, i.e., to the solutions (x, y, z) in coprime integers.

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1.3. Density in the 1-dimensional case. – As anticipated, we now state the main theorem concerning curves, obtained by combining results of Siegel’s (1929) and Faltings (1983). A smooth algebraic curve C defined over the complex number field is topologically characterized by two discrete invariants: its genus g and the number d = ]( ˜C C ) of its points at infinity in a smooth completion ˜C (so d equals zero if C is projective). Recall that we defined the Euler characteristic of C to be the number χ = χ( C ) = 2g − 2 + d. It is a homotopy invariant of the topological space C (C). The mentioned combination for Siegel’s and Falting’s theorems reads as follows: Theorem 1.4 (Siegel-Faltings Theorem). – Let C = projective) curve over a number field κ, where D infinity. Let O S ⊂ κ be a ring of S-integers. If the

˜C

D be an irreducible (affine or ⊂ ˜C (¯ κ) is the set of its points at set C ( O S ) is infinite, then χ ≤ 0.

This result is best-possible, in view of the following theorem: Theorem 1.5. – Let C be a (affine or projective) curve with χ( C ) ≤ 0, defined over a number field κ. There exists a finite field extension κ0 of κ and a ring of S-integers O S ⊂ κ0 such that C ( O S ) is infinite. We say that the integral points on a curve are potentially dense, if the conclusion of the above theorem holds; in higher dimensions, we shall require Zariski density (after finite extension of the ring of S-integers). The proof of Theorem 1.5 consists in analyzing one by one all the possible cases of curves with χ ≤ 0. We start with the projective ones, where, we recall, integral points coincide with rational ones. In the projective case, the Euler characteristic, if ≤ 0, can only be −2 and 0. The first case χ = −2 corresponds to a curve of genus 0; after performing a suitable quadratic extension of κ, such a curve becomes isomorphic to the line P1 , which possesses infinitely many rational points. The case χ = 0 corresponds to a genus 1 curve; after enlarging if necessary the number field κ, we can suppose that there exists a rational point, hence we obtain the structure of an elliptic curve. If we find an algebraic point of infinite order on this elliptic curve, we can then choose a number field κ0 so that the elliptic curve (including its origin) and the given point of infinite order are defined over κ0 . Hence, over κ0 the curve in question will have infinitely many rational points. Now, to prove that not all algebraic points have finite order, we dispose of several methods, none of which is completely obvious. First, we can prove that the absolute height of torsion points is bounded, so any point of sufficiently large height is necessarily of infinite order. Alternatively, one can argue p-adically, proving that a point sufficiently close to the origin in the p-adic sense cannot be torsion, unless it coincides with the origin; again, this property provides algebraic points of infinite order.

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Concerning open curves, the inequality χ ≤ 0 holds for the affine line A1 = P1 {∞}, for which χ = −1, and for the complement of two points on P1 , for which χ = 0. In the first case, over a suitable ring of S-integers (or after changing the integral model), we have infinitely many integral points. In the second case, after possibly a quadratic extension we can achieve the rationality of the points at infinity, so the curve will become P1 {0, ∞} ' Gm and again, after a suitable enlargment of O S if necessary (so that the group O ∗S becomes infinite) we obtain infinitely many integral points. It is worthwile to notice some alternative formulations of the inequality χ ≤ 0, as well as some coincidences with hyperbolicity results in the sense of Picard’s theorem. Namely, we can restate Theorem 1.4 (together with its converse, Theorem 1.5) as follows: For an (affine or projective) smooth algebraic curve C the following properties are equivalent: (i) C ( O S ) is Zariski-dense, for a suitable ring of S-integers O S ; (ii) C is a homogeneous space for an algebraic group; (iii) the fundamental group of the topological space C (C) is abelian; (iv) there exists a non-constant holomorphic map C → C (C); (v) the degree of the divisor (D + K ˜C ), where D is the divisor at infinity and K ˜C is a canonical divisor of the complete curve ˜C , is ≤ 0. In other words, the negation of any of the properties (ii),. . . ,(v) implies that for every ring of S-integers O S the set C ( O S ) is finite. 1.4. The higher dimensional case. – In higher dimensions, it is natural to try to figure out which of the above properties (ii),. . . ,(v) (after suitable reformulation of (v)) implies the potential density of integral points, and which ones are implied by that density. Let us analyze the possibility of such implications. It is rather easy to see (although not completely obvious) that on every homogenous space the set of integral points is potentially dense; for instance, in the case of principal homogeneous spaces, this fact amounts to saying that the integral points on an algebraic group are potentially dense. The crucial fact consists in proving that not all algebraic points on an algebraic group (of positive dimension) are torsion, and we have already discussed the case of elliptic curves. However, in dimension ≥ 2 the implication (i) ⇒ (ii) does not hold; for instance, the rational points on a elliptic surface can be Zariski-dense, and in general there are no non-trivial algebraic group actions on such surfaces. An alternative to asking that the variety be acted on by a single algebraic group is that it is covered by images of non-constant maps from algebraic groups (which can then be chosen to be commutative). It was asked (e.g., by Vojta) whether the Zariski closure of the set of integral points on a variety X is the union of a finite set and the

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images of non-constant maps G → X, where G is a commutative algebraic group). The above assertion might be viewed as a substitute to the implication (i) ⇒ (ii). Concerning relations between (iii) and (i), neither implication holds. It is rather easy to construct examples both of (smooth) algebraic varieties with non-abelian fundamental group and a Zariski-dense set of integral points as well as varieties with abelian fundamental group and degenerate integral points. Some examples of the first class are represented by hyperelliptic surfaces (2); for examples in the other direction, one can take the complement of four or more lines in general position on the plane: its fundamental group is abelian by a theorem of Zariski, while its integral points are degenerate by the S-unit Equation Theorem (see §6). Hence property (iii) neither implies nor is implied by the potential density of integral points. The relations between conditions (i) and (iv) in higher dimensions have been intensively investigated. As a consequence of Campana’s Conjecture (to be stated below), conditions (i) and (iv) should be equivalent in every dimension, with the provisio that in higher dimensions (iv) is reformulated as the existence of an entire curve with Zariski-dense image (in dimension one, this is equivalent to being non-constant). Concerning (v), we need to reformulate the condition on the positivity of the degree of divisor; a possibility is the notion of bigness: a divisor D on a (smooth complete) ˜ is said to be big if h0 (X, ˜ nD)  ndim X˜ . This condition amounts to a positive variety X multiple of D being linearly equivalent to the sum of an ample and an effective divisor. One of the most important problems in this field, raised by Vojta after combining previous formulations suggested by Bombieri and Lang, aims at providing a substitution for the implication (i) ⇒ (v) and reads as follows: ˜ be a smooth projective variety defined over a number Vojta’s Conjecture. – Let X ˜ field κ. Let D ⊂ X be a possibly reducible hypersurface, defined over κ, with nor˜ D. Letting K ˜ be a canonical mal crossing singularities (if any) and put X = X X ˜ divisor for X, suppose that the sum D + KX˜ is a big divisor. Then X( O S ) is not Zariski-dense. The complete varieties whose canonical bundle is big are said to be of general type. ˜ D, where D has normal crossing The smooth open varieties of the form X = X singularities and KX˜ + D is big are called varieties of log-general type. The condition for an open variety X of being of log-general type depends only on X, not on its ˜ compactification X. It is easy to check that Theorem 1.4 implies the positive solution of Vojta’s Conjecture in the one-dimensional case. On the other hand, Theorem 1.5 asserts that when˜ is not big, the integral points on X = X ˜ D ever the divisor D + KX˜ on a curve X are potentially dense. (2)

Here is an exemple: given an elliptic curve E and a torsion point T ∈ E(κ) of order 2, consider the automorphism of order four E 2 → E 2 sending (P, Q) → (Q, P + T ). The fundamental group of the quotient variety is a non-trivial extension of {±1} by Z4 , hence it is non-abelian. For a discussion of hyper-elliptic varieties and their fundamental groups, see [12].

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We now present some consequences of Vojta’s conjecture. (i) Let A be an abelian variety, D ⊂ A an hypersurface which is an ample divisor (3). Put X := A D. Since KA = 0, KA + D = D which is ample by assumption, Vojta’s Conjecture predicts the degeneracy of the integral points of X. Actually a stronger result was proved by Faltings [34], namely the finiteness of such points. (ii) Let A be an abelian variety, X ⊂ A be a closed proper algebraic subvariety. If A is simple, then X is of general type (in general it can be isomorphic to the product of a variety of general type by an abelian variety). In this case, Vojta’s conjecture, already formulated by Weil, was again proved by Faltings in [34], whose result implies that the Zariski closure of the set of rational points is a finite union of translates of algebraic subgroups contained in X. In particular, if A is simple, the set of rational points is finite. (iii) Consider now an irreducible closed algebraic subvariety X of a torus Grm . The set of its S-integral points can be dense only if X is a translate of a subtorus. This follows from the S-unit equation theorem, and was proved before Vojta’s Conjecture was formulated. These three examples, together with some applications, will be discussed in detail in §6. Note that again X is a product of a variety of log-general type by a sub-torus. ˜ ⊂ P3 . It is of general type if and only (iv) Consider a smooth algebraic surface X if its degree is ≥ 5. We dispose of no example of any such surface for which the degeneracy of rational points is proved (over arbitrary number fields). Note that the smooth hypersurfaces in P3 are simply connected, so they cannot be embedded into an abelian variety (more generally, any rational map from such surfaces to an abelian variety is constant). Hence Faltings’ theorem discussed ˜ is ≤ 3, it is known above cannot be applied. When the degree of the surface X that the set of rational points is potentially dense. This follows from the fact ˜ becomes rational after an extension of the scalars. However, the case of that X degree four is still open. Examples are known of smooth quartic surfaces with a Zariski-dense set of rational points (for instance when they admit an elliptic fibration, see e.g., Swinnerton-Dyer’s paper on the quartic Fermat surface [63]) and it is widely believed that the rational points are always potentially dense. (v) Consider an affine surface obtained from the projective plane by removing a (possibly reducible) curve with normal crossing singularities. We obtain a surface of log-general type whenever the degree of this curve is ≥ 4. The degeneracy of the corresponding integral points is proved only when such a boundary curve has at least four components. In that case, it admits non trivial maps to G3m , hence the S-unit equation theorem can be applied (see §6). The first open case arises for a curve consisting of a conic and two lines in general position. It (3)

It is always the case when A is simple.

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implies for instance the following (still unknown) assertion: the pairs of S-units (u, v) ∈ O ∗S × O ∗S such that 1 + u + v is a perfect square are not Zariski dense in the plane. ˜ ⊂ P3 of degree d ≤ 4. (vi) Consider again a smooth irreducible hypersurface X ˜ When the degree is 4, the canonical divisor of X is equivalent to zero, and so the complement of a hyperplane section (with normal crossing singularities) is of log-general type. The same is true for the complement of two hyperplane sections in a cubic surface, and of three hyperplane sections on a quadric. In general, the degeneracy of the integral points in these situations is still unproven, but partial results (especially in the cubic case) are provided in [24] and will be discussed in §7. (vii) Let A be an abelian surface, with origin O. A rational point P ∈ A(κ) is S-integral with respect to O if for no valuation (outside S) it reduces to O. It can be conjectured that whenever A(κ) is Zariski-dense, the subset of integral points with respect to O is also Zariski-dense. This example can also be reduced to an instance of integrality with respect to a divisor, after blowing-up the origin O on A and removing the resulting exceptional divisor. The sum of the canonical divisor plus the divisor at infinity turns out to be linearly equivalent to twice that divisor, hence not big. As mentioned, an even more challenging conjecture was formulated by F. Campana, who proposed the following definition: Definition. – A smooth projective algebraic variety X is special if for every p = 1, . . . , dim X, every line sub-bundle L ⊂ ΩpX and every integer m ≥ 1, the image of the rational map X 99K PM , M = h0 (L⊗m ) − 1, associated to the line bundle L⊗m , has dimension < p. Then Campana’s conjecture (in the special case where rational points are concerned) reads: Campana’s Conjecture. – Let X be a smooth projective variety over a number field. Then the following are equivalent: (i) X is a special variety; (ii) the rational points on X are potentially dense; (iii) there exists an entire curve C → X(C) with Zariski-dense image. All K3 surfaces are special in this sense, so according to the above conjecture its rational points should be Zariski-dense (after suitable extension of the ground field). Also, the rationally connected varieties are special, so one expects, if he believes Campana’s conjecture, that on these varieties too the rational points are potentially dense. Note that in dimension one, the special varieties are exactly the non hyperbolic curves, those of Euler characteristic ≤ 0, so we find again the theorem of Faltings and our previous consideration on the potential density of rational points on elliptic curves.

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2. Heights, Diophantine approximation 2.1. Valuations and heights. – We recall the standard vocabulary and fix the notation that will be used throughout. Let κ be a number field. For every place ν of κ, the corresponding absolute values differ logarithmically by a positive constant: namely, if |·|ν and k·kν are two equivalent absolute values of κ there exists a positive real number δ such that for every x ∈ κ, |x|ν = kxkδν . We are looking for a canonical normalization, which will simplify the notation in the formulation of results from Diophantine approximation. One natural choice would be simply to choose the ν-adic absolute values extending the natural ones already defined in the rational number field Q. However, there is another possibility, which is less canonical since it depends on the number field κ, but has the advantage that by adopting this new convention, the generalization and extensions of Roth’s theorem will be easier to state. We proceed to define this second normalization. For each place ν (i.e., equivalence class of absolute values of κ) we normalize the corresponding absolute value | · |ν of κ in the following way: if ν is ultrametric, lying above the prime p of Z, we set for every α ∈ Q, [κν :Qp ]

|α|ν = |α|p [κ:Q] , where | · |p denotes the ususal p-adic absolute value of Q. If ν is archimedean, corresponding to an embedding κ ,→ C, we put |α|ν = |α|

[κν :R] [κ:Q]

,

where | · | denotes the usual complex absolute value. With these normalizations, the Weil height of an algebraic number α can be expressed as Y H(α) = max(1, |α|ν ), ν

or in logarithmic form h(α) =

X

log+ |α|ν .

ν

Here the sum (and the product in the previous formula) runs over the places of any number field containing α, and the result turns out to be independent of such a ¯ → R+ . (Here R+ number field. Hence the height can be defined as a function h : Q denotes the semigroup of non-negative real numbers.) The fundamental property of Weil height is represented by the following finiteness statement Theorem 2.1 (Northcott Theorem). – For each pair of numbers d ≥ 1, c ≥ 0, the set of ¯ such that algebraic numbers α ∈ Q [Q(α) : Q] ≤ d,

h(α) < c

is finite.

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¯ The height satisfies the following properties: for every α ∈ Q h(αn ) = |n| · h(α), h(α) = 0

⇔

{0},

∀n ∈ Z, α

is a root of unity.

These properties can be restated by saying that the Weil height is a normalized height ¯ ∗ /Tors(Q ¯ ∗) on the multiplicative group Gm . It defines a norm on the quotient group Q (see [10], Chap. V ). Another class of one-dimensional algebraic groups we shall be interested in is provided by elliptic curves. Given an elliptic curve E over a number field κ, its set of rational points E(κ) has the structure of an abelian group. Letting x ∈ κ(E) be a non-constant function (for instance the x-coordinate in a Weierstrass model), one can define the “naive” height associated to the rational function x by letting h(P ) = hx (P ) := h(x(P )),

∀P ∈ E(¯ κ),

where the last height is the one already defined in P1 (¯ κ) (putting h(∞) = 0). The crucial point in the construction of the Néron-Tate height is the following proposition Proposition 2.2. – Let E be an elliptic curve defined over the field of algebraic num¯ Then the function E(Q) ¯ →R bers Q. P 7→ |h(2P ) − 4h(P )| is bounded. ˆ ) be its limit, It follows that the sequence n 7→ h(2n P )/4n converges. Letting h(P ¯ we obtain the so-called Néron-Tate height on E, i.e., a function E(Q) → [0, +∞) with the following properties ˆ ) ≥ 0 ∀P ∈ E(Q) ˆ ) = 0 if and only if P is a torsion point. ¯ and h(P (i) h(P ˆ ˆ ) for all P ∈ E(Q) ¯ and all integers n. (ii) h(nP ) = n2 h(P ˆ ˆ ) − h(Q) ˆ (iii) The function (P, Q) 7→ h(P + Q) − h(P is a non-degenerate bilinear ¯ ⊗Z R. form on the real vector space E(Q) ˆ ) = hx (P ) + O(1) (iv) h(P ¯ there exists constants c1 (A), c2 (A) such that ∀P ∈ E(Q), ¯ (v) For each A ∈ E(Q), ˆ + A) − h(P ˆ )| ≤ c1 (A) · h(P ˆ )1/2 + c2 (A). |h(P Note that from (iv) it easily follows the finiteness of points of bounded height which are defined over a fixed number field. A remark on (v): if A is a torsion ˆ ) = h(P ˆ point, h(P + A) due to (ii); then for a fixed torsion point A, the difference |h(P + A) − h(P )| between the naive heights is uniformely bounded. Note that if for some positive integer n, nA = O, O being the neutral element for the group law on E, then the divisors nA and nO on E are linearly equivalent, so the associated

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heights differ by a O(1)-term (see e.g., [65], Proposition 1.2.9). In general, the degreeone divisors A and O, as well as its multiples, are only algebraically equivalent, not linearly equivalent. 2.2. The Chevalley-Weil and Mordell-Weil theorems. – Let X, Y be algebraic varieties defined over a number field κ, and let F : X → Y be a morphism, also defined over κ. Then each κ-rational point p ∈ X(κ) will be sent to a κ-rational point F (p) ∈ Y (κ). If the morphism F is (generically) finite of degree d, the pre-image of a rational point in Y is (generically) formed by d algebraic points; one expects that in fact these points have degree d and consequently form a unique orbit for the Galois action of G al(¯ κ/κ). (4) However, there are cases of morphisms F : X → Y of degree > 1 between irreducible varieties, with Y (κ) Zariski-dense, where the pre-images of rational points will automatically be rational, or at least they will be all together defined on a finite degree extension of κ. This happens when the morphism is unramified, and is the content of the Chevalley-Weil theorem below. Since we are interested only on varieties in characteristic zero, we give a topological definition of unramified morphism. We say that a morphism F : X → Y between smooth quasi-projective varieties over a field κ ⊂ C is unramified if the corresponding continuous map X(C) → Y (C) is a covering in the topological sense. In particular, each point p ∈ Y (C) admits exactly deg F pre-images. The mentioned theorem of Chevalley and Weil (which can be found in this formulation e.g., in [16], Ch. 5 or [25], Chap. III, §2), reads: Theorem 2.3 (Chevalley-Weil). – Let F : X → Y be a finite unramified morphism between smooth quasi-projective varieties over a number field κ. Then — there exists a number field κ0 such that for each point p ∈ X(¯ κ) with F (p) ∈ Y (κ), p lies in X(κ0 ). — there exist finitely many κ-varieties X (i) , i = 1, . . . , n, and morphisms Fi : X (i) → Y defined over κ such that: (a) for each i = 1, . . . , n there exists an isomorphism Gi : X (i) → X, defined over κ ¯ , with F ◦ Gi = Fi and (b) n [ Y (κ) = Fi (X (i) (κ)). i=1

A typical example is provided by isogenies between algebraic groups; other examples in the compact case, which necessarily concern higher dimensions, are provided in [25], Ch. III, §8. In the affine case, a crucial instance is provided by isogenies of linear tori. Take for instance the squaring map Gm → Gm sending x 7→ x2 . The variety Gm = P1 {0, ∞} is affine and its integral points, over a ring of S-units O S , form the (4)

The so called Hilbert Irreducibility Theorem asserts precisely that for a κ-rational variety Y and a generically finite morphism F : X → Y there always exists a Zariski-dense set of rational points of Y whose pre-images consist of a single Galois orbit. See [16], Ch. 4 or [25], Ch. III.

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group of units O ∗S . As a consequence of Dirichlet’s unit theorem, this group is finitely generated, hence its subgroup of squares has finite index. It follows that there exist units 1 , . . . , k ∈ O ∗S = Gm ( O S ) such that each element of Gm ( O S ) is of the form i u2 for some u ∈ Gm ( O S ). Then the two conclusions of Theorem 2.3 easily follow: the √ √ first one by putting κ0 = κ( 1 , . . . , k ); the second one by taking X (i) = Gm for each i = 1, . . . , k and Fi : X (i) → Gm being the morphism Fi (x) = i x2 . An example of this kind in the compact case is provided by elliptic curves. Start with the Legendre model of an elliptic curve E y 2 = x(x − 1)(x − λ), where λ ∈ κ {0, 1}, κ being a number field. Let us choose a finite set of places S so large that it contains the archimedean places and λ and 1 − λ are both S-units. Then for each rational point (a, b) ∈ E(κ) and each valuation ν outside S the ν-adic valuation of a and of a − 1 is even. Then the square roots of a and of a − 1 generate an extension of κ which is unramified at ν. Since, by Minkowski’s theorem, there are only finitely extensions of fixed degree and unramified outside a given finite set, all the square roots of a and a − 1 lie in a number field κ0 . The Chevalley-Weil Theorem is the first tool in the proof of the finite generation of the group of rational points on an elliptic curve (Mordell-Weil theorem). The second tool is the theory of heights on elliptic curves (Néron-Tate height). We can now prove the Mordell-Weil Theorem. Fix an elliptic curve E over a number field κ. Consider the multiplication-by-2 map E → E; being an unramified cover of E, we can apply the Chevalley-Weil theorem deducing the existence of finitely many κ-twists of this morphism such that each rational point on E lifts to at least one of them. In concrete terms, there exist finitely many points A1 , . . . , Ak ∈ E(κ) such that each rational point P ∈ E(κ) is of the form P = Ai + 2Q, for some rational point Q ∈ E(κ). From properties (ii) and (v) of the canonical height it follows that there exists ˆ ) > H and P = 2Q + Ai , h(Q) ˆ ˆ ). Let a number H such that whenever h(P < h(P now Γ ⊂ E(κ) be the subgroup generated by A1 , . . . , Ak and all rational points of height ≤ H. We claim that this group coincides with E(κ). Actually, suppose not and ˆ ) > H and so P can let P be the rational point of smallest height outside Γ. Then h(P ˆ ˆ ), so that, by minimality of P , be written as P = 2Q + A with A ∈ Γ and h(Q) < h(P also Q must belong to Γ. Then P too belongs to Γ, and this contradiction concludes the proof. 2.3. Diophantine approximation on the line. – In this section, we present without proof classical material about Diophantine approximation, mainly following [16]. More details and complete proofs can be found for instance in [58], [59], [10], [14]. We are primarly interested in the rational approximation to algebraic numbers; more precisely, we are interested in estimating the accuracy in the approximation to such numbers with respect to the denominator of the approximant. The following theorem gives the best possible result for an arbitrary irrational number.

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Theorem 2.4 (Dirichlet). – Let α ∈ R Q be a real irrational number. There exist infinitely many rational numbers a/b (a, b coprime integers, b > 0) such that a 1 α − < 2 . b b For instance, one can take for a/b the truncated continued fraction expansion of α. Some irrational numbers P∞ can be approximated to a higher degree; for instance, Liouville’s number α := n=1 10−n! has the property that for every positive µ there exist infinitely many rationals a/b (a, b coprime integers, b > 0) such that a 1 α − < µ . b b Such numbers are never algebraic; actually, a theorem of Liouville, admitting an elementary proof, states that: Theorem 2.5 (Liouville). – Let α be a real irrational algebraic number of degree d over Q. There exists a positive number c(α) such that for all rational numbers a/b a c(α) α − ≥ d . b b A theorem due to Roth (1955) [54], which is much harder to prove, improves on Liouville’s exponent d: Theorem 2.6 (Roth’s Theorem). – Let α be a real algebraic number,  > 0. For all but finitely many rational numbers a/b, the following inequality holds: a 1 (2.7) α − > 2+ . b b In an other formulation: if α is algebraic irrational, there exists a positive real number c(α, ) such that for all rational numbers a/b, a c(α, ) (2.8) α − > 2+ . b b Roth’s proof is ineffective, in the sense that it does not provide any means of finding the finitely many rational numbers a/b which violate the inequality (2.7). Looking at its second formulation, by the ineffective nature of Roth’s proof it is not possible to calculate the function c(α, ). Roth’s theorem is best possible as far as the exponent is concerned in view of the mentioned result of Dirichlet (Theorem 2.4). However, one can try to improve on Roth’s exponent after restricting the approximations to suitable classes of rational numbers. For instance, one can consider the set of rational numbers which, once written in base ten, have only finitely many digits. These numbers form the ring 1 ] = Z[ 12 , 15 ]. of S-integers Z[ 10 For these approximations, Ridout [52] improved Roth’s bound by proving that: for every irrational algebraic number α and every positive real  > 0, there are only finitely many pairs of integers (a, n) ∈ Z × N such that |α − 10an | < 10−(1+)n .

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A similar result holds whenever the numerators of the approximations are supposed to be of special type, e.g., products of powers of primes from a fixed finite set. When both numerators and denominators are subject to lie in a finitely generated multiplicative semi-group, then the exponent can be lowered to “” (see Corollary 2.13). In another direction, one can try to replace the rational number field Q by an arbitrary number field κ ⊂ C. Of course, the expected exponent should change; for instance, if κ ⊂ R and has degree d = [κ : Q] over the rationals, a variation of Dirichlet’s theorem asserts that each real number α ∈ R κ can be approximated to a degree −2d with respect to the height. However, our care in chosing the normalization of absolute values and heights assure that, with respect to our choice, the exponent in Roth’s theorem remains the same, as in the following statement: Theorem 2.9. – Let κ be a number field, ν be a place of κ and α ∈ κν be an element of the topological closure of κ, algebraic over κ but not lying in κ. Let | · |ν denote the absolute value normalized with respect to κ and extended to κν . Then for every positive real number  > 0 there exists a positive number c(α, ν, ) such that for all β∈κ |α − β|ν > c(α, ν, ) · H(β)−2− . Let us consider the particular case where ν is archimedean and κ ⊂ κν = R. While generic real numbers can be approximated by a sequence of rationals with an error bounded by Dirchlet’s Theorem, we expect that using as approximants elements of κ instead of only rational numbers the degree of approximability of any real number will increase. Since κ is a vector space of dimension [κ : Q] over Q, it should be possible to make the error in the approximation as little as the height of the approximant to the power −2[κ : Q]. Actually this is true, and can be proved via the classical pigeon-hole principle. However, in Theorem 2.9 above the usual exponent 2 appears; taking into consideration our normalization, the same inequality written with respect to the usual real absolute value would show precisely the exponent −2[κ : Q]; so Theorem 2.9 states that for algebraic numbers no improvement on Dirichlet’s exponent can be obtained. The most general version of Roth’s Theorem, encompassing both Ridout’s theorem and the above Theorem 2.9, was formulated by Lang in [44]: Theorem 2.10. – Let κ be a number field; let S be a finite set of places of κ. Let, for every ν ∈ S, | · |ν be the extension of the ν-adic absolute value to κν , normalized with respect to κ and let αν ∈ κν be an algebraic number. For every  > 0 there exists a positive number c = c(S, (αν )ν∈S , ) such that for all β ∈ κ with β 6= αν for every ν ∈ S, Y |αν − β|ν > c · H(β)−2− . ν∈S

Notice that interesting cases arise even when some, or even all, the αν lie in κ. Indeed, another equivalent formulation of the general Roth’s Theorem 2.10 involves only κ-rational points. It appears e.g., in [14] and reads as follows:

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Theorem 2.11. – Let κ be a number field, d ≥ 1 an integer, α1 , . . . , αd be pairwise distinct elements of κ. Let S1 , . . . , Sd be pairwise disjoint finite sets of absolute values. Finally, let  > 0 be a positive real number. Then for all but finitely many elements β ∈ κ, (2.12)

d Y Y

|αh − β|ν > H(β)−2− .

h=1 ν∈Sh

The above theorem can be further generalized, by allowing also points at infinity as target of the approximation. This will be useful in order to deduce the mentioned theorem of Ridout. Precisely, for α = ∞ and any absolute value ν, let us define the ν-adic distance from α to β ∈ κ, provided β 6= 0, by putting |α − β|ν = |∞ − β|ν := |β|−1 ν . Then the condition that a rational number β ∈ Q be of the form β = a/b where b is a product of primes from a fixed set T can be expressed by the inequality Q −1 ; if |β| ≤ 1 we also have H(β) = |b| so the arithmetic ν∈T min(1, |β − ∞|ν ) ≤ |b| condition that β lies in a fixed ring of S-integers is equivalent to the inequality Y min(1, |β − ∞|ν ) ≤ H(β)−1 , ν∈T

where T ⊂ S is the set of ultrametric places in S. Actually, the generalization of Theorem 2.11 with one point α allowed to be at infinity follows formally from the present version of Theorem 2.11 itself: observe that applying projective transformations Φ : P1 → P1 of the form Φ(x) =

ax + b , cx + d


where ac db ∈ GL2 (κ) one can send the given set of target points {αν }ν∈S ⊂ P1 (κ) = κ ∪ {∞} to a subset of κ = P1 (κ) {∞}. For instance, in the special case in which the set of {αν , ν ∈ S} consists of the three rational points 0, 1, ∞ ∈ P1 (κ), the above Theorem 2.11 implies: Corollary 2.13. – Let Γ ⊂ κ∗ be a finitely generated multiplicative group. Let T be a finite set of places of κ and  > 0 a positive real number. Then for all but finitely many γ ∈ Γ Y (2.14) |γ − 1|ν > H(γ)− . ν∈T

The proof of the deduction from Theorem 2.11 can be found e.g., in [16], Ch. II. Let us recall that from the theory of linear forms in logarithms (Baker’s method) one can prove a stronger result, replacing the right-hand side term in (2.14) by some (negative) power of the logarithmic height of the approximant. In the rational case, we state the following further corollaries:

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Corollary 2.15 (Theorem of Ridout). – Let {p1 , . . . , pl }, {q1 , . . . , qm } be two set of prime numbers; let λ, µ be real numbers in the closed interval [0, 1]. Let us consider the set B of rational numbers β of the form β = p/q where pa1 1 · · · pal l · p∗

p

=

q

bm = q1b1 · · · qm · q∗ ,

where a1 , . . . , al , b1 , . . . , bm are integers with ai ≥ 0, bj ≥ 0 and p∗ , q ∗ satisfy p∗

≤

q∗

≤ q 1−µ .

p1−λ

Let α ∈ R be a real algebraic number and let  > 0 be a positive real number. Then for all but finitely many β ∈ B, |α − β| > H(β)−2+λ+µ− . Corollary 2.16. – Let p be a prime number, α ∈ Zp a p-adic algebraic integer. For every  > 0 there exist only finitely many integers n ∈ Z such that |n − α|p < |n|−1− . We end this section by providing yet another version of Roth’s theorem; we shall present it as a lower bound for homogeneous linear form. Theorem 2.17 (Homogeneous Roth’s Theorem). – Let κ be a number field, S be a finite set of absolute values of κ. For each ν ∈ S, let L1,ν (X, Y ), L2,ν (X, Y ) be linearly independent linear forms with coefficients in κ. Finally, let  > 0 be a positive real number. For all but finitely many (x : y) ∈ P1 (κ) the following inequality holds: Y |L1,ν (x, y)|ν |L2,ν (x, y)|ν (2.18) · > H(x/y)−2− . max(|x|ν , |y|ν ) max(|x|ν , |y|ν ) ν∈S

Note that, due to the appearance of the denominator max(|x|ν , |y|ν ), the left handside term is invariant by multiplication of x and y by a non-zero constant, so it only depends on the projective class (x : y) of (x, y). This is consistent with the right-hand side term, which only depends on the ratio x/y. The left-hand side term in the inequality of Theorem 2.17 can be viewed as the product of distances from the approximating points (x : y) ∈ P1 (κ) to the points defined by the vanishing of the linear forms L1,ν , L2,ν . If Q = (a : b) ∈ P1 (κ) is a point and L(x, y) = bx − ay is a linear form vanishing on (a : b), we can define the distance between a point P := (x : y) ∈ P1 and the point Q to be (2.19)

distν (P, Q) =

|L(x, y)|ν . max(|x|ν , |y|ν )

Of course this quantity depends on the chosen equation for Q, but this choice affects the outcome just by a multiplicative constant independent of P .

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Coming back to Theorem 2.17, where for each place ν two ν-adic linear forms are involved, let us remark that by the triangle’s inequality, only one of the linear forms can be “small” at one single point (x : y). If Q1 , Q2 are the zeros of L1 (X, Y ), L2 (X, Y ) respectively, and the sequence (x : y) converges to Q1 , then |L2 (x,y)|ν asymptotically max(|x| → distν (Q1 , Q2 ) > 0. ν ,|y|ν ) Hence Theorem 2.17 can be rephrased by saying that given rational points Qν for ν ∈ S, for all  > 0 the lower bound Y distν (P, Qν ) > H(P )−2− ν∈S

holds for all but finitely many rational points P ∈ P1 (κ). 2.4. Diophantine approximation on elliptic curves. – Given an elliptic curve E over a number field κ, and a place ν of κ, one can define a distance on the compact topological space E(κν ). Several possibilities are available, all being equivalent for our purposes: one can for instance define a metric in the projective plane P2 (κν ) and take the induced one on E(κν ). Alternatively, if the place ν is archimedean, and corresponds to an embedding κ ,→ C of κ into the complex number field C, one can view E(C) as a quotient C/Λ of the complex plane C by a lattice Λ and define locally the metric as the one induced from the archimedean metric in C. In any case, given a sequence {Pn }n∈N in E, converging ν-adically to a point Q ∈ E(κν ), a distance will be fixed in a neighborhood of Q in such a way that for a local parameter t at Q, |t(Pn )|ν  dist(Pn , Q)  |t(Pn )|ν . Now, the standard Roth’s theorem on the line, e.g., in the version of Theorem 2.6, immediately provides a lower bound of the form (2.20)

dist(Pn , Q)  H(Pn )−2−

for every sequence Pn converging (in an archimedean place, say) to an algebraic target. Since the Euler characteristic of an elliptic curve is 0, while for the projective line P1 it is −2, it is natural that the exponent −2 −  of Roth’s Theorem is replaced in the elliptic case by a − exponent. This refined inequality can in fact be proved, by combining the Mordell-Weil Theorem with Roth’s Theorem: Theorem 2.21. – Let E be an elliptic curve over a number field κ, ν a valuation of κ and Q ∈ E(κν ) be an algebraic point. For every  > 0 there exists a real number c > 0 such that for every point P ∈ E(κ), P 6= Q, (2.22)

distν (P, Q) ≥ c · H(P )− .

Proof. – Let  > 0 be given. Suppose by contradiction that (2.22) does not hold for any real number c > 0. Then in particular there would exist infinitely many rational points P ∈ E(κ) such that (2.23)

distν (P, Q) < H(P )− .

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Choose an integer m > 0 such that 4 < . m2

(2.24)

Since the quotient group E(κ)/mE(κ) is finite, we can find a point F ∈ E(κ) and infinitely many solutions P to (2.23) of the form P = F + mP 0 , for some P 0 ∈ E(κ). By compactness of the topological group E(κν ), there exists a number c1 > 1 such that 0 0 0 c−1 1 distν (mP , Q − F ) < distν (F + mP , Q) < c1 distν (mP , Q − F ),

so that the solutions to (2.23) with P = F + mP 0 give rise to solutions to the equation distν (mP 0 , Q − F ) < c1 H(P )− . Let now Q1 , . . . , Qm2 ∈ E(¯ κ) be the solutions X to the equation mX = Q − F ; if a sequence of points of the form mP 0 converges to Q − F , then the corresponding sequence of the points P 0 admits a subsequence converging to one of the points Q1 , . . . , Qm2 (after suitably extending the valuation ν to κ ¯ ). Choose one such point Q0 ∈ {Q1 , . . . , Qm2 } ∩ E(κν ). Since the map E(κν ) 3 X 7→ mX ∈ E(κν ) is unramified, we have (using again the compactness of E(κν )) that for some number c2 > 1 0 0 0 0 0 c−1 2 distν (P , Q ) ≤ distν (mP , Q − F ) ≤ c2 distν (P , Q ).

Then the solutions to (2.23) give rise to infinitely many solutions to the equation distν (P 0 , Q0 ) ≤ c3 H(P )− = c3 H(mP 0 + F )− , for some real number c3 . Taking logarithms (for simplicity of notation) we write (2.25)

− log distν (P 0 , Q0 ) ≥ h(mP 0 + F ) − log c3 .

But we know from the properties of the Néron-Tate height, there exist numbers c4 = c4 (F ), c5 (F ), independent of P 0 , such that p h(mP 0 + F ) ≥ m2 h(P 0 ) − c4 m h(P 0 ) − c5 . From the inequality (2.25) and the above inequality it follows that p − log distν (P 0 , Q0 ) ≥ m2 h(P 0 ) − c4 m h(P 0 ) − c5 − log c3 , and from the the inequality (2.24) satisfied by m we have p − log distν (P 0 , Q0 ) ≥ 4 h(P 0 ) − c4 m h(P 0 ) −  c5 − log c3 > 3 h(P 0 ) whenever h(P 0 ) is sufficiently large. Hence, from the infinitude of the set of solutions to (2.23) we obtain infinitely many solutions to the inequality distν (P 0 , Q0 ) < H(P 0 )−3 contradicting (2.20).

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The above proof carries out also on the multiplicative group, giving rise to an alternative proof of Theorem 2.13, which can then be formally deduced from Roth’s Theorem 2.6 (making use of the finite generation of the group of S-units). Also, a weaker version of Roth’s Theorem would be sufficient; actually, the first proof of 2.13 is due to Gelfond [38], who already in 1952 proved the inequality (2.14), using an approximation result of his own in place of the yet unavailable Roth’s Theorem. Once again, the exponent − appearing in (2.14) instead of −2 −  is justified by the fact that the Euler characteristic of Gm is 0. 2.5. The gap principle on the line and on elliptic curves. – The so-called gap principle in Diophantine approximation is the elementary but crucial fact that the ratio of the heights of two “very good” rational approximations to a single real number can be bounded from below (see inequality (2.26)). An important theorem of Mumford, which constitutes a first step toward the Vojta-Bombieri proof of Mordell’s conjecture (see e.g., [10]), provides a compact analogue to that inequality. Suppose we have two rational numbers p1 /q1 , p2 /q2 , where p1 , q1 (resp. p2 , q2 ) are coprime integers, with 0 < q1 < q2 and suppose that α ∈ R is a real number. If for some exponent µ > 0 the two inequalities α − p1 < 1µ , α − p2 < 1µ q1 q1 q2 q2 hold, then by the triangle’s inequality p1 − p2 < 1µ + 1µ ≤ 2µ . q1 q2 q1 q2 q1 Writing the left-hand side above with a common denominator, one obtains |p1 q2 − p2 q1 | 2 < µ. q1 q2 q1 On the other hand, the determinant at the numerator is non-zero, due to the fact that the approximations p1 /q1 , p2 /q2 are distinct, so its absolute value is at least 1. We deduce 1 (2.26) q2 > · q1µ−1 . 2 Now, if the exponent µ satisfies µ > 2 the inequality is non-trivial and leads to the following: Theorem 2.27 (Gap Principle). – Let α ∈ R be any real number. Let µ > 2 be a real number and p1 /q1 , p2 /q2 , . . . a sequence of rational numbers with 0 < q1 < q2 < · · · , satisfying for all n = 1, 2, . . . α − pn < 1µ . qn qn Then lim inf n→∞

log qn+1 ≥ µ − 1. log qn

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We remark at once that we do not suppose that the target α is algebraic. Also, the result remains true, but trivial, if µ ≤ 2, whenever when µ > 2 it says that the sequence of the approximations grows at least exponentially with n. Note that we are considering approximations by rational numbers, i.e., by points on P1 , and that χ(P1 ) = −2; this is the reason why the theorem requires that µ > 2 in order to be non-trivial. The p-adic versions of Roth’s theorem also admit corresponding gap principles. Take a prime number p; Theorem 2.9 states that for every algebraic p-adic number α ∈ Qp and every  > 0, there are only finitely many rational solutions a/b, a, b ∈ Z, b 6= 0, to the inequality a α − < max(|a|, |b|)−2− . b p Take now an arbitray p-adic number α, possibly transcendental. Then the above inequality can admit infnitely many solutions. However, given two solutions a1 /b1 , a2 /b2 with max(|a1 |, |b1 |) < max(|a2 |, |b2 |) to the above inequality, we obtain that |a2 b1 − a1 b2 |p < max(|a1 |, |b1 |)−2− , while clearly |a2 b1 −a1 b2 | ≤ 2 max(|a1 |, |b1 |)·max(|a2 |, |b2 |). It follows from the product formula (i.e., from the fact that no power of p can divide any non-zero number which is smaller than that power) that max(|a1 |, |b1 |)−2− · 2 max(|a1 |, |b1 |) · max(|a2 |, |b2 |) ≥ 1, i.e., max(|a2 |, |b2 |) > 12 max(|a1 |, |b1 |)1+ . This is the sought gap inequality. If, on the other hand, we are interested in approximating a p-adic integer α ∈ Zp by rational integers, as it was the case in Corollary 2.16 to Ridout’s theorem, we shall consider an inequality of the type (2.28)

|α − m|p < |m|−1− ,

to be solved in integers. Suppose 0 < m < n are two solutions. From the (ultra-metric) triangle’s inequality we obtain |n − m|p ≤ m−1− . On the other hand, the maximal power of p dividing the non-zero integer n−m cannot exceed n − m. We then obtain n − m ≥ m1+ , so in particular n > m1+ . The difference between the two last situations is that the approximating numbers in the second case are integers, so they are automatically close to infinity in the infinite place of Q. From another view point, we are doing approximation on the affine line, which has Euler characteristic −1, hence the gap principle only requires inequality (2.28) to give a non-trivial conclusion.

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A generalization, involving several places and arbitrary number fiels, reads as follows: Theorem 2.29. – Let κ be a number field, S a finite set of places of k. For each place ν ∈ S, let αν be a point in κν and µν a positive real number. Suppose that X µν = 2 +  > 2. ν∈S

Let β1 , β2 ∈ κ be two solutions of the system of inequalities (2.30)

|αν − β|ν < max(1, |2|ν )−1 · H(β)−µν

with 0 < h(β1 ) ≤ h(β2 ). If β1 6= β2 then h(β1 ) < h(β2 ) and h(β2 ) ≥ 1 + . h(β1 ) The proof mimics the three particular cases already analyzed. From this result, one can deduce a gap principle for the solutions to the slightly different inequality of the form appearing in Theorem 2.10. Namely one can prove the following Corollary 2.31. – Let κ, S be as above, and for each ν ∈ S, αν be as before a point in the completion κν . Let  > 0 be a real number. Let β1 , β2 , . . . be a sequence of rational points in κ with h(β1 ) ≤ h(β2 ) ≤ · · · satisfying Y (2.32) |αν − β|ν < H(β)−2− . ν∈S

Then there exists a real number δ > 0 and an integer N = N (|S|, , δ) such that for all large integers n h(βn+N ) > 1 + δ. h(βn ) The idea for deducing Corollary 2.31 from Theorem 2.29 is that the inequality (2.32) implies one of the finitely many systems of inequalities like (2.30), up to “shrinking ”. The details, in quantitative form, appear e.g., in §3.4 of [14]. Clearly, doubling N one can replace δ by 2δ + δ 2 = (1 + δ)2 − 1, so the conclusion of the Corollary holds for every δ. As mentioned in the introduction and just explained above, the reason for the exponent −2 in Equation (2.32) is that the approximation takes place on the projective line, whose Euler characteristic is precisely −2. It is then natural to expect that on elliptic curves the exponent can be lowered to “0 + ”; actually, given a reasonable notion of distance on elliptic curves one expects the following theorem to hold:

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Theorem 2.33. – Let E be an elliptic curve over a number field κ. Let ν be a place of κ and A ∈ E(κν ) be a point defined over the corresponding completion κν . Let  > 0 be a positive real number and let P1 , P2 , . . . ∈ E(κ) be a sequence of κ-rational points of E satisfying h(P1 ) < h(P2 ) < · · · and distν (A, Pn ) < H(Pn )−

(2.34)

for all n = 1, 2, . . .. Then there exist an integer N ≥ 1 and a real δ > 0 such that for all n h(Pn+N ) > 1 + δ. h(Pn ) As for Corollary 2.31, the conclusion could be rephrased by saying that for every positive number C there exists an integer N such that for all solutions P1 , P2 , . . . to the inequality (2.34) (ordered by increasing height), h(Pn+N ) > C · h(Pn ). We provide a detailed proof of Theorem 2.33, since we cannot locate this statement, nor its proof, anywhere in the literature. We start by proving the following Proposition, from which Theorem 2.33 will follow rather formally: Proposition 2.35. – Let E be an elliptic curve over a number field κ, ν a valuation of κ and A ∈ E(κν ) and m ≥ 2 an integer. Let  > 0 be a positive real number. There exists a number c = c(E, ν, m, ) such that for all but finitely many rational points ˆ ) < h(Q), ˆ P, Q ∈ E(κ) satisfying distν (A, P ) < H(P )− , distν (A, Q) < H(Q)− , h(P and P − Q ∈ mE(κ), ˆ h(Q) ≥ m2 − 2. ˆ ) h(P

(2.36)

The result is non-trivial whenever m2 > 3, i.e., for all sufficiently large values of m. Proof. – By assumption, there exists a rational point B ∈ E(κ) such that P, Q can be written in the form P = B + mP 0 ,

Q = B + mQ0 ,

for rational points P 0 , Q0 ∈ E(κ). Moreover, this point B can be chosen in any set of representatives of the group E(κ)/mE(κ), which is finite by the weak Mordell-Weil Theorem. In the sequel of the proof, we let C1 , C2 , . . . denote positive numbers depending only on E, m, κ and ν (as well as, of course, on the notion of ν-adic distance). By the properties of distances, we have distν (A, mP 0 + B) < H(P )− ⇒ distν (A − B, mP 0 ) < C1 H(P )− and analogously for Q, so, using for simplicity the logarithmic notation, under the hypotheses of the proposition − log distν (A−B, mP 0 ) > h(P )−log C1 ,

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− log distν (A−B, mQ0 ) > h(Q)−log C1 .

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ˆ ) < h(Q), ˆ By the triangle’s inequality and the fact that h(P we obtain (taking also into account that the logarithmic canonical height differs from the logarithmic naive height by a bounded function) ˆ )) − log C2 . − log distν (mP 0 , mQ0 ) > (h(P Recall that the pre-image of mP 0 by the multiplication-by-m map is formed by m2 points of the form P 0 + F , where F is a point of m-torsion. Now, since the multiplication-by-m map and translations are unramified, we obtain from the above displayed inequality that, for a suitable m-torsion point F , (2.37)

ˆ ) − log C3 . − log distν (P 0 , Q0 + F ) ≥ h(P

Now we shall use Liouville’s inequality, which we write in terms of the logarithmic Néron height as ˆ 0 ) + h(Q ˆ + F 0 )) + log C4 = (h(P ˆ 0 ) + h(Q ˆ 0 )) + log C4 , − log distν (P 0 , Q0 + F ) ≤ (h(P since F is of finite order. Comparing with (2.37), we get (2.38)

ˆ ) < h(P ˆ 0 ) + h(Q ˆ 0 ) + log C5 . h(P

Now recall that P = mP 0 + B, so, by the properties of heights, taking into account that B can be chosen in a finite set depending only on m, we obtain   q C 6 ˆ ) ≥ m2 · h(P ˆ 0 ) − C6 · h(P ˆ 0 ) = m2 · h(P ˆ 0 ) 1 −  q h(P 2 0 ˆ m h(P )   ˆ 0 ) 1 − q C7  = m2 · h(P ˆ 0) h(P (recall that C6 , C7 depend on m) and the same comparison the heights q holds for q 0 0 ˆ ˆ 0 ) are of Q, Q . The above inequalities are meaningful only if h(P ) and h(Q both > C7 , which we can certainly assume up to disregarding finitely many pairs ˆ ), we get P, Q ∈ E(κ). Inserting in (2.38), multiplying by m2 and dividing by h(P  −1  −1 ˆ C7  h(Q) C7  m2 + 1 − q (2.39) m2 < 1 − q + log C5 · . ˆ ) ˆ ) h(P h(P ˆ 0) ˆ 0) h(P h(Q q q ˆ 0 ), C7 / h(Q ˆ 0 )); then the above inequality implies Set now t = max(C7 / h(P ! ˆ h(Q) m2 m2 < (1 − t)−1 1 + + log C5 · . ˆ ) ˆ ) h(P h(P ˆ ) → ∞, and the same is true of the last Observing that t tends to zero for h(P ˆ ) is addend, we see immediately that the above inequality implies (2.36) whenever h(P sufficiently large.

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Proof of Theorem 2.33. – We can now finish the proof of the elliptic Gap Principle (Theorem 2.33). Let m > 1 be an integer such that m2  > 3. We let {A1 , . . . , Ah } be a set of representatives of E(κ) modulo mE(κ). Set N = h; then in every finite sequence Pn , Pn+1 , . . . , Pn+N of rational points in E(κ) there are two points Pi =: P and Pj =: Q, with n ≤ i < j ≤ n + N such that P − Q is divisible by m in E(κ). Any ˆ ˆ ) applies a fortiori to the ratio h(P ˆ n+N )/h(P ˆ n ). lower bound for the ratio h(Q)/ h(P 2 Now, fix a number δ with 0 < δ < m − 3. Proposition 2.35 provides the lower bound ˆ ˆ ) ≥ m2 −2 which implies, for large values of h(P ), that h(Q)/ ˆ ˆ ) > (1+δ), h(Q)/ h(P h(P concluding the proof.

3. Higher dimensional Diophantine approximation In higher dimensions, we shall be interested in approximating hyperplanes defined by linear forms with algebraic coefficients by rational points. We shall adopt the language and notation of projective geometry for simplicity, as in the homogeneous version of Roth’s Theorem given in Theorem 2.17. The main result of this section is the so-called Subspace Theorem, first proved, in a particular case, by W. M. Schmidt in the seventies. Here we formulate the generalization provided by H.-P. Schlickewei, which is the natural extension of Roth’s theorem to higher dimensions. We need an extension of the notion of height to algebraic points in projective spaces. Let κ be a number field, x = (x0 , . . . , xN ) ∈ κN +1 {0} a non-zero vector. For every place ν of κ, its ν-adic norm kxkν is defined to be kxkν = max(|x0 |ν , . . . , |xN |ν ). The height of the associated projective point, still denoted by x = (x0 : · · · : xN ) ∈ PN (κ), is set to be Y H(x) = kxkν , ν

where the product runs over all the valuations of κ. With these conventions, Schmidt’s Subspace Theorem reads: Theorem 3.1 (Subspace Theorem). – Let N ≥ 1 be positive, κ be a number field and S a finite set of places of κ. Let, for every ν ∈ S, L0,ν (X0 , . . . , XN ), . . . , LN,ν (X0 , . . . , XN ) be linearly independent linear forms with algebraic coefficients in κν . Then for each  > 0 the solutions x = (x0 : · · · : xN ) ∈ PN (κ) to the inequality (3.2)

N YY |Li,ν (x)|ν ν∈S i=0

kxkν

< H(x)−N −1−

lie in the union of finitely many hyperplanes of PN , defined over κ.

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For N = 1, the conclusion provides the finiteness of the solutions to the inequality (3.2); so we recover Roth’s Theorem. In higher dimension, however, the finiteness conclusion does not hold: for instance, when the point x lies in the hyperplane defined by the vanishing of one linear form, the left-hand side term in (3.2) vanishes, so the inequality is satisfied. It is worth noticing, however, that the exceptional hyperplanes containing the infinite families of solutions are not necessarily the zero sets of the involved linear forms, as the following example shows: Example. – Let α be a real irrational algebraic number, with 0 < α < 1; consider a “good” rational approximation p/q ∈ Q to α. By this we mean that p, q are coprime integers, q > 0, and α − p < 1 ; q q2 we know from Dirichlet’s Theorem that there exist infinitely many of them. Since α < 1, for infinitely many good approximations p/q one has max(|p|, |q|) = |q|, so we can write the above inequality as α − p < max(|p|, |q|)−2 . q For each such pair (p, q) we have the upper bound |qα − p| |qα − p| ≤ < max(|p|, |q|)−2 . max(|p|, |q|) |q|

(3.3)

Now take N = 2, κ = Q and S consisting of the archimedean absolute value of Q and define the three linear forms Li (X0 , X1 , X2 ) (i = 0, 1, 2) as follows: L0 (X0 , X1 , X2 ) = X0 − αX2 ,

L1 (X0 , X1 , X2 ) = X1 − αX2 ,

L2 (X0 , X1 , X2 ) = X2 .

Now, with each good approximation p/q to the number α as above we associate the point (x0 : x1 : x2 ) = (p : p : q). Then the double product in (3.2) becomes 2  N YY |Li,ν (x)|ν |q| |p − qα| · . = kxk max(|p|, |q|) max(|p|, |q|) ν i=0 ν∈S

By the above inequality (3.3) and the trivial estimate |q| ≤ max(|p|, |q|), we have the upper bound N YY |Li,ν (x)|ν < max(|p|, |q|)−4 , kxk ν i=0 ν∈S

which means that inequality (3.2), with e.g.,  = 1/2, admits infinitely many solutions (x0 : x1 : x2 ) = (p : p : q) on the projective line of equation X0 = X1 . So, the degeneracy conclusion of Theorem 3.1 cannot be replaced by a finiteness one, even after assuming Li,ν (x) 6= 0. It will prove useful to have an ‘affine version’ of the Subspace Theorem, of which Theorem 3.1 represents the projective, or homogeneous, version. Here is such a version, which can be formally deduced from Theorem 3.1:

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Theorem 3.4. – Let κ be a number field, S a finite set of places containing the archimedean ones, N ≥ 2 an integer. Let, for each ν ∈ S, Lν,1 (X1 , . . . , XN ), . . ., Lν,N (X1 , . . . , XN ) be linearly independent linear forms with algebraic coefficients in κν . Then the solutions (x1 , . . . , xN ) ∈ O N S to the inequality N YY

|Lν,i (x)|ν < H(x)−

ν∈S i=1

lie in the union of finitely many proper linear subspaces of κN . The Subspace Theorem, like Roth’s theorem, is ineffective; however, the number of the higher dimensional components of the Zariski-closure of the set of solutions to (3.2) can be bounded (see [32]). An interesting issue on higher dimensional Diophantine approximation theory concerns approximation to non-linear hypersurfaces. Given a hypersurface D ⊂ Pn , defined by a homogenous equation F (x0 , . . . , xn ) = 0, and a place ν of a field, we can define the distance from a point P = (p0 : · · · : pn ) to the hypersurface D relatively to the place ν as |F (p0 , . . . , pn )|ν distν (P, D) = ; F k(p0 , . . . , pn )kdeg ν changing the equation for D affects the distance function by a multiplicative constant. In this context, Vojta’s Main Conjecture (see [65] or [13]) predicts the following: Conjecture (Vojta’s Main Conjecture). – . Let S be a finite set of places of a number field κ; for each ν ∈ S, let Dν be a hypersurface of Pn (possibly reducible) with normal crossing singularities, defined over κ. Let  > 0 be a positive real number. There exists a proper closed subvariety Z ⊂ Pn and a number c > 0 such that for all rational points P ∈ (Pn Z)(κ) Y distν (P, Dν ) > c · H(P )−n−1− . (3.5) ν

This is a reformulation of a particular case of Conjecture 3.4.3 from [66] (the Main Conjecture in [66]) or Conjecture 15.5 from [13]). The original Vojta’s conjecture applies to hypersurfaces of any smooth projective variety; in that case the right-hand side takes into consideration the canonical bundle of the variety. The following theorem has been proved independently by Evertse-Ferretti in [30] and by Corvaja-Zannier in [21]: Theorem 3.6. – Let S be a finite set of places of a number field κ, and for each place ν ∈ S, Fν (x0 , . . . , xn ) ∈ κ[x0 , . . . , xn ] be a homogeneous form. Let  > 0 be a positive real number. Then for all rational points P = (x0 : · · · : xn ) outside a proper Zariski closed subset, the following inequality holds: ! Y |Fν (x0 , . . . , xn )|ν1/ deg Fν > H(P )−n−1− . k(x0 , . . . , xn )kν ν∈S

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In the case of polynomials of degree 1 the result is best-possible, and is a particular case of the Subspace theorem. As for the subspace theorem, one can consider approximating several hypersurfaces with respect to a same valuation. Also, one can try to improve on the exponent on the right-hand side working with approximants on a fixed algebraic subvariety. The most general result obtained so far is the following theorem of Evertse and Ferretti from [30]: Theorem 3.7. – Let X ⊂ Pn be a projective variety over a number field κ. Let S be a finite set of places of κ. Let F1 , . . . , Fq ∈ κ[x0 , . . . , xn ] be homogeneous forms with coefficient in κ such that the hypersurfaces of X defined by the vanishing of F1 , . . . , Fq are in general position. Let  > 0 be a positive real number. Then for all rational points P = (x0 : · · · : xn ) ∈ X(κ) outside a proper Zariski closed subset of X the following inequality holds: ! q 1/ deg Fν YY |Fν (x0 , . . . , xn )|ν min 1, > H(P )− dim X−1− . k(x , . . . , x )k 0 n ν i=1 ν∈S

It is easy to deduce from the above statement analogues lower bounds for nonhomogeneous polynomials and integral points. For instance, the following result appears in [21] Theorem 3.8. – Let X ⊂ An be an affine algebraic variety defined over a number field κ. For each place ν in a finite set of places S, containing the archimedean ones, let fν ∈ κ[x1 , . . . , xn ] be a polynomial of degree d > 0. For each  > 0 there are only finitely many integral points x ∈ X( O S ) such that Y 0< |fν (x)|ν < H(x)−d(dim X−1)− . ν∈S

In the above statement, the (affine) height H(x) of x = (x1 , . . . , xn ) ∈ An coincides with the projective height of the point (1 : x1 : · · · : xn ) ∈ Pn . By known argument involving Galois conjugates of polynomials and places, one can deduce from the above the following Corollary 3.9. – Let X ⊂ An be an affine algebraic variety defined over Q. ¯ 1 , . . . , xn ] be a polynomial with algebraic coefficients. The set Let f (x1 , . . . , xn ) ∈ Q[x of integral points x = (x1 , . . . , xn ) ∈ Zn ∩ X(Q) such that 0 < |f (x)| < H(x)−d(dim X−1)− is not Zariski-dense in An . We stress that in the above formula, although the polynomial f has algebraic irrational coefficients, the absolute value must be normalized with respect to Q, i.e., it must be the ordinary real or complex absolute value. Liouville bound would give |f (x)|  H(x)−d[κ:Q] , where κ is the field generated by the coefficients of f . Hence the result is non-trivial whenever [κ : Q] > dim X − 1.

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In the same way, Theorem 3.6 implies the following generalization of Thue’s inequality: Theorem 3.10. – Let F (x0 , . . . , xn ) ∈ Q[x0 , . . . , xn ] be an irreducible homogeneous ¯ in the product of m factors of degree d. Let X ⊂ Pn polynomial, which splits over Q be an algebraic variety, defined over Q, not contained in the zero set of F . Let  > 0 be a positive real number. Then for each rational point P = (x0 : · · · : xn ) ∈ X(Q) outside a proper closed Zariski subset of X, the following holds: |F (x0 , . . . , xn )| > H(P )−d(dim X+1)− . kxkdeg F Note that for n = 1 each polynomial splits into linear factors. Hence, putting X = P1 we reobtain Thue-Roth’s inequality, from which Thue’s theorem on the finiteness of solutions to the equation F (x0 , x1 ) = 1 follows immediately. If n > 1, how¯ 0 , . . . , xn ]. ever, the homogeneous polynomial F can remain irreducible in the ring Q[x In that case, the above inequality is trivial (d being the degree of F , taking integral coprime coordinates for P , the inequality boils down to |F (x0 , . . . , xn )| > max(|x0 |, . . . |xn |)−d dim X− which is weaker than Liouville’s). In particular, one cannot prove the analogue of Thue’s theorem, namely the finiteness (or the degeneracy) of the solutions to F (x0 , . . . , xn ) = 1 in integers (x0 , . . . , xn ) ∈ Zn+1 . Recent developments on these topics have been carried out by Min Ru [55] (see also the preprint by P. Vojta and M. Ru [56]). A very different problem consists in studying the approximation of points by (sequences of) points in higher dimensional algebraic varieties. This topic has been investigated by D. Mc Kinnon and M. Roth (see [48]). 4. A proof of Siegel’s theorem for integral points on curves In this section we prove Siegel’s Theorem using the approach developed in [19], which is based on the Subspace Theorem. We recall the statement of Siegel’s theorem, in the generalized version for rings of S-integers, whose original proof is due also to the contribution by Mahler [47]. The most general version, equivalent to the one below, appears probably for the first time in a paper of Lang [43]. Theorem 4.1 (Siegel’s Theorem). – Let C be an affine curve of Euler characteristic χ, defined over a number field κ. Let O S ⊂ κ be a ring of S-integers. If χ > 0 then C ( O S ) is finite. The above theorem encompasses, in particular, Thue’s 1909 finiteness result on the equations of the type (4.2)

F (x, y) = c,

where F (X, Y ) ∈ Z[X, Y ] is a homogeneous form of degree d ≥ 3, with no repeated factors, and c ∈ Z {0} a non-zero constant.

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Note that the affine curve defined by an equation as above has genus (d−1)(d−2)/2 and has d points at infinity, so its Euler characteristic is χ = d2 − 2d and is positive precisely whenever d ≥ 3. We shall prove, using the Subspace Theorem treated in the previous section, the particular case below of Siegel’s theorem. The full Siegel’s Theorem will follow by reducing the general case to the special one via the Chevalley-Weil theorem. Theorem 4.3. – Let κ be a number field, O S ⊂ κ a ring of S-integers. Let C be an affine algebraic curve over κ with at least three points at infinity. Then C ( O S ) is finite. Some remarks are in order: (1) the number of points at infinity depends on a compactification of C , i.e., on embeddings C ,→ An ,→ Pn ; the statement means that if in some embedding this number is ≥ 3, then the conclusion follows. The maximal number of points at infinity occurs for a compactification for which the points at infinity are all smooth. (2) We did not suppose that the affine curve C is smooth, neither in Theorem 4.1 nor in Theorem 4.3; however, by taking a normalization C 0 → C , the finiteness of C 0 ( O S ) would imply the same conclusion for C ( O S ). Hence one can reduce to the case when C is smooth. (3) No general finiteness result can hold for all curves with just one or two points at infinity, as shown by the case of smooth rational ones (resp. A1 and Gm ). Before proving Theorem 4.3, let us show how to use the Chevalley-Weil theorem to deduce Theorem 4.1 from Theorem 4.3. We have already remarked that we can reduce to the smooth case. If a (smooth) curve C has positive Euler characteristic but only one or two points at infinity, then its genus must be positive. Then its smooth completion ˜C is not simply connected; more precisely, it admits connected unramified covers of any degree. Consider an unramified 0 0 cover ˜C → ˜C of degree ≥ 3. Then the pre-image in ˜C of the set ˜C C consisting of the points at infinity has cardinality ≥ 3, hence Theorem 4.3 applies to C 0 . The finiteness of any set of S-integer points on C 0 implies, via the Chevalley-Weil Theorem, the same assertion for C . We first look at the example of the Thue’s equation, which we can write in the form (4.4)

F (x, y) = m ·

d Y

(x − αi y) = c,

i=1

¯ ∗ are conjugate algebraic numbers and c, m ∈ Z {0} are nonwhere α1 , . . . , αd ∈ Q zero rational integers. Suppose that d ≥ 3 and, by contradiction, there is an infinite set of integral solutions to (4.4). We write C ⊂ A2 for the algebraic curve defined by Equation (4.4) and embed ¯ contains the d points at infinity Qi := (αi : 1 : 0), C ,→ ˜ C ⊂ P2 , where ˜ C (Q) i = 1, . . . , d.

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By the compactness of the topological space ˜C (R), from any infinite sequence of solutions (xn , yn ) ∈ Z2 we can extract a sequence convergent to one of the points at infinity, say Q1 = (α1 : 1 : 0). Consider the rational function ϕ := x − α1 y ∈ κ( C ), where κ = Q(α1 ). We can view ϕ also as a morphism ˜C → P1 , of degree d, sending (α1 : 1 : 0) to 0 (with multiplicity d − 1). Clearly, for every solution (u, v) ∈ Z2 of (4.4) we have |u − α1 v| =

|c| 1 · . |m| |u − α2 v| · · · |u − αd v|

Now, for all but finitely many solutions of the sequence of solutions converging to Q1 , we have the lower bound min(|α1 − αi |, |1 − α1−1 αi |) · max(|u|, |v|), 2 valid for every i = 2, . . . , d. By the two above inequalities there exists a positive real number C such that for all but finitely many solutions in our sequence |u − αi v| >

|ϕ(u, v)| = |u − α1 v| ≤ C · max(|u|, |v|)−d+1 ≤ C · max(|u|, |v|)−2 . This inequality contradicts Roth’s theorem. Little modification is needed to recover the full Thue-Mahler theorem, where S-integer solutions to a general equation of the form (4.4) are considered. Let us now come to the general case of Theorem 4.3: C is an arbitrary affine algebraic curve; let us embed it into an affine space An so that its completion ˜C in Pn is smooth at infinity. Let d ≥ 3 be the number of points at infinity, which are labeled Q1 , . . . , Qd . Suppose by contradiction that C ( O S ) is infinite. Let κ be a number field containing a field of definition for the curve and for the points at infinity. We denote again by O S a ring of S-integers of κ containing the given ring (appearing in Theorem 4.3). Q By compactness of the topological space ν∈S ˜C (κν ) we can extract an infinite sequence of integral points P1 , P2 , . . . converging with respect to the places of S. Let, for each place ν ∈ S, Rν ∈ ˜C (κν ) be the ν-adic limit of the sequence P1 , P2 , . . . Since the height of Pn tends to infinity, and the points Pn are S-integers, some of the limit points Rν must lie at infinity. Let S 0 ⊂ S be the set of such places. The idea is to replace the morphism ϕ : ˜C → P1 used in the proof of Thue’s theorem by a morphism Φ : ˜C → PM for some (large) dimension M . We give the details, following closely the original paper [19] and the books [16], [25]. For every ‘large’ integer N , put VN = H0 ( ˜C , N (Q1 + · · · + Qd )) = {f ∈ κ ¯ [ C ] : (f ) ≥ −N (Q1 + · · · + Qr )}. Let f0 , . . . , fM , where M + 1 = h0 (N (Q1 + · · · + Qd )) = dN + O(1), be a basis of VN . Since the divisor Q1 + · · · + Qd is defined over κ, we can choose f0 , . . . , fM defined over κ, i.e., with fi ∈ VN ∩ κ[ C ] for i = 0, . . . , M .

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After multiplying the fj by a suitable constant, we can suppose that fj (Pn ) ∈ O S for all j, n. For every ν ∈ S, consider the filtration VN = Wν,1 ⊃ Wν,2 ⊃ · · · defined as Wj = Wν,j = {f ∈ VN : ordRν f ≥ j − 1 − N }. We have dim(Wj /Wj+1 ) ≤ 1 for each j; in particular dim Wj ≥ M − j + 1. Now, for each ν ∈ S 0 , choose a basis of VN containing a basis of each subspace Wν,j (for each j such that Wν,j 6= {0}). These functions can be expressed as linear combinations of the basis (f0 , . . . , fM ), i.e., as values of linear forms Lν,j (f0 , . . . , fM ), where Lν,j (X1 , . . . , XM ) has its coefficients in κ ¯ . Clearly ordRν Lν,j (f0 , . . . , fM ) ≥ j − N. For ν ∈ S S 0 we just put Lν,j (f0 , . . . , fM ) = fj . For each ν ∈ S 0 choose a local parameter tν ∈ κ( C ) at Rν . The above displayed inequality implies that |Lν,j (f0 (Pn ), . . . , fM (Pn ))|ν  |tν (Pn )|j−N . ν Now, observe that we dispose of M +1 = dN +O(1) rational functions Lν,j (f0 , . . . , fM ), of which at most N have poles and Q approximately (d − 1)N have zeros at Rν . Estimating the order of the product j Lν,j (f0 , . . . , fM ) we have that M X

ord(Lν,j (f0 , . . . , fM )) ≥

j=0

M X

(−N + j) = −N (M + 1) +

j=0

M (M + 1) 2

M ((d − 2)N + O(1)), 2 so this order is positive, and actually > (d − 2)N 2 /2 + O(N ) for large N . +1 Put x = (f0 (Pn ), . . . , fM (Pn )) ∈ O M and let as before kxkν be its sup-norm in S the ν-adic absolute value. Observing that for ν 6∈ S 0 the absolute values of fj (Pn ) are uniformly bounded, we can deduce that =

M YY

|Lν,j (x)|ν 

ν∈S j=0

Y

2

(d−2) N2 +O(N )

(|tν (Pn )|)

On the other hand, the height is easily estimated by H(x)  Finally we obtain, dividing the exponents by N , M YY

.

ν∈S 0

Q

ν∈S 0

−N

(|tν (Pn )|)

.

N

|Lν,j (x)|ν  H(x)(2−d) 2 +O(1) .

ν∈S j=0

Since d is assumed to be ≥ 3, the exponent is negative for sufficiently large values of N . Then the Subspace Theorem, in its version 3.4, implies that infinitely many vectors x lie on a hyperplane; this is impossible, since the functions f0 , . . . , fM are linearly independent, so every non-trivial linear combination of f0 , . . . , fM can have only finitely many zeros.

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Historical note. – The original proof of Siegel’s theorem appeared in [62], and treated only the case in which the ring of S-integers coincides with the ring of algebraic integers of κ. An English translation accompanied by the original German version is reproduced in [69]. For a discussion on this proof, see the papers by S. Lang [43], C. Fuchs and U. Zannier [69] and by Zannier [71]. A different proof, in the spirit of Dyson’s proof of his Diophantine approximation theorem [28], is due to C. Gasbarri [37]. Still another approach, using the language of non-standard analysis, appears in work of Robinson and Roquette [53].

5. The generalized Fermat equation and triangle groups In this section, we show yet another example of a situation in which a hyperbolicity condition implies a finiteness result for a Diophantine equation. The main results are due to H. Darmon and A. Granville [26]. Let p, q, r be a triple of natural numbers with 1 ≤ p ≤ q ≤ r. The aim of this section is the study of the Diophantine equation (5.1)

xp + y q = z r

to be solved in coprime integers x, y, z ∈ Z. More generally, we shall treat the equations axp + by q = cz r , where the coefficients a, b, c are non-zero integers. We shall also consider the corresponding equations in a ring of S-integers. It will turn out once again that the results (finiteness or density according to the cases) will heavily depend on the sign of a kind of Euler characteristic which we now define. Given the triple (p, q, r), the Euler characteristic χ of the triple will be the rational number 1 1 1 χ = χ(p, q, r) := 1 − − − . p q r In accordance to this position, we say that the triple is hyperbolic, elliptic or euclidean (alternatively: parabolic) according to the sign of the difference 1 − p1 − 1q − 1r , so:

(5.2)

1 1 1 + + >1 p q r 1 1 1 + + =1 p q r 1 1 1 + + 0, and construct a geodesic triangle in the hyperbolic plane with angles (π/p, π/q, π/r) (this is possible precisely because χ > 0; by Lambert’s theorem on hyperbolic triangles, its area will be π · χ). The reflections with respect to the sides of the triangles generate an infinite discrete group of hyperbolic isometries; the index-two subgroup of orientation preserving isometries turns out to be isomorphic to the abstract triangle group T (p, q, r). The main arithmetic result is the following theorem due to H. Darmon and A. Granville [26]: Theorem 5.3. – Let (p, q, r) be a hyperbolic triple. Then for every non-negative integers a, b, c there exist only finitely many solutions (x, y, z) ∈ Z3 with gcd(x, y, z) = 1 to the equation (5.4)

axp + by q = cz r .

On the contrary, for euclidean or elliptic triples, we have

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Theorem 5.5. – Let (p, q, r) be a triple with χ(p, q, r) ≤ 0. Then there exist non-zero integers a, b, c such that the solutions (x, y, z) ∈ Z3 in coprime integers to the Equation (5.4) are Zariski-dense in the surface defined by the above equation. Our approach to the proof of Theorem 5.3 constitutes a simplification of the original one, and makes essential use of the Chevalley-Weil theorem. Before starting the proofs of the above statements, we pause for a remark on the condition of coprimality of x, y, z. Since Equations (5.1), (5.4) are not homogeneous, there is no way in general to pass from a solution (x, y, z) to one with coprime coordinates. For instance, the solution (8, 4, 2) to the equation x2 + y 3 = z 7

(5.6)

does not produce, at least in any obvious way, any solution with coprime coordinates. We dispose of a geometric formulation of the coprimality condition: let us denote by S the quasi projective surface in A3 defined by the system of equation and inequality ( axp + by q = cz r (5.7) (x, y, z) 6= (0, 0, 0). Then the integral points on S are precisely the integral solutions to (5.4) with coprime coefficients. ¯ be the projective closure in P3 of the quasi-projective Yet in other words: let S surface S defined by (5.7); blowing-up the origin of A3 ⊂ P3 we obtain a new surˆ in the projective 3-space blown-up at one point. Let D be the curve obtained face S ˆ with the union of the pull-back of the plane at infinity and the by intersecting S ˆ which are integral with respect to D exceptional divisor. The rational points on S correspond to the solutions to (5.4) with coprime coordinates. We have seen that the coprimality condition is not a trivial one, and the existence of a solution to (5.1) does not lead automatically to a new one with coprime coordinates. However, every solution (x, y, z) ∈ Z3 to Equation (5.1) (or to the Equation (5.4)) gives rise to infinitely many of them, whose coordinates are not coprime. These are obtained in the following way: let (u, v, w) ∈ N3 be a generator of the one-dimensional lattice in Z3 formed of the vectors (l, m, n) ∈ Z3 with pl = qm = rn (if p, q, r are coprime, then (u, v, w) = (qr, rp, pq)). Then the group Gm acts on the surface defined by (5.4) via (5.8)

Gm × S 3 (λ, (x, y, z)) 7→ (λu x, λv y, λw z) ∈ S .

It follows that for each integral solution (x, y, z) ∈ Z3 to (5.4) one can produce infinitely many solutions by taking integral values of λ in (5.8) above. The above considerations lead to the following versions of Theorems 5.3 and 5.5 for rings of S-integers Theorem 5.9. – Let (p, q, r) be a triple of positive integers. The following are equivalent:

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(i) There exists a ring of S-integers O S such that the set of integral points on the surface of Equation (5.7) is Zariski-dense; (ii) we have χ(p, q, r) ≤ 0. Let us start with the proof of Darmon-Granville’s finiteness theorem (Theorem 5.3). Let S ⊂ A3 {(0, 0, 0)} be the surface defined by the Equation (5.7) and let β : S → P1 be the morphism axp . S 3 (x, y, z) 7→ cz r Note that if gcd(p, q, r) = 1 then the fibers of β are all isomorphic to Gm and are precisely the orbits for the Gm -action described in (5.8). However, the projection β : S → P1 does not define a bundle over P1 ; one does obtain a (principal) Gm -bundle over P1 {0, 1, ∞} after removing from S the pre-images of 0, 1, ∞, which are the multiple fibers for β. Let us now consider a Galois cover C → P1 of the projective line ramified over {0, ∞, 1} of order (p, q, r); if gcd(p, q, r) = 1, these covers are obtained from any non-trivial finite quotient of the triangle group T (p, q, r): take a finite index normal subgroup ∆CT (p, q, r) distinct from the triangle group itself; then define the compact Riemann surface C to be the quotient of the hyperbolic plane H by the action of ∆; ∆ is necessarily free and so acts freely on H , and the quotiemt map H → C turns out to be the universal cover of C . Recall that the quotient H /T (p, q, r) of the hyperbolic plane by the action of the full triangle group is the projective line P1 . The corresponding map π : C → P1 , induced from the surjective map H → P1 , ramifies precisely over three points, which correspond to the vertices of the triangles composing the tiling of H . The ramifications indices are precisely p, q and r. If the integers p, q, r fail to be coprime, the construction is similar up to the provisio that we must avoid that the quotient T (p, q, r) → (T (p, q, r)/∆) factors through another triangle group of the form T (p/m, q/m, r/m), for any common divisor m > 1 of p, q, r. The Riemann surface defined analytically in the above way turns out to be an algebraic curve defined over the field of algebraic numbers; also the map π : C → P1 can be defined over the field of algebraic numbers. The crucial point in the proof of Theorem 5.3 is the following Lemma 5.10. – In the above setting, let F = C ×P1 S be the normalization of the fiber product of π : C → P1 and β : S → P1 . Then the natural morphism π ¯ : F → S is unramified. This lemma permits to apply the Chevalley-Weil theorem. Proof. – The statement is local, so we shall argue locally in the complex topology. Let us denote by β¯ : F → C the projection to C ; it satisfies π ◦ β¯ = β ◦ π ¯. Let s ∈ S be any point; if β(s) 6∈ {0, ∞, 1} then there exists a neighborhood U of β(s) in P1 (C) such that π : π −1 (U ) → U is a topological cover. In that case, the

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surface F can be locally defined as the fiber product β −1 (U ) ×U π −1 (U ), which is ¯ −1 (U ) → β −1 (U ) is a topological cover (the a smooth complex space, so π ¯ : (π ◦ β) −1 pull-back via β : β (U ) → U of the topological cover π : π −1 (U ) → U ). The problem arises when β(s) ∈ {0, 1, ∞}. In these cases we exploit the multiplicity of the fibers of β : S → P1 to kill the ramification of the map π : C → P1 . Note that in that case the (set-theoretic) fiber product of S → P1 and C → P1 is singular above β(s), as we now show (but the normalization that we called F is smooth). Suppose for instance that s lies over the point 0 on the line, so that the coordinate x vanishes (the argument is symmetric if β(s) = ∞ or β(s) = 1). Local parameters at s on the smooth surface S are for instance the regular functions x, z − z(s), while the y function can be expressed in term of x, z as  r 1/q cz − axp (5.11) y= , b where the q-th root is a well-defined function in a neighborhood of s and the choice of the branch is the one compatible with the y-coordinate at s. Take a point f in the pre-image π ¯ −1 (s). Since the point 0 is ramified of order p under the Galois cover π : C → P1 , there is a local parameter t on C (in the analytic sense) ¯ ) =: γ ∈ C which is a p-th root of the function π at the point β(f ¯ ∗ (β). The ramified p cover C → P1 will be locally defined by t 7→ t = β. Now locally at f the surface F is birationally defined by adding the function t to the local parameters x, z − z(s) and to the function y defined in (5.11); the algebraic relation satisfied by t is axp . cz r Note that the above equation defines a singular, and non-normal, variety; the local ring at f of F , which is integrally closed, is generated over the local ring of S at s by an element which we shall denote again by t, of the form x t= , (cz r /a)1/p tp = β =

where again the p-th root is well-defined and the choice of its branch depends on the choice of the point f ∈ F lying over s. Note that there are p choices of points f ∈ F corresponding to the point (s, γ) ∈ S × C , while in the set-theoretic fiber product S ×P1 C there would be just one; these p points correspond to the p possible branches of the p-th root of the function appearing in the denominator in the above formula. Clearly t and π ¯ ∗ (z − z(s)) =: w are local parameters at f and the map π ¯ can be defined by sending (t, w) 7→ (x, y, z) where x = t(cz r /a)1/p , y is defined by (5.11) and z = w + z(s). Hence it is a local biholomorphism, which implies that the cover is unramified at f .

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We note (although this fact will not be used) that the projection β¯ : F → C defines a principal Gm -bundle on the curve C . We can now conclude the proof of Theorem 5.3: we have already remarked that the integral solutions (x, y, z) with coprime coordinates to the Equation (5.4) correspond to the integral points on the surface S defined in (5.7). By the Chevalley-Weil theorem, these integral points lift to integral points of F , defined over a fixed ring of S-integers. But now, the rational points on F , in particular the integral ones, project via β¯ to rational points on C , which is a curve of genus ≥ 2. By Faltings theorem, C contains only finitely many rational points, which implies that the rational ¯ This last fact, in turn, implies points on F accumulate on finitely many fibers of β. that the integral points on S accumulate on finitely many fibers for β; but a fiber of β can contain only finitely many points with integral coprime coordinates, which ends the proof. We remark that the coprimality assumption appearing in Theorem 5.3 is used only in the application of the Chevalley-Weil theorem. It is an open problem to decide whether, for a hyperbolic triple (p, q, r), the integral solutions to (5.4) are contained in finitely many Gm -orbits for the action (5.8). A conjecture of F. Campana predicts this finiteness. We have just seen that hyperbolic triples always lead to finitely many solutions (with coprime coordinates). We now turn to the inverse direction: if the triple is not hyperbolic, can we have infinitely many coprime solutions to Equation (5.4), at least for one choice of the non-zero coefficients a, b, c? The answer turns out to be affirmative, as claimed in Theorem 5.5. We divide the proof of Theorem 5.5 into two parts, one for the elliptic and one for the parabolic case. Elliptic case. – The elliptic case is divided into four sub-cases, the first cooresponding to the dihedral groups, the following ones to the three groups associated to the regular solids. Dihedral case. – Suppose first the triple is (2, 2, n), for some n ≥ 2. Then the equation (with the choice (a, b, c) = (1, 1, 1)) becomes x2 + y 2 = z n . It can be written as (x + iy)(x − iy) = |x + iy|2 = z n . It then boils down to finding infinitely many Gaussian integers α ∈ Z[i] which are n-th powers in Z[i] and whose real and imaginary parts are coprime. For instance, writing (1 + ki)n = pn (k) + iqn (k) where pn (T ), qn (T ) ∈ Z[T ] are coprime polynomials, one observes that (pn (0), qn (0)) = (1, 0) so pn (k), qn (k) take coprime values for infinitely many integers k. The above pair of polynomials corresponds to parametrizing the rational curve A1 3 t 7→ (x, y, z) = (pn (t), qn (t), 1 + t2 )

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lying on the surface of equation x2 + y 2 = z n . Specializing the variable t to integers produce infinitely many integral points on the surface; to produce a Zariski-dense set of integral solutions just use two parameters, replacing 1 + ki by h + ki. Tetrahedral case. – Consider now the triple (2, 3, 3) (associated to the Tetrahedron). We shall prove that the equation x2 + y 3 = z 3

(5.12)

admits infinitely many solutions (x, y, z) ∈ Z3 with coprime coordinates, and that these triples are indeed Zariski-dense in the surface defined by the above equation. Note that Equation (5.12) is equivalent to the equation (z − y)(z 2 + zy + y 2 ) = x2 , and that the latter is certainly solved in coprime integers whenever we find coprime integers z, y such that both z − y and z 2 + zy + y 2 are perfect squares. The condition that z 2 + zy + y 2 be a perfect square can be expressed by the following equation z 2 + zy + y 2 = v 2 which represents a smooth projective conic with at least one rational point (e.g., the point (v : y : z) = (1 : 1 : 0)). We then obtain a rational parametrization (5.13)

(v : y : z) = (t2 − ts + s2 : t2 − s2 : 2ts − t2 ).

Whenever t, s are coprime integers, we obtain coprime values of y, z unless t ≡ −s (mod 3) in which case the gcd(y, z) = 3. Now, the condition that z − y be also a square leads to the equation s2 + 2ts − 2t2 = u2

(5.14)

which again represents a smooth projective conic with a rational point (e.g., the point (s : t : u) = (1 : 0 : 1)). We then obtain infinitely many other rational points, via the parametrization (5.15)

(s : t : u) = (2λ2 + µ2 : 2λ2 + 2λµ : 2λ2 − 2λµ − µ2 ).

By letting λ, µ vary among the integers we obtain integer values of s, t, u satisfying (5.14). If we choose coprime integers λ, µ with (λ, µ) ≡ (0, 1) (mod 6) we obtain coprime values of s, t with (s, t) ≡ (1, 0) (mod 3), hence coprime values for z, y as wanted. This concludes the proof for the triple (2, 3, 3). We note that the bulk of this proof was the geometric fact that the surface S admits a degree 2 (unramified) cover S 0 → S with S 0 ' P1 × Gm (over the complex number field). Since this last surface admits a Zariski dense set of integral points, the same will be true of the surface S .

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Octahedral case. – We now consider the triple (2, 3, 4), associated to the octahedron (or the cube). We sall prove that the equation (5.16)

x2 + y 3 = z 4

admits infinitely many integral solutions with coprime coordinates. Again, the equation is tranformed into (z 2 − x)(z 2 + x) = y 3 whose solutions can be obtained from the solutions to the system ( z 2 − x = u3 z2 + x = v3 , which is equivalent to the single equation (5.17)

v 3 + u3 = 2z 2

(any solution (u, v, z) of (5.17) gives rise to the solution (v 3 − z 2 , u, v, z) to the system above). The Equation (5.17) is of Fermat-type with an elliptic exponent vector (2, 3, 3), as the one considered before. Certainly it will admit infinitely many integral solutions with coprime coordinates in a suitable ring of S-integers, since the surface it defines is isomorphic over the complex number field to the one defined by Equation (5.12). However, we can easily see that the above equation admits infinitely many integral solutions with coprime coordinates already over the ring Z. Again, we can factor the left-hand side in (5.17) as (v + u)(v 2 − uv + v 2 ) and reduce to finding u, v such that for some integers ξ, η ( u2 − uv + v 2 = ξ 2 u + v = 2η 2 . This system is treated as before. The first equation defines the same smooth conic with one rational point as in the previous case, so admits a parametrization with quadratic polynomials (e.g., (u, v, ξ) = (t2 − s2 , t2 − 2ts : t2 − ts + s2 ) ). The second equation becomes then 2t2 − 2ts − s2 = 2η 2 which again represents a smooth conic with a rational point (e.g., (s : t : η) = (0 : 1 : 1)) so it admits infinitely many rational points, parametrized by values of quadratic polynomials in new variables. Again, suitable specializations of the variables give rise to coprime solutions (x, y, z) to the original equation. This completes the proof in the octahedral case. The link between the octahedral and the tetrahedral cases admits the following geometric interpretation: letting S (2,3,3) and S (2,3,4) be the surfaces associated to the tetrahedral and the octahedral equations respectively, the above calculations provide a degree 3 unramified cover S (2,3,3) → S (2,3,4) . Icosahedral case. – As expected, the icosahedral triple (2, 3, 5) leads to the most difficult case. The corresponding equation was implicitely solved by Klein in his famous

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book on the icosahedron [42]. He considered the degree sixty Galois cover P1 → P1 given by (−z 20 + 228 z 15 − 494 z 10 − 228 z 5 − 1)3 z 7→ , 1728 z 5 (z 10 + 11 z 5 − 1)5 which admits a Galois group isomorphic to the triangle group T (2, 3, 5) ' A5 . He found three fundamental invariants x, y, z, of respective degrees 30, 20, 12 (the number of edges, faces and vertices of an icosahedron),which in homogenous coordinates u, v (where z = u/v) can be written as x = 126 (u30 + v 30 + 522(u25 v 5 − u5 v 25 ) − 10005(u20 v 10 + u10 v 20 )) y = 124 (−u20 − v 20 + 228(u15 v 5 − u5 v 15 ) − 494u10 v 10 z = 123 uv(u10 + 11u5 v 5 − v 10 ). These three fundamental invariants satisfy the Fermat-type icosahedral equation x2 + y 3 = z 5 ; the integral coprime specializations of (u, v) give rise, after clearing out the twelfth power of twelve, an integral solution to the equation x2 + y 3 = 1728 · z 5 , which then turns out to admit infinitely many integral solutions (x, y, z) with coprime coordinates. Parabolic case. – We must now consider the three parabolic triples (2, 4, 4), (3, 3, 3) and (2, 3, 6). Not surprisingly, to produce infinitely many integral solutions one is led to producing infinitely many rational points on elliptic curves (recall that the elliptic curves are complete curves of parabolic type). • Let us start from the triple (2, 4, 4). The Diophantine equation x4 +y 4 = z 2 has no coprime integral solutions outside those with one vanishing term, as proved already by Fermat via his infinite descent method, introduced precisely for solving that equation. However, as we now show, the equation admits infinitely many integral solutions over suitable number fields, and suitable twisted forms of it admit infinitely many integral solutions already in Z. We interpret the coordinates x, y as homogeneous coordinates in P1 . Let ˜C be a complete curve which covers the line with a degree two morphism ramified over the zero set of the binary quartic form x4 + y 4 , i.e., over the points (1 : ξ), (1 : iξ), (1 : −ξ) and (1 : −iξ), where ξ 4 = −1. Then necessarily ˜C has genus one, and can be defined in an affine model by the equation u2 = 1 + v 4 . It will have infinitely many rational points over some number field κ. Writing u = z/δ with z, δ coprime S-integers on a suitable principal ring of S-integers of κ, we see that δ must be a square; writing δ = x2 we have z 2 = x4 + (xv)4 , so that xv =: y is an S-integer. We have then solved the original equation. If one wants the solutions already in the ring Z of rational integers, it suffices to change the coefficients for the monomials. For instance, the Diophantine equation 15 z 2 = x4 − y 4

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admits infinitely many solutions in coprime integers. • The triple (3, 3, 3) already defines an elliptic curve, if we interpret the coordinates in the projective sense. Also in this case, the Q-rank is zero, as proved by Euler, but putting as coefficient for z the famous ‘taxi-cab number’ 1729 leads to the elliptic curve x3 + y 3 = 1729 z 3 , admitting, in addition to the trivial solution (1 : −1 : 0), the two non-trivial solutions (1, 12, 1) and (9, 10, 1), which provide infinitely many rational points. • Finally, let us study the case of the (2, 3, 6): the Diophantine equation x2 + y 3 = z 6 is equivalent to the system ( x2 + y 3 = z 3 z = w2 . Recall that the first equation is of tetrahedral type, and was solved via the parametrizations (5.13) and (5.15), from which it follows that z can be expressed as a quartic form in two parameters. We then reduce to the triple (2, 4, 4) already considered. We end this paragraph by noting that all our calculations could be expressed in the language of weighted projective space, which might be considered simpler by some readers.

6. Algebraic groups and the S-unit equation theorem Throughout this chapter, κ is a fixed number field and S a finite set of absolute values of κ, containing the archimedean ones. Unless otherwise stated, all algebraic varieties are defined over the number field κ. 6.1. The S-units equation. – In this section we will be interested in the Diophantine equation (6.1)

u1 + · · · + un = 1

to be solved in S-units u1 , . . . , un ∈ O ∗S . The above equation defines an irreducible algebraic sub-variety V ⊂ Gnm , not a translate of a subgroup. Its set V ( O S ) of S-integral points corresponds to the set of solutions in S-units to the Equation (6.1). Since V is a variety of log-general type, after Vojta’s conjecture it is expected that its set of integral points is not Zariski-dense. This assertion is in fact a theorem, proved by Evertse, van der Poorten and Schlickewei, after preliminary work by Dubois and Rhin, actually in the stronger form given here Theorem 6.2. – Equation (6.1) admits only finitely many solutions (u1 , . . . , un ) ∈ ( O ∗S )n for which no sub-sum of the ui vanishes.

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Of course, if n = 2 then no subsum can vanish, so we obtain unconditional finiteness (and, by the way, this was the Siegel-Mahler theorem for integral points on P1 {0, 1, ∞}). If, on the contrary, n ≥ 3, then there are certainly infinite families of solutions; for instance, for n = 3 the family (u, −u, 1), where u ∈ Gm . These families correspond to sub-tori contained in V . The three-dimensional case (n = 3 in the above theorem) corresponds to the complement of four lines in general position on the projective plane, as we now explain. Using suitable projective coordinates X, Y, Z for P2 , we can suppose that the four lines, labeled L1 , . . . , L4 , are defined by the equations X = 0, Y = 0, Z = 0 and X + Y = Z. The complement of the union of the first three lines is isomorphic to G2m ; actually, putting u = X/Z and v = Y /Z, we see that the integral points on this complement correspond to the S-unit values of u and v. The condition that the point (X : Y : Z) does not reduce to the line X + Y = Z modulo any prime amounts to saying that u+v 6≡ 1 modulo any prime. This means precisely that w =: 1−u−v is a unit, and this leads to the linear equation u + v + w = 1, to be solved in S-units. The mentioned infinite families (there are three of them in this case) correspond to the three lines in P2 intersecting the union of the four lines L1 , . . . , L4 in only two points. These lines are images of non-constant morphisms Gm → P2 (L1 ∪ · · · ∪ L4 ). It is easy to see that there are no other curves in the affine surface P2 (L1 ∪ · · · ∪ L4 ) of Euler characteristic ≤ 0, hence no other infinite family of solutions. Proof. – The theorem can be restated as follows: for every infinite sequence of solutions to (6.1), there is a sub-sum vanishing infinitely often. Also, the theorem is equivalent ot the following statement: for every infinite set of solutions, some ratio uh /uk takes infinitely often the same value. We shall prove the theorem under this formulation. We follow the pattern given in Chapter II of [60] and Chapter II of [25]. We argue by induction on n, the case n = 1 being obvious. (1) (1) (2) (2) Let P1 = (u1 , . . . , un ), P2 = (u1 , . . . , un ), . . . be an infinite sequence of solutions to (6.1). For each place ν ∈ S and each index j = 1, 2 · · · , let ijν be such (j) (j) that |uij | = maxi {|ui |ν }. Up to extracting a subsequence, we can suppose that the ν

index ijν ∈ {1, . . . , n} does not depend on j. Hence we denote it by iν . Let us define linear forms Lν,i , for i = 1, . . . , n, ν ∈ S, by putting i 6= iν

Lν,i (X1 . . . , Xn ) = Xi , and

Lν,iν (X1 , . . . , Xn ) = X1 + · · · + Xn . Let us estimate the double product n Y Y

|Lν,i (Pj )|ν =

i=1 ν∈S

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n Y Y i=1 ν∈S

! (j) |ui |ν

·

Y |u(j) + · · · + u(j) n |ν 1 . kPj kν

ν∈S

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(j)

Due to the fact that the ui are S-units, factor equals 1; also, due to the Q the first −1 Equation (6.1), the second factor equals ν∈S kPj kν = H(Pj )−1 . An application of the Subspace Theorem in the form of Theorem 3.4 provides the existence of a linear equation of the form (j) a1 u1 + · · · an u(j) n = 0, valid for infinitely many indices j. Let us consider from now on only these indices j. Dividing out by un and putting bi = −ai /an for i = 1, . . . , n − 1, we obtain another S-unit equation b1 v1 + · · · + bn−1 vn−1 = 1 −1 satisfied by (v1 , . . . , vn−1 ) = (u1 u−1 n , . . . , un−1 un ) for infinitely many solutions (u1 , . . . , un ) of (6.1). Up to enlarging the set S, we can suppose that all the nonvanishing coefficients bi are S-units, so also the addends bi vi in the above equation are S-units. By the inductive hypothesis, a single value for some ratio bi vi /bj vj is attained infinitely often. Then the same is true for the ratio ui /uj , finishing the proof. A generalization of the S-unit equation theorem, obtained via a similar proof, reads as follows: Theorem 6.3. – Let V ⊂ Gnm be an algebraic sub-variety of a torus. Then the Zariskiclosure of the set V ( O S ) of integral points of V is a finite union of translates of algebraic sub-groups of Gnm contained in V . For the proof, see e.g., [60], [10] or [25]. 6.2. Applications of the S-unit equation theorem. – The S-unit equations appear in different contexts, so that Theorem 6.2 (and Theorem 6.3) admits numerous applications. For a survey on some of these applications, the reader is addressed to [31]. We want just to explain here the geometric pattern inherent to any application of the S-unit equation theorem. Suppose we are studying the integral points on a quasi-projective variety V , and that on V one can find a regular never vanishing function f . Then (after possibly multiplying f by a fixed non-zero constant), for every S-integral point P ∈ V ( O S ), the value f (P ) of f at P is a unit. Suppose now that we dispose of several such functions f1 , . . . , fn , with n > dim V , and that these functions are multiplicative independent modulo constants. Certainly f1 , . . . , fn are algebraically dependent, and they satisfy at least n−dim V independent algebraic relations Pi (f1 , . . . , fn ) = 0, for i = 1, . . . , n−dim V , where P1 , . . . , Pn−dim V are polynomials in n variables. These equations define a proper closed sub-variety W of Gnm , which is not a translate of an algebraic subgroup. Moreover, the dominant map F = (f1 , . . . , fn ) : V → W 0 (where W 0 is a component of W , image of V ) sends integral points of V to integral points of W 0 . An application of Theorem 6.3 (with W 0 replacing the variety V appearing in that theorem) enables to conclude that the integral points on V are not Zariski-dense.

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We can conclude by saying that the S-unit equation theorem applies to varieties V admiting dominant maps to a sub-variety of a torus, not (isomorphic to) a torus itself. Actually, the use of the Chevalley-Weil Theorem permits sometimes to apply the S-unit theorem in a more general situation: namely, suppose that an algebraic variety V admits an étale cover V 0 → V with a variety V 0 admitting such a map to a sub-variety of a torus. Then one can prove via the S-unit theorem the degeneracy of integral points on V 0 , and deduce via Chevalley-Weil the same conclusion for V . In the case of curves, the above described technique has been used for the first time by X [67] to prove the finiteness of the integral solutions to the hyper-elliptic equation (6.4)

y 2 = f (x),

where f (x) ∈ κ[x] is a polynomial without quadratic factors of degree ≥ 3. The affine curve C defined by the above equation admits in general no morphism to Gm , and in any case any morphism C → Gnm factors through a morphism C → Gm . Hence the S-unit equation theorem cannot be applied directly. However, there always exists an étale cover C 0 → C such that the curve C 0 admits a map C 0 → Gnm , for some n ≥ 2, such that the image curve has positive Euler characteristic. ¯ Here are the details: let us factor the polynomial f (X) in Q[X] as f (X) = c ·

d Y

(X − αi ),

i=1

where d ≥ 3 is the degree of f (X) and α1 , . . . , αd are pairwise distinct algebraic numbers. Put κ0 = κ(α p1 , . . . , αd ). The function field of the affine curve C defined by (6.4) over κ0 is κ0 (x)( f (x)). Since the rational functions (x − α1 ), . . . , (x − αd ) have no common zeroes on C and their product is a perfect square, each factor is a square √ √ in C[ C ]. Hence the field extension κ0 ( C )( x − α1 , . . . , x − αd )/κ0 ( C ) is unramified. Hence by Chevalley-Weil the S-integral points of C lifts to solutions over a fixed number field of the system of equations  2  x − α1 = y1 (6.5) x − α2 = y22   x − α3 = y32 √ √ √ (here we used just the intermediate extension κ0 ( C )( x − α1 , x − α2 , x − α3 )/κ0 ( C ), omitting the other square roots). From the above system of equations it follows that  2 2  α1 − α2 = y2 − y1 = (y2 − y1 ) · (y2 + y1 ) α2 − α3 = y32 − y22 = (y3 − y2 ) · (y3 + y2 )   α3 − α1 = y12 − y32 = (y1 − y3 ) · (y1 + y3 ). Enlarging S so that αi − αj becomes an S-unit for 1 ≤ i < j ≤ 3 we obtain that the three S-integers u1 := y3 − y2 , u2 := y1 − y3 and u3 := y2 − y1 are S-units. Since their sum vanishes, we apply the S-unit equation theorem and conclude easily

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the finiteness of the S-integral solutions to Equation (6.4). The appearance of the S-units u1 , u2 , u3 is due to the fact that the affine curve C 0 defined by the system (6.5) (endowed with an unramified map C 0 → C ) admits a morphism C 0 → G2m . A simpler application of the S-unit equation in two variables appears already with Thue’s Equation (4.2), as follows: letting C be the affine curve defined by Thue’s Equation (4.2), which we can re-write as d Y

(x − αi y) = c,

i=1

for α1 , . . . , αd pairwise distinct algebraic numbers, consider the map C

→ P1

{(α1 : 1), . . . , (αd : 1)} ,→ P1

{(α1 : 1), (α2 : 1), (α3 : 1)} ,→ G2m ,

where the first arrow sends C

3 (x, y) 7→ (x : y) ∈ P1 .

Then apply the S-unit equation theorem to the image of C into G2m . 6.3. Semi-abelian varieties and quasi-Albanese maps. – In this section, we partially follow Chap. III, §5.1 of [25]. Recall that a semi-abelian variety is an extension of an abelian variety by a torus, i.e., an irreducible algebraic group A sitting in an exact sequence {0} → Grm → A → A0 → {0}. Automatically, A is commutative. If A is defined over a number field κ and O S is the ring of S-integers of κ, the group of integral points A( O S ) is finitely generated; this fact follows formally by the combination of Mordell-Weil Theorem, applied to the abelian variety A0 , and Dirichlet’s Unit Theorem, applied to the torus Grm (recalling that the set of S-integral points on a torus Gm coincides with the group of S-units). The mentioned theorem of Vojta, generalizing a previous one by Faltings, states the following: Theorem 6.6. – Let A be a semi-abelian variety defined over a number field κ, X ⊂ A an algebraic subvariety. Then the set X( O S ) = X ∩A( O S ) is a finite union of translate of subgroups. It then follows formally that each of these translate is actually the set of integral points of a translate of an algebraic group entirely contained in X. In particular, if X contains no translate of algebraic subgroups of positive dimension, then X( O S ) is finite. Theorem 6.6 can be restated without mentioning integrality nor rationality: simply, the intersection Γ∩X between a finitely generated subgroup Γ ⊂ A(C) and an algebraic variety X ⊂ A is a finite union of cosets of Γ. We pause to discuss the applicability of the above theorem.

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Let us consider first the compact case. An abelian variety A of dimension g admits g linearly independent invariant 1-forms, trivializing the sheaf Ω1A . Whenever a (compact) variety X can be embedded into A, the restriction to X of these 1-forms are regular 1-forms on X; they remain linearly independent unless X is contained in a translate of an algebraic subgroup of A (corresponding to the linear subspace of the Lie algebra of A determined by the linear relations on the restrictions of the g forms). More generally, whenever a variety X admits a morphism X → A, whose image is not contained in a translate of an algebriac subgroup, then H 0 (X, Ω1X ) =: q(X) ≥ dim A. Vice-versa, one can produce an abelian variety and a morphism X → A starting from the holomorphic 1-forms on X: letting ω1 , . . . , ωg , where g = q(X), be a basis of H 0 (X, Ω1X ), and P ∈ X a point of X, consider the integration map Z Q ! Z Q ωg ω1 , . . . , (mod Λ), (6.7) X 3 Q 7→ P

P

where Λ ⊂ Cg is the Z-module of the periods, i.e., the integrals over the loops on X(C). The quotient Cg /Λ turns out to be an abelian variety, defined over the same field of definition for X and P . This abelian variety is called the Albanese variety of X, denoted by Alb(X); it has the universal property that every map from X to an abelian variety B (actually to any algebraic group) factors through the Albanese of X via a group homomorphism Alb(X) → B (possibly composed with a translation). In the case of curves, one obtains Abel’s construction of the jacobian variety. Hence the ‘maximal’ abelian variety A such that X admits a morphism to A whose image is not contained in a translate of a proper algebraic group is the Albanese Alb(X), which has dimension q(X) = h0 (X, Ω1X ); this number is also called the irregularity of X. When q(X) > dim X, one can apply Faltings Theorem (Theorem 6.6 in the compact case) which implies the degeneracy of the rational points on the image of X in Alb(X), hence also the degeneracy of the rational points on X. For non complete algebraic varieties, a parallel theory is possible. We address the interested reader to Chapter 5 of the book by Noguchi-Winkelmann [51]. Here we just sketch the main idea. Every smooth quasi-projective (complex) algebraic variety X can be realized as ˜ D for a smooth complete variety X ˜ and a normal crossing divithe complement X ˜ sor D ⊂ X. ˜ along D We say that a regular 1-form ω on X has logarithmic singularities on X (or that ω is a logarithmic 1-form) if locally at any point p of D where D admits an equation f1 · · · fh = 0, for (f1 , . . . , fh ) a subset of a coordinate system at p, ω can be written as df1 dfh ω = a1 + · · · + ah + regular form, f1 fh where a1 , . . . , ah are holomorphic functions in a neighborhood of the point p. It is well known that the logarithmic 1-forms are closed (see e.g., [49]). Also, they always form a finite-dimensional vector space.

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Hence one can repeat the construction performed in the compact case, by integrating a basis of logarithmic 1-forms, obtaining a map of the form (6.7) to a complex Lie group, which turns out to be a semi-abelian variety, called the quasi Albanese (or ˜ with a torus of generalized Albanese) of X. It is an extension of the Albanese of X dimension equal to the dimension of the quotient {log 1 − forms}/{regular 1 − forms}. Let us analyze some cases. Suppose D1 , D2 are irreducible linearly equivalent di˜ Then X = X ˜ (D1 + D2 ) admits a morphism F : X → Gm , where visors on X. f is a function with all its poles on D1 and all its zeros on D2 . The corresponding logarithmic 1-form is df /f ; in that case the quasi Albanese variety of X is simply the ˜ by Gm . product of the Albanese variety of X However, it is not always the case that the quasi-Albanese is a trivial extension of the Albanese. Already in the case of curves, the extension in general does not split. Consider a non-rational complete (smooth) curve ˜C . Let P ∈ ˜C be a point on ˜C . Every 1-form which is regular on C = ˜C {P } having a pole of order ≤ 1 on P is in fact regular everywhere; this follows by considering the sum of the residues, which must vanish. (Alternatively, one can compare the abelianized fundamental groups of C and ˜C : if g is the genus of ˜C , then the fundamental group is generated by 2g elements subject to one commutation relation; eliminating from ˜C a single point P the commutation relation disappears, and the fundamental group of the complement C of P turns out to be freely generated by 2g generators. Hence the inclusion C ,→ ˜C induces an isomorphism between the abelianized fundamental groups, showing that the abelian covers of the two curves correspond bijectively in a natural manner. In particular H 1 ( C (C), R) ' H 1 ( ˜C (C), R) ' R2g ). Consider now the complement of two points P1 , P2 on ˜C . By Riemann-Roch 0 ˜ h ( C , Ω1˜C (P + Q)) = g + 1 > h0 ( ˜C , Ω1˜C ), hence we obtain an extra logarithmic 1-form which is not a regular one. Whenever the class of [P1 ] − [P2 ] in the jacobian of ˜C (identified with Pic0 ( ˜C )) is torsion, one can find a function f having all its zeros on P1 and its poles on P2 , thus providing a morphism C = ˜C {P1 , P2 } → Gm . If, however, [P1 ] − [P2 ] is not torsion on the jacobian, such a morphism does not exist, although a logarithmic 1-form with poles at P1 , P2 still exists. Note that the two points P1 , P2 on ˜C are in any case algebraically equivalent, and this fact suffices to produce the logarithmic 1-form. This principle has been exploited by Vojta and Noguchi-Winkelmann to produce maps to a semi-abelian variety from an algebraic variety with sufficiently many components at infinity compared with the rank of the Néron-Severi group. See for instance the main theorem in [50]. We now exploit this same principle to a curious Diophantine problem, also considered in [25]. We first introduce some notation: let an elliptic curve over Q be defined by a Weierstrass equation (6.8)

y 2 = x3 + ax + b,

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where a, b ∈ Z are integers with 4a3 −27b2 6= 0. For a rational solution P = (x, y) ∈ Q2 of the above equation, one can write the rational numbers x, y in a unique way as v u x = 2, y = 3, d d for integers u, v, d without common factor, d > 0. Denote by d(P ) the positive number d appearing in the above formulae. The primes dividing d(P ) are precisely the primes modulo which the point reduces to the (unique) point at infinity of the completion of the curve defined by the above equation. We propose the following conjecture, inspired by a result in complex analysis (see the main theorem in [18]), which would follows from Vojta’s Conjecture, as we shall explain in a moment: Conjecture. – Let E1 , E2 be two elliptic curves defined over the rational integers by a Weierstrass equation. If there exist infinitely many pairs (P1 , P2 ) ∈ E1 (Q) × E2 (Q) with (6.9)

d(P1 ) = d(P2 ),

then E1 is isomorphic to E2 over the rationals and for all but finitely many such pairs P1 = ±P2 . (In the above equation the symbol d(P1 ) denotes the denominator-function attached to E1 while d(P2 ) denotes the denominator-function attached to E2 ). Note that this statement, already in the particular case E1 = E2 , represents a strong generalization of Siegel’s finiteness theorem for integral points on curves. In fact, Siegel’s theorem is equivalent to saying that d(P ) can be 1 only finitely many times. It is easy to derive from Siegel’s theorem that each value of d(P ) can be attained only finitely often. The proposed conjecture implies that only finitely many values of d(P ) can be attained more than two times (i.e., for other points than ±P ). Let us show how to reduce to a question on integral points on a quasi-projective variety, to which Vojta’s Conjecture can be applied. For this purpose, we first note that the condition d(P1 ) = d(P2 ) can be restated by saying that P1 reduces to the origin O1 ∈ E1 modulo some prime only if P2 reduces to the origin O2 ∈ E2 modulo the same prime, with the same multiplicity. So the pair (P1 , P2 ) ∈ E1 × E2 does not reduce, modulo any prime, to the divisor {O1 } × E2 + E1 × {O2 } unless it reduces to the single point (O1 , O2 ). ˜ be the surface obtained by blowing-up the origin (O1 , O2 ) in E1 × E2 . Let then X ˜ Let D1 ⊂ X be the strict transform of the divisor {O1 } × E2 and D2 the strict transform of the divisor E1 × {O2 }. Then the solutions to Equation (6.9) correspond ˜ (D1 + D2 ). to the integral points on X := X ˜ → E1 × E2 , the canonical Letting L be the exceptional divisor of the blow-up X ˜ divisor of X turns out to be (linearly equivalent to) L. Then the sum of the divisor at infinity of X plus the canonical divisor is D1 + D2 + L. Note that 2(D1 + D2 + L) = (D1 + D2 ) + (D1 + D2 + 2L) and that the second addend is the pull-back of the ample divisor {O1 } × E2 + E1 × {O2 }; hence (D1 + D2 + 2L) is a big divisor, and so is

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the sum 2(D1 + D2 + L) and D1 + D2 + L. Hence Vojta’s Conjecture applies, and provides (conjecturally) the degeneracy of the integral points on X. To deduce the strong conclusion of our Conjecture, we need to classify the possible infinite families of solutions, corresponding to integral points on curves on X. By Siegel’s theorem, such curves must be non-hyperbolic; we conclude via the following Lemma 6.10. – In the above notation, if the elliptic curve E1 is not isomorphic to E2 , the only non-hyperbolic curve on X is the intersection with X of the ex˜ → E1 × E2 . If E1 is isomorphic to E2 ceptional divisor L of the blow-up X (over C) the quasi-projective variety contains complete non-hyperbolic curves, which are all obtained as pre-image in X of algebraic subgroups in E1 × E2 of the form {(P, Q) ∈ E1 × E2 | Q = Φ(P )} for some isomorphism Φ : E1 → E2 . If E1 = E2 has no complex multiplication, the only such subgroups are defined by equations of the form P = ±Q. The proof is rather easy; we address to Ch. 4, §5.1 of the book [25] for the details. Note that the extra non-hyperbolic curves arising in case of complex multiplication are irrelevant for our problem, since the Q-rational points on such curves cannot be Zariski-dense. Unfortunately, we cannot prove the degeneracy of the integral points on that surface X, so our conjecture is still an open problem. However we can prove, using Theorem 6.6, the following weaker result: Theorem 6.11. – Let E1 , E2 be two elliptic curves in Weierstrass equation, with origins O1 , O2 respectively. Let A1 6= O1 (resp. A2 6= O2 ) be a rational point on E1 (resp. E2 ). Suppose there are infinitely many pairs (P1 , P2 ) ∈ E1 (Q) × E2 (Q) such that (6.12)

d(P1 ) = d(P2 )

and

d(P1 − A1 ) = d(P2 − A2 ).

Then E1 is isomorphic to E2 over Q and after suitably identifying E1 ' E2 we have A1 = A2 and, unless 2A1 = O, for all but finitely solutions of (6.12), P1 = P2 . If A1 = A2 has order two, then for all but finitely many solutions P1 = ±P2 . Proof. – Let us denote by Oi , i = 1, 2, the point at infinity of the curve Ei , which we take as neutral element for the group law. We start by noting that for (P1 , P2 ) ∈ (E1 × E2 )(Q), the condition (6.12) means that “for every integer m, P1 reduces to O1 modulo m if and only if P2 reduces to O2 modulo m; also P1 reduces to A1 modulo m if and only if P2 reduces to A2 modulo m”. In geometric terms, the point (P1 , P2 ) reduces to the divisor {O1 } × E2 + E1 × {O2 } only when (P1 , P2 ) reduces to the point (O1 , O2 ) and it reduces to the divisor {A1 } × E2 + E1 × {A2 } only if it reduces to the point (A1 , A2 ). In more geometric terms, define Y˜ → E1 × E2 to be the blow-up of the abelian surface E1 × E2 over the two points (O1 , O2 ), (A1 , A2 ). Let D1 (resp. D2 ) be the strict transform of {O1 } × E2 (resp. E1 × {O2 }) and C1 (resp. C2 ) the strict transform of the divisor {A1 } × E2 (resp. E1 × {A2 }).

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Then a rational point R ∈ Y˜ (Q), not belonging to the exceptional divisors, lying over a point (P1 , P2 ) ∈ (E1 × E2 )(Q) is integral with respect to D1 + D2 + C1 + C2 if and only if (P1 , P2 ) is a solution to Equation (6.12). Hence the solutions to our Equation (6.12) correspond to the integral points on the quasi-projective surface Y := Y˜ (D1 + D2 + C1 + C2 ). Note the natural morphism Y → X, where X is the surface defined in the above discussion, sending integral points on Y to integral points on X. A generic integral point on X, however, might lift to a rational non-integral point on Y . We would like to apply Theorem 6.6 to the quasi projective variety Y . The presence of a dominant map Y → E1 × E2 guarantees that the logarithmic irregularity of Y is at least 2 = dim Y . Moreover, the complete variety Y˜ has irregularity exactly 2, being birational to an abelian surface. Our aim is to exploit the divisors that we removed to produce a 1-form with logarithmic singularities along the removed divisors, thus producing a map to a semi-abelian variety of dimension 3. As we remarked, on the first elliptic curve E1 one can construct a meromorphic 1-form ω1 with simple poles at O1 and A1 ; automatically, the two residues will be the opposite one of the other; we can suppose that the residue at O1 is 2πi, while at A1 is −2πi. We can do the same on the second elliptic curve E2 , producing a meromorphic 1-form ω2 with simple poles at O2 , A2 and corresponding residues 2πi, −2πi. Denoting by πi : Y˜ → Ei the canonical projections, let us compute the pole divisor of the 1-form on Y˜ ω := π1∗ ω1 − π2∗ ω2 . Certainly, it has simple poles at D1 , D2 , C1 , C2 and is regular at any point not sent to {A1 , O1 , A2 , O2 } by the two projections. It only remains to check what happens over the exceptional divisors of the blow-up. We claim that these divisors are not poles of ω. Let us make the explicit calculation in local coordinates. Let t be a local parameter at O1 in E1 and s a local parameter at O2 in E2 . Up to a regular term, the forms ω1 , ω2 are expressed locally as ω1 =

dt , t

ω2 =

ds . s

The blow-up of the point (O1 , O2 ) on the surface E1 × E2 can be locally described by the equation tη = sξ, (t, s) ∈ C2 , (ξ : η) ∈ P1 , and the exceptional divisor lies over (t, s) = (0, 0). Over the open set (ξ : η) 6= (1 : 0), we can put η = 1 and use the coordinates s, ξ, while t = s · ξ. Then the 1-form ω can be written as dt ds sdξ ξds ds dξ ω= − = + − = , t s sξ sξ s ξ

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which is regular. Hence ω is regular on Y , with logarithmic poles at infinity. It follows that the generalized Albanese variety of Y is three-dimensional, being an extension of E1 × E2 by Gm . Then Vojta’s Theorem 6.6 applies and implies the degeneracy of the integral points on Y . Now, the possible infinite families of solutions, corresponding to curves on Y , also provide infinite families on X, and these have been already classified in Lemma 6.10. We obtain that if such infinite families do exist, E1 and E2 are isomorphic and after identifying E1 with E2 the pairs (P1 , P2 ) satisy P1 = ±P2 . But the curve defined by P1 = P2 gives rise to a complete curve on Y (hence non-hyperbolic) only if A1 = A2 ; so if such infinite family of solutions exist, we must have A1 = A2 . Otherwise, the only infinite family must be that of the form P1 = −P2 , which again can exist only if A1 = −A2 . In that case, after applying to E2 the automorphism P 7→ −P , we obtain another identification between E1 and E2 under which A1 coincides with A2 . If A1 = A2 is of order 2, then both infinite families are present. We end the discussion on Theorem 6.11 by linking it with a classical arithmetical problem of Erdős and Woods. For a natural number n ∈ N, denote by P (n) its set of its prime divisors: suppose that two natural numbers m, n satisfy the two equalities of sets: P (m) P (m + 1)

= P (n) = P (n + 1).

Can one derive the equality m = n? The answer is known to be negative, as shown by the infinite family of pairs m = 2(2h − 1),

n = 2h+2 (2h − 1),

so that m + 1 = 2h+1 − 1 and n + 1 = 22h+2 − 2h+2 + 1 = (m + 1)2 . However, no infinite families of pairs m < n with P (m)

= P (n)

P (m + 1)

= P (n + 1)

P (m + 2)

= P (n + 2)

are known. Erdős and Woods conjectured that there exists an integer k such that, given two natural numbers m, n, the equalities P (m+i) = P (n+i) for i = 0, . . . , k−1 implies the equality x = y. Of course, in the equality of sets P (m) = P (n) one does not take into account the multiplicities with which the primes appear in the factorizations of m and n. If one wants to take into account these multiplicities, it is necessary to disregard a finite set of primes, in order to avoid trivialities. Then a natural analogue with multiplicity of the Erdős-Woods problem might be asking whether several consecutive ratios x/y, (x + 1)/(y + 1),. . . , (x + k)/(y + k) can consist of S-units.

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In this respect, we can prove the following Theorem 6.13. – Let O S ⊂ Q be a finitely generated ring. If the group of units O ∗S is infinite, there exist infinitely many pairs of distinct natural numbers m < n such that ( ∗ n m ∈ OS ∗ n+1 m+1 ∈ O S . For every finitely generated ring O S ⊂ Q, the system  ∗ n   m ∈ OS n+1

m+1   n+2 m+2

∈ O ∗S ∈ O ∗S

has only finitely many solutions. The first part of the Theorem can be interpreted as a density result for the integral points on a certain surface, and will be discussed in the last section. The finiteness statement can be easily deduced from the S-units equation theorem in three variables, i.e., once again can be interpreted as a result on integral points on surfaces, the topic of next section.

7. Integral points on surfaces We have already encountered some problems reducing to questions on integral points on surfaces. We consider here further problems of that type, partially following the presentation in [16]. We first consider the problem of rational points. In that case we can consider surfaces up to birational isomorphism. The birational classification of (complex) algebraic surfaces was carried out by the Italian school at the end of the 19th century, and led to the following list — Rational surfaces. These are the surfaces birationally isomorphic to the plane; it is the case of all smooth hypersurfaces of degree ≤ 3 in projective 3-space. — Ruled surfaces, i.e., surfaces birationally isomorphic to a product ˜C × P1 , where C

is a curve (if C is the line, then the resulting surface will be rational).

— Elliptic surfaces. They can be thought of as elliptic curves over a 1-dimensional function field; in other words they are surfaces admitting a dominant map ˜ 99K ˜C whose generic fiber has genus one. They can belong to other families X (e.g., they can be rational, or so-called Enriques surfaces). — Abelian surfaces, i.e., abelian varieties of dimension two.

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— K3 surfaces. These are (smooth projective) surfaces which are simply connected and whose canonical bundle is trivial. Being simply connected, they admit no non-zero regular 1-forms, so their cotangent bundle is certainly not trivial, unlike what happens for abelian surfaces. They might admit a fibration to P1 , with elliptic generic fiber, so they can be elliptic in our sense. All smooth quartics in P3 are K3 surfaces, as well as the smooth hypersurfaces of multi-degree (2, 2, 2) in P31 . — Kummer, bielliptic (or hyper-elliptic) surfaces. They are obtained as quotients of abelian surfaces. For instance a Kummer surface is the normalization of the quotient of the form A/ ± Id, where A is an abelian surface and −Id is the involution of A sending P 7→ −P . — Surfaces of general type: all the remaining ones. They are characterized by having a canonical divisor which is big. It is the case for all smooth hypersurfaces of P3 of degree ≥ 5, as well as those with irregularity ≥ 2 which are not abelian varieties. According to Bombieri’s Conjecture (Lang-Vojta’s Conjecture in the case of compact surfaces) the set of rational points on surfaces of general type should be degenerate. Let us then analyze the other classes of surfaces. The rational points on rational surfaces are clearly potentially dense. The same is true of the ruled surfaces with elliptic base. Elliptic surfaces with rational base can have a Zariski-dense set of rational points, e.g., whenever they admit a section of infinite order (with respect to the group law on the fibers). Abelian varieties admit algebraic points which generate a Zariski dense group: hence the rational points are potentially dense. Little is known in general on K3 surfaces, but in several cases (e.g., when they admit elliptic fibrations) one can show the potential density of rational points, which is believed to hold in general. Kummer and bielliptic surface always satisfy potential density of rational points, since they are dominated by abelian varieties. Finally the Enriques surfaces, which always admit elliptic fibrations, are known to satisfy potential density of rational points (see [8]). We shall concentrate from now on on integral points on surfaces. We have already remarked, while discussing Vojta’s Conjecture at §1.4, that the complement in P2 of a curve of degree ≥ 4, with normal crossing singularities, is conjectured to have degenerate sets of integral points. We said that this question is open, the only general result being proved when the curve has at least four components. We add that in some cases the degeneracy of integral points has been proved on the complement of an irreducible curve (see [35], [70], [46]) ; however the method of proof, consisting on increasing the number of components at infinity after taking an unramified cover, only works for highly singular curves, never for curves with normal crossing singularities.

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The degeneracy on the complement of a curve D on P2 with r ≥ 4 components is proved by mapping P2 D to Gr−1 m . This map is constructed from functions having zeros and poles in the support of D, and there exist r − 1 multiplicative independent functions of that type, since any two divisors on P2 are linearly dependent in the Picard group. The same strategy holds if one removes four (or more) divisors on any algebraic surface whenever they define a rank-1 subgroup in the Picard group. This number four can be compared with the number three in Siegel’s Theorem 4.3: recall that a basically equivalent formulation of Siegel’s Theorem on curves states that on every affine curve with at least three points at infinity, the set of integral points is finite. However, in higher dimensions, one cannot expect to prove any degeneracy result valid for all surfaces with four divisors removed; actually, for every number n one can easily construct an affine surface whose set of integral points is Zariski dense and whose divisor at infinity consists of n components: simply starting from the affine plane A2 , viewed as a complement of a line in the projective plane; after blowing-up n − 1 points at infinity, the same affine plane becomes the complement in a projective surface of a set of n − 1 curves (and the full divisor at infinity admits normal crossing singularities). The aim of the next section is to provide a criterion involving the intersection matrix of the divisor at infinity. 7.1. A Subspace Theorem approach. – Most of the results and proofs in this section are based on the paper [20]. Here is the announced general statement on integral points on surfaces: ˜ be a smooth projective surface, D1 , . . . , Dr be irreducible curves, Theorem 7.1. – Let X no three of them intersecting. Assume there exist positive integers p1 , . . . , pr such that — the divisor D = p1 D1 + · · · + pr Dr is big and numerically effective; — for each i = 1, . . . , r, letting ξ be the minimal (real) solution to the equation Di2 ξ 2 − 2(D.Di )ξ + D2 = 0, (which, by Hodge index theorem, admits real solutions, since D is big and nef ) the inequality 2ξD2 > (D.Di )ξ 2 + 3pi D2

(7.2) holds.

˜ |D| such that for every Then there exists a (possibly reducible) curve Y ⊂ X := X ring of S-integers O S , the set X( O S ) Y ( O S ) of the S-integral points on X not lying on Y is finite.

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In particular, the set X( O S ) is not Zariski-dense, but the conclusion of the theorem is stronger, since it implies that the 1-dimensional part of the Zariski-closure of the set of integral points is independent of O S (for a sufficiently large ring O S ). The idea of the proof is the same as the one for Siegel’s theorem: consider a finite ˜ |D|. dimensional vector space of regular functions on X = X Suppose we dispose of an infinite sequence P1 , P2 , . . . of S-integral points of X. Letting f1 , . . . , fd be a basis for this vector space; after possibly multiplying each function by a non-zero integer, we obtain that f1 , . . . , fd take S-integral values at the S-integral points of X. ˜ ν ) is compact for every valuation, in particular for every valuation in S, Since X(κ from any sequence of integral points one can extract a sequence of points converging in each valuation of S. Since the values of the function fi at S-integral points are bounded by 1 in each valuation outside S, the height of fi (P ), for P an S-integral point, must be given by some absolute values in S. This means that the sequence converges to some point at infinity Qν ∈ |D|(κν ) in at least one valuation ν of S. Let us find a new basis g1 , . . . , gd of the κ-vector space generated by f1 , . . . , fd , such that the product g1 (P ) · · · gd (P ) is as small as possible at the place ν. Recall that the fi might have poles at the divisor at infinity; however, by making suitable linear combinations of them, we can hope to find functions g1 , . . . , gd whose products has more zeros than poles at each component at infinity containing Qν . Expressing the gi = gi,ν as linear combination, gi,ν = Li,ν (f1 , . . . , fd ), where Li,ν (T1 , . . . , Td ) is a linear form with rational coefficients, we obtain “small” values, with respect to the place ν, of the form Li (f1 (P ), . . . , fd (P )), for each S-integral point P of the selected infinite sequence. If this procedure can be performed at every valuation of S, then the double product d YY |Li,ν (f1 (P ), . . . , fd (P ))|ν ν∈S i=1

maxj (|fj (P )|ν )

will be smaller than a suitable negative power of the height of the point (f1 (P ), . . . , fd (P )). An application of the Subspace Theorem will permit to conclude that some fixed non-zero linear form in f1 , . . . , fd vanishes for infinitely many point of the sequence. Since this must hold for every subsequence of the given sequence, the degeneracy follows easily. The main difference between dimension one and higher dimensions lies in the fact that the irreducible components of a divisor on a curve are single points, while on a surface they are curves. Now, given a vector space of regular functions (say) on a neighborhood of a point P on a curve, the subspace of those having at that point a zero of order k has codimension ≤ k on the whole space. Replacing the point P on a curve with a curve C on a surface, things change dramatically. The subspace of the functions vanishing on C is no more a hyperplane

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on the whole space. Its codimension depends on the geometry of the curve C relatively to the vector space of rational functions we are considering. In the case of our concern, when the vector space of functions is the full linear system attached to a divisor, the codimension of those functions vanishing on C can be estimated via the following lemma: ˜ be a smooth complete surface. Let D be a divisor on X ˜ and C Lemma 7.3. – Let X ˜ Then an irreducible curve on X. ˜ O ˜ (D))/H 0 (X, ˜ O ˜ (D − C))) ≤ max(0, 1 + D · C). dim(H 0 (X, X X In the above formula, the symbol D · C denotes the intersection product of D and C. The lemma can be applied also when C is a component of D; it then give an estimate ˜ O ˜ (D)) formed by those functions having of the codimension of the subspace of H 0 (X, X ˜ O ˜ (D)). a pole on C of lesser order than that of a generic function in the set H 0 (X, X The proof follows by taking the cohomology of the short exact sequence 0 → O X˜ (D − C) → O X˜ (D) → O X˜ (D)|C → 0. The first steps of the long cohomology sequence ˜ O ˜ (D − C)) → H 0 (X, ˜ O ˜ (D)) → H 0 (X, ˜ O ˜ (D) ) → · · · 0 → H 0 (X, X X X |C provides an embedding ˜ O ˜ (D))/H 0 (X, ˜ O ˜ (D − C)) ,→ H 0 (X, ˜ O ˜ (D) ) = H 0 (C, O ˜ (D) ). H 0 (X, X X X X |C |C The last term is the space of global sections of a line bundle of degree D · C on an irreducible curve. Hence its dimension is bounded by 1 + D · C and the lemma follows. The above estimates are responsible for the apparence of the intersection products on the statement of Theorem 7.1. The details of the proof can be found in [20], [16], [25] or in Bilu’s Bourbaki lecture [7]. Let us comment on the condition expressed by the inequalities (7.2). Whenever the divisors D1 . . . , Dr are algebraically equivalent (or more general algebraically equivalent up to multiplicative constant), it turns out that one can find some weights p1 , . . . , pr verifying (7.2) for all i = 1, . . . , r if and only if r ≥ 4. This fact is in accordance with the fact that removing three lines on the plane one still obtain a surface with potentially dense integral points. On the other hand, A. Levin and P. Autissier (see [46] or [7]) proved that whenever the Di are ample divisors, the hypothesis of Theorem 7.1 again reduces to r ≥ 4. As ˜ observed by Levin in his paper [46], for every smooth projective algebraic surface X, the complement of four ample divisors is of log-general type (while, once again, the example of P2 (three lines) shows that the number 4 cannot be lowered to 3). Another interesting case to which Theorem 7.1 can be applied is shown in the next statement, proved in [22].

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Corollary 7.4. – Let D1 , D2 , D3 be three irreducible, ample, algebraically equivalent ˜ Let D4 be any effective divisor and suppose divisors on a smooth complete surface X. ˜ (D1 ∪ · · · ∪ D4 ) that no three of them intersect. Then the integral points on X = X are not Zariski-dense. The method introduced in [20] has been much developed in higher dimensions by Levin and Autissier (see [46], [1], [2], [7]). 7.2. Integral points on certain rational surfaces. – In view of the fact that the set of rational points on a rational surface is potentially dense, and in some sense denser than on any other kind of surfaces (at least according to Manin’s conjecture) it is tempting to investigate the behavior of integral points on affine (or quasi-projective) rational surfaces. We have already discussed in some detail the case of the complement of a curve in the projective plane. After the projective plane, the first natural example of a rational surface is constituted by (smooth) quadric surfaces on P3 . These surfaces can be identified with P1 × P1 and the divisors on it are identified modulo linear equivalence by their bidegree. The canonical divisors have bi-digree (−2, −2), hence it is conjectured that the removal of a divisor of bidegree (a, b) with a ≥ 3, b ≥ 3 (and with only normal crossing singularities) produces an affine surface with degenerate sets of integral points. Again, this is not settled in general, but can be proved whenever the divisor at infinity has at least four components. However, one case of a divisor with three components was provided in [22]: ˜ ⊂ P3 be a smooth quadric surface, H1 , H2 , H3 irreducible hyTheorem 7.5. – Let Q perplane sections sharing a common point where they intersect transversally. Then ˜ (H1 ∪ H2 ∪ H3 ) are not Zariski-dense. the integral points on Q The proof consists on reducing to the situation of Corollary 7.4 after blowing ˜ denote the new surface, Di , up the point of intersection of H1 , H2 , H3 . Letting X for i = 1, 2, 3, the strict transform of Hi and D4 the exceptional divisor, we can ˜ (D1 ∪ D2 ∪ D3 ∪ D4 ) apply the mentioned corollary. Since the integral points on X ˜ correspond bijectively to the integral points on Q (H1 ∪ H2 ∪ H3 ) the conclusion of Corollary 7.4 implies the conclusion of the theorem. ˜ (H1 ∪H2 ∪H3 ) can be viewed as the solutions The integral points on the surface Q to a divisibility problem, as we now explain, following [22], §4. ˜ is given in P3 by the homogeneous equation Suppose that Q X0 (X1 + X2 + X3 ) = Q(X1 , X2 , X3 ), for a quadratic form Q in three variables (chosen so that the above equation defines a smooth surface). Take for Hi the hyperplane section defined by xi = 0, for i = 1, 2, 3. The three curves H1 , H2 , H3 meet at the point (1 : 0 : 0 : 0). A rational point (x0 : x1 : x2 : x3 ) ∈ P3 (κ) is integral with respect to these three divisors if coordinates

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can be chosen such that y := x0 is an S-integer, while u := x1 , v := x2 , w := x3 are S-units. We then obtain the Diophantine equation y(u + v + w) = Q(u, v, w), which is equivalent to the integrality condition Q(u, v, w) ∈ OS . u+v+w Several other problems on integral points on rational surfaces can be reduced to divisibility problems. Even in dimension one, Siegel’s finiteness result, in the case of rational curves, can be expressed in terms of divisibility: given two coprime polynomials f (X), g(X) ∈ O S [X], if for infinitely many S-integers x ∈ O S f (x) divides g(x) in the ring O S , then f has at most one complex root. Considering polynomials in two variables, some extensions of the above statement are possible. The S-unit equation theorem for three variables is an example. Solving the equation u + v + w = 1 in S-units amounts to finding two S-integers x, y ∈ O S such that x | 1, y | 1, (1 − x − y) | 1, so the values of three polynomials, namely X, Y, 1 − X − Y , divide the values of three more polynomials, in this case all taken to be the constant 1 polynomial. By applying once again Theorem 7.1, it was proved in [24] the following Theorem 7.6. – Let, for i = 1, 2, 3, (fi (X, Y ), gi (X, Y )) three pairs of non-zero polynomials satisfying deg fi ≥ deg gi . Suppose they satisfy the general position assumptions below. Then the set of pairs of S-integers (x, y) ∈ O 2S such that fi (x, y) | gi (x, y) for i = 1, 2, 3, are not Zariski-dense in the plane. The general position assumptions: — for each 1 ≤ i < j ≤ 3 the curves of equation fi = 0 and fj = 0 do not meet at infinity (under the canonical embedding A2 ,→ P2 ); — there exist no common zero to the three polynomials f1 , f2 , f3 ; — for each i such that the polynomial gi is not constant, the two affine curves fi = 0 and gi = 0 intersect transversely; — for 1 ≤ i < j and h ∈ {i, j}, the three curves fi = 0, fj = 0 and gh = 0 have no points in common. As mentioned, the S-unit equation theorem is the case deg fi = 1 and deg gi = 0 for each i = 1, 2, 3. Already the case in which deg fi = deg gi = 1 for i = 1, 2, 3 escapes from any attempt based on the S-unit equation theorem, and leads to the following result.

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Theorem 7.7. – Let L1 , . . . , L4 be four lines in general position on the projective plane. Let Pi ∈ Li , for i = 1, 2, 3, be a point of Li , not belonging to other lines on the ˜ be the surface obtained by blowing-up the three points P1 , P2 , P3 configuration. Let X and let Di be the strict transform of Li , for i = 1, 2, 3, 4. Then the integral points ˜ (D1 ∪ D2 ∪ D3 ∪ D4 ) are not Zariski-dense. on X = X Let us show the link between the above statement and divisibility problems; this example will show how to connect in general divisibility questions to problems on integral points on varieties. Suppose L4 is the line at infinity, so that the integral points with respect to L4 are the pairs of S-integers (x, y) ∈ A2 ( O S ). If fi = 0 is the equation of Li , for i = 1, 2, 3, the points Pi can be defined by a system of equations fi = gi = 0. Now the condition that gi (x, y)/fi (x, y) be an integer, amounts to saying that no prime divides fi (x, y) unless it divides also gi (x, y), and in that case it must divide gi (x, y) with at least the same multiplicity. Geometrically, this means that after blowing-up the point Pi , the corresponding point induced by (x, y) on the blown-up surface does not reduce to the strict transform of the line fi = 0. An interesting feature of the surface appearing in Theorem 7.7 is that the surface X in Theorem 7.7 is simply connected. As we explained in the discussion following Theorem 6.6, when a smooth (projective or quasi-projective) variety is simply connected, no method based on S-unit equations or on abelian varieties can be applied, since the log-irregularity vanishes. 7.3. Final remarks around Vojta’s Conjecture. – In this last section, we present some remarks around Vojta’s conjecture. We shall show via a simple example that the normal crossing condition in Vojta’s conjecture cannot be removed. Consider the case of a conic and two non-tangent lines intersecting on the conic. After a coordinate change we can suppose that the lines are defined by ZX = 0 and the conic by XY + Z 2 = Y Z, so the three components meet at the point (0 : 1 : 0). The integral points on the complement of the pair of lines correspond to pairs (u, y) with u ∈ O ∗S and y ∈ O S . The further integrality requirement, due to the removal of the conic, amounts to imposing that uy − y + 1 be an S-unit. We then obtain the equation v = uy − y + 1, which can be written in the form of a divisibility problem (7.8)

v−1 ∈ OS . u−1

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Clearly, over a sufficiently large ring of S-integers, the solutions are Zariski dense (in the plane). For instance, there exist infinitely many pairs of natural numbers (m, n) such that 3m − 1 ∈ Z. 2n − 1 It suffices to chose an odd number n, so that 2n − 1 is coprime with 3, and to set m to be the order of 3 modulo 2n − 1. The first part of Theorem 6.13 is reduced to the density of the solution of a divisibility relation like (7.8). As a second remark, we point out a curious coincidence: consider a smooth irreducible curve D ⊂ P2 in the plane. As we said, Vojta’s Conjecture predicts the degeneracy of the integral points on the complement P2 \ D when deg D ≥ 4. Also, the same conjecture, referred to lower dimension and to the compact case, which is Falting’s theorem, predicts the degeneracy of the rational points on D precisely whenever deg D ≥ 4. When D is reducible, say the union of two smooth curves D1 , D2 intersecting transversally, the condition of Vojta’s conjecture becomes deg D1 + deg D2 ≥ 4. In that case, one should compare with the condition on Vojta’s conjecture for integral points on D1 relative to the divisor D1 ∩D2 and on D2 relative to the divisor D1 ∩D2 . It turns out that the two affine curves D1 (D1 ∩ D2 ) and D2 (D1 ∩ D2 ) satisfy simultaneously the hypothesis of Siegel’s theorem, so that their set of integral points is finite, if and only if the pair (P2 , D) satisfies the hypothesis of Vojta’s conjecture on surfaces, predicting the degeneracy of the integral points on P2 with respect to D. For example, if D1 , D2 are conics intersecting on four points, then the affine curves Di (D1 ∩ D2 ) are isomorphic to the complement of four points on the line. If, on the other hand, D1 is a cubic and D2 a line, then D1 (D1 ∩ D2 ) is a genus one curve deprived of three points while D2 (D1 ∩ D2 ) ' P1 {0, 1, ∞}. The same happens in arbitrary dimension with any number of components (e.g., with hyperplanes in general position in Pn ). Vojta’s conjecture has been revisited by F. Campana in a series of paper (see e.g., [11]). We present here a (very) simplified version of a conjecture proposed by Campana. ˜ ˜ if Define a point p ∈ X(κ) to be half-integral with respect to a divisor D ⊂ X p∈ / D and for every prime ideal P ⊂ O S , if p reduces modulo P to some point of D, then p reduces modulo P 2 to a point of D. The mentioned simplified version of Campana’s conjecture is the following: ˜ be a smooth projective variety over a number field κ, O S ⊂ κ a Conjecture. – Let X ring of S-integers. Let D = D0 + D00 be a reduced effective normal crossing divisor ˜ Let K ˜ be a canonical divisor for X. ˜ If the Q-divisor on X. X 1 KX˜ + D0 + D00 2

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˜ D which are S-integral with respect to D0 is big, then the set of rational points of X 00 and half-S-integral with respect to D is not Zariski-dense. For example, given a square-free polynomial f (X) ∈ Z[X], the integers x ∈ Z such that f (x) is a ‘powerfull number’ (all its prime factors appear with multiplicity ≥ 2),are half-integral with respect to the zero set of f (X). According to Campana’s conjecture, there should be only finitely many such numbers, whenever the degree of f (X) is at least three. This would be also a consequence of the abc conjecture. Note that the condition that an integer number n is powerfull can be expressed in term of the solvability of the Diophantine equation n = x2 y 3 . Some cases of Campana’s conjecture can be proved over function fields: for instance, the Stother-Mason abc inequality for polynomials is an example in dimension 1. In dimension two, some cases are settled in [23], and a very general result has been proved ˜ of arbitrary dimension n with q(X) ˜ ≥ n. by Yamanoi [68] for compact varieties X However, in the arithmetic context, little is known, due to the well-known difficulty of exploiting the ramification term in the Diophantine inequalities. Acknowledgments. – The author would like to thank Frédéric Campana for useful discussions on these topics. He is also very grateful to an anonymous referee, who pointed out several inaccuracies in a previous version, and suggested some improvements. References [1] P. Autissier – “Géométrie, points entiers et courbes entières”, Ann. Sci. Éc. Norm. Supér. 42 (2009), p. 221–239. [2]

, “Sur la non-densité des points entiers”, Duke Math. J. 158 (2011), p. 13–27.

[3] A. Baker (ed.) – New advances in transcendence theory, Cambridge Univ. Press, 1988. [4] A. Beauville – Surfaces algébriques complexes, Société Mathématique de France, 1978, Avec une sommaire en anglais, Astérisque, No. 54. [5] F. Beukers – “Ternary form equations”, J. Number Theory 54 (1995), p. 113–133. [6] Y. Bilu, M. Strambi & A. Surroca – “Quantitative Chevalley-Weil theorem for curves”, Monatsh. Math. 171 (2013), p. 1–32. [7] Y. F. Bilu – “The many faces of the subspace theorem [after Adamczewski, Bugeaud, Corvaja, Zannier. . .]”, Séminaire Bourbaki, vol. 2006/2007, exposé no 967, Astérisque 317 (2008), p. 1–38. [8] F. A. Bogomolov & Y. Tschinkel – “Density of rational points on Enriques surfaces”, Math. Res. Lett. 5 (1998), p. 623–628. [9] E. Bombieri – “The Mordell conjecture revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), p. 615–640. [10] E. Bombieri & W. Gubler – Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge Univ. Press, 2006.

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[11] F. Campana – “Orbifolds, special varieties and classification theory”, Ann. Inst. Fourier 54 (2004), p. 499–630. [12] F. Catanese – “Topological methods in moduli theory”, Bull. Math. Sci. 5 (2015), p. 287–449. [13] J.-L. Colliot-Thélène, P. Swinnerton-Dyer & P. Vojta – Arithmetic geometry, Lecture Notes in Math., vol. 2009, Springer; Fondazione C.I.M.E., Florence, 2011. [14] P. Corvaja – “Autour du théorème de Roth”, Monatsh. Math. 124 (1997), p. 147–175. [15]

, “Problems and results on integral points on rational surfaces”, in Diophantine geometry, CRM Series, vol. 4, Ed. Norm., Pisa, 2007, p. 123–141.

[16]

, Integral points on algebraic varieties, Institute of Mathematical Sciences Lecture Notes, vol. 3, Hindustan Book Agency, New Delhi, 2016.

[17] P. Corvaja, A. Levin & U. Zannier – “Integral points on threefolds and other varieties”, Tohoku Math. J. 61 (2009), p. 589–601. [18] P. Corvaja & J. Noguchi – “A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties”, Math. Ann. 353 (2012), p. 439–464. [19] P. Corvaja & U. Zannier – “A subspace theorem approach to integral points on curves”, C. R. Math. Acad. Sci. Paris 334 (2002), p. 267–271. [20]

, “On integral points on surfaces”, Ann. of Math. 160 (2004), p. 705–726.

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, “Some cases of Vojta’s conjecture on integral points over function fields”, J. Algebraic Geom. 17 (2008), p. 295–333.

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, Applications of Diophantine approximation to integral points and transcendence, Cambridge Tracts in Mathematics, vol. 212, Cambridge Univ. Press, 2018.

[26] H. Darmon & A. Granville – “On the equations z m = F (x, y) and Axp + By q = Cz r ”, Bull. London Math. Soc. 27 (1995), p. 513–543. [27] O. Debarre – Higher-dimensional algebraic geometry, Universitext, Springer, 2001. [28] F. J. Dyson – “The approximation to algebraic numbers by rationals”, Acta Math. 79 (1947), p. 225–240. [29] J.-H. Evertse & R. G. Ferretti – “Diophantine inequalities on projective varieties”, Int. Math. Res. Not. 2002 (2002), p. 1295–1330. [30]

, “A generalization of the Subspace Theorem with polynomials of higher degree”, in Diophantine approximation, Dev. Math., vol. 16, Springer, 2008, p. 175–198.

[31] J.-H. Evertse, K. Gyory, C. Stuart & R. Tijdeman – “S-unit equation and their applications”, in Proceedings of the Symposium on Transcendental Number Theory held at the University of Durham, Durham, July 1986 (A. Baker, ed.), Cambridge Univ. Press, 1988.

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[32] J.-H. Evertse & H. P. Schlickewei – “A quantitative version of the absolute subspace theorem”, J. reine angew. Math. 548 (2002), p. 21–127. [33] G. Faltings – “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. math. 73 (1983), p. 349–366. [34]

, “Diophantine approximation on abelian varieties”, Ann. of Math. 133 (1991), p. 549–576.

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, “A new application of Diophantine approximations”, in A panorama of number theory or the view from Baker’s garden (Zürich, 1999), Cambridge Univ. Press, 2002, p. 231–246.

[36] C. Fuchs & U. Zannier – “Integral points on curves: Siegel’s theorem after Siegel’s proof”, in On Some Applications of Diophantine Approximations (U. Zannier, ed.), Publications of the Scuola Normale Superiore, vol. 2, Springer, 2014, p. 139–157. [37] C. Gasbarri – “Dyson’s theorem for curves”, J. Number Theory 129 (2009), p. 36–58. [38] A. O. Gel0 fond – Transcendental and algebraic numbers, Dover Publications, 1960. [39] B. Hassett & Y. Tschinkel – “Density of integral points on algebraic varieties”, in Rational points on algebraic varieties, Progr. Math., vol. 199, Birkhäuser, 2001, p. 169– 197. [40] G. Heier & M. Ru – “Essentially large divisors and their arithmetic and functiontheoretic inequalities”, Asian J. Math. 16 (2012), p. 387–407. [41] M. Hindry & J. H. Silverman – Diophantine geometry, Graduate Texts in Math., vol. 201, Springer, 2000. [42] F. Klein – Lectures on the icosahedron and the solution of equations of the fifth degree, revised ed., Dover Publications, 1956. [43] S. Lang – “Integral points on curves”, Inst. Hautes Études Sci. Publ. Math. 6 (1960), p. 27–43. [44]

, Fundamentals of Diophantine geometry, Springer, 1983.

[45] A. Levin – “One-parameter families of unit equations”, Math. Res. Lett. 13 (2006), p. 935–945. [46]

, “Generalizations of Siegel’s and Picard’s theorems”, Ann. of Math. 170 (2009), p. 609–655.

[47] K. Mahler – “Über die rationalen Punkte auf Kurven vom Geschlecht Eins”, J. reine angew. Math. 170 (1934), p. 168–178. [48] D. McKinnon & M. Roth – “Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties”, Invent. math. 200 (2015), p. 513–583. [49] J. Noguchi – “A short analytic proof of closedness of logarithmic forms”, Kodai Math. J. 18 (1995), p. 295–299. [50] J. Noguchi & J. Winkelmann – “Holomorphic curves and integral points off divisors”, Math. Z. 239 (2002), p. 593–610. [51]

, Nevanlinna theory in several complex variables and Diophantine approximation, Grundl. math. Wiss., vol. 350, Springer, Tokyo, 2014.

[52] D. Ridout – “The p-adic generalization of the Thue-Siegel-Roth theorem”, Mathematika 5 (1958), p. 40–48.

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[53] A. Robinson & P. Roquette – “On the finiteness theorem of Siegel and Mahler concerning Diophantine equations”, J. Number Theory 7 (1975), p. 121–176. [54] K. F. Roth – “Rational approximations to algebraic numbers”, Mathematika 2 (1955), p. 1–20; corrigendum, 168. [55] M. Ru – “On a general Diophantine inequality”, Funct. Approx. Comment. Math. 56 (2017), p. 143–163. [56] M. Ru & P. Vojta – “A birational Nevanlinna constant and its consequences”, Amer. J. Math. 142 (2020), p. 957–991. [57] W. M. Schmidt – Approximation to algebraic numbers, Secrétariat de l’Enseignement Mathématique, Université de Genève, 1972. [58]

, Diophantine approximation, Lecture Notes in Math., vol. 785, Springer, 1980.

[59]

, Diophantine approximations and Diophantine equations, Lecture Notes in Math., vol. 1467, Springer, 1991.

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, Diophantine approximations and Diophantine equations, Lecture Notes in Math., vol. 1467, Springer, 1991.

[61] J-P. Serre – Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. [62] C. L. Siegel – “Ueber einige Anwendungen diophantischer Approximationen”, Abh. Pr. Akad. Wiss. 1 (1929), p. 209–266 (Ges. Abh., I, 209–266). English translation in [69]. [63] H. P. F. Swinnerton-Dyer – “A4 + B 4 = C 4 + D4 revisited”, J. London Math. Soc. 43 (1968), p. 149–151. [64] A. Thue – “Über Annäherungswerte algebraischer Zahlen”, J. reine angew. Math. 135 (1909), p. 284–305. [65] P. Vojta – Diophantine approximations and value distribution theory, Lecture Notes in Math., vol. 1239, Springer, 1987. [66]

, “Integral points on subvarieties of semiabelian varieties. I”, Invent. math. 126 (1996), p. 133–181.

[67] X – “The Integer Solutions of the Equation y 2 = axn + bxn−1 + · · · + k”, J. London Math. Soc. 1 (1926), p. 66–68. [68] K. Yamanoi – “Holomorphic curves in algebraic varieties of maximal Albanese dimension”, Internat. J. Math. 26 (2015), 1541006, 45. [69] U. Zannier (ed.) – On some applications of Diophantine approximations, Publications of the Scuola Normale Superiore, vol. 2, Springer, 2014. [70] U. Zannier – “On the integral points on the complement of ramification-divisors”, J. Inst. Math. Jussieu 4 (2005), p. 317–330. [71]

, “Roth’s theorem, integral points and certain ramified covers of P1 ”, in Analytic number theory, Cambridge Univ. Press, 2009, p. 471–491.

Pietro Corvaja, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Udine, Via delle Scienze, 206, 33100 Udine (Italy)

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FIBRATIONS IN ALGEBRAIC GEOMETRY AND APPLICATIONS by Claire Voisin

Abstract. – We give a survey of various methods for constructing rational fibrations on algebraic varieties (i.e., dominant rational mappings of normal varieties that induce an algebraically closed extension of function fields), and their applications. These fibrations are a major tool in the classification theory of algebraic varieties. The most important among them are the Iitaka fibration, the MRC fibration, and the Gamma fibration. We present them together with several concrete modes of use. We discuss finally the core fibration, introduced more recently by Campana, which is a conjectural bridge between these algebraically defined fibrations and hyperbolicity.

0. Introduction This paper is devoted to a crucial tool for the study of algebraic varieties, namely fibrations. We will in this survey paper use the following definition. Definition 0.1. – A (rational) fibration on a projective variety or compact complex manifold X is a dominant (rational or meromorphic) map X 99K Y with connected general fiber. The obvious interest of a fibration is that it allows to deduce properties of the total space from properties of the base and of the fibers. For example, the following facts hold: 1. If the base is Brody (resp. algebraically) hyperbolic, and all the fibers are Brody (resp. algebraically) hyperbolic then the total space is Brody hyperbolic. 2. If the base is of general type and the fiber is of general type, then the total space is of general type (see [32], [49]). 3. If the base is rationally connected and the fiber is rationally connected, so is the total space (see Section 3.2, [28]).

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Note however that a general fibration, especially rational fibration, may not be very useful : Indeed, starting from any smooth projective variety X, and choosing a e 99K P1 Lefschetz pencil of high degree hypersurfaces in X, we get a rational map X which does not say much about X since the base has Kodaira dimension −∞ and the fiber is of general type. What happens in this case is the fact that the blown-up locus is of codimension 2 in X but of codimension 1 and ample in the fibers. The case of a morphism is much better but an interesting morphism to a smaller dimensional basis does not always exist and many interesting fibrations are only defined as rational maps. This paper will rather describe carefully constructed fibrations for which the fibers are instead simpler than the total space, reducing in principle the study to phenomena on the base. There are two ways of constructing fibrations: The first way, which is more classical in birational geometry, consists in exploiting the presence of sections of the adequate tensors, or divisors. This method was initiated by Iitaka for sections of line bundles, and the exploitation of holomorphic contravariant tensors has been developed over the time by Castenuovo-de Franchis, Bogomolov, Catanese, Campana. We will describe this method in Section 1. The second way appears in [1], [13], [35] and consists in constructing geometrically the fiber through a general point x by imposing that it contains all points that can be reached from x by composing a certain number of allowed geometric processes. The two striking instances of this approach are the MRC fibration, where the fiber through a very general point contains all rational curves passing through this point, and the Shafarevich map or Γ-reduction, for which the fiber through a very general point contains all the varieties passing through this point and having a fundamental group with small image in the ambient space. This is described in Sections 3.2, 3.3. A third method presented in [16] beautifully combines both approaches to produce the so-called core fibration which will be described in Section 4. This fibration into special varieties is conjecturally the one which allows to understand the degeneracy of the Kobayashi pseudodistance at the general point of a variety. For a long period, standard conjectures about the Kobayashi pseudodistance ([37], [33]) were the following: Conjecture 0.2. – A variety of general type has its Kobayashi pseudometric nondegenerate at a general point. Very nice recent evidences for this conjecture have been obtained by DiverioMerker-Rousseau [24] and Brotbek [11], who work in the case of hypersurfaces in projective space, and by Demailly [23]. Conjecture 0.3. – A variety with trivial canonical bundle has a vanishing Kobayashi pseudodistance. We refer to [51] for examples and discussions concerning Conjecture 0.3. A beautiful progress on this conjecture has been obtained recently by Verbitsky [48] in the case of hyper-Kähler manifolds (see also Diverio’s contribution to this volume). The two

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conjectures together suggest a parallel between the Kodaira dimension of an algebraic variety and the degeneracy level of its Kobayashi pseudometric or pseudodistance. The great novelty of Campana’s ideas developed in [16], [17] is to suggest that apart from the two extreme cases presented in the two conjectures above, the relationships between these two measures of positivity of the cotangent bundle is much more subtle. Indeed, Campana makes the following conjecture (cf. [16]); Conjecture 0.4. – Special varieties have vanishing Kobayashi pseudodistance. As we will discuss in Section 4, special varieties may have all possible Kodaira dimensions except for the maximal one, i.e., they cannot be of general type. For example, most elliptic fibrations over Pn−1 will have Kodaira dimension n − 1 and will be special. An essential feature of projective or compact Kähler geometry which makes the second strategy work is the fact that parameter spaces for compact closed algebraic (or analytic) subsets of a projective (or compact Kähler) manifold is compact. We refer for Section 2.1 for details. If we drop the Kähler assumption, then the local deformation theory of closed analytic subschemes is unchanged, but we completely loose the compactness, as shows the example of the twistor family of a K3 surface S (see Section 2.1, Example 2.2).

1. Fibrations and holomorphic forms This section is devoted to the construction of rational fibrations using sections of line bundles or differential forms. The main applications concern the geometry of the canonical line bundle, with the beginning of a birational classification of algebraic varieties by the Kodaira dimension. 1.1. General facts on fibrations and holomorphic forms. – Let f : X → Y be a surjective morphism, where X and Y are smooth complex projective varieties or compact Kähler manifolds. Remark 1.1. – Starting from a morphism f : X → Y , up to replacing f by its Stein factorization fSt : X → YSt , one may assume the fibers are connected, but then YSt is only normal and some extra work is needed if one also wants smoothness of Y . The morphism f is proper and we have: Lemma 1.2. – The following properties hold: (i) The morphism f∗ : π1 (X) → π1 (Y ) has image of finite index (and is surjective if the fibers are connected). (ii) The morphisms f ∗ : H i (Y, Q) → H i (X, Q) are injective morphisms of Hodge structures.

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(iii) The morphisms f ∗ : H 0 (Y, ΩiY ) → H 0 (X, ΩiX ) are isomorphisms for i ≥ 0 if the smooth fibers Xs of f are connected and satisfy H 0 (Xs , ΩjXs ) = 0 for all j such that i ≥ j > 0. Proof. – (i) follows from the fact that there is a dense Zariski (or analytic-Zariski) open set U ⊂ Y such that the restriction fU : XU → U is smooth, hence a locally topologically trivial fibration. Then the statements hold for fU ∗ : π1 (XU ) → π1 (U ) and we conclude using the fact the map π1 (U ) → π1 (Y ) is surjective because Y is smooth. For items (ii) and (iii), we use the fact that there is a left inverse on cohomology with real coefficients given by α 7→ f∗ (ω d ^ α), where d = dim X − dim Y is the relative dimension and ω is the class of a Kähler form on X with volume 1 along the fibers of f . It follows that both maps f ∗ are injective. Finally the proof of (iii) uses the cotangent bundle sequence along smooth fibers (1)

0 → f ∗ ΩY,s → ΩX|Xs → ΩXs → 0,

which induces a filtration of ΩiX|Xs , showing that under our assumptions H 0 (Xs , ΩiX|Xs ) = H 0 (Xs , f ∗ ΩiYs ). We then conclude that any holomorphic differential i-form on X is a section of f ∗ ΩiY , at least over the open set Y 0 of regular values of f and more precisely is of the form f ∗ β, where β is a holomorphic differential i-form on Y 0 . But then β extends to Y as β 0 = f∗ (ω d ∧ α), and we thus conclude that α = f ∗ β 0 on X 0 = f −1 (Y 0 ) hence everywhere. Remark 1.3. – The final argument given above fails if we replace holomorphic forms by pluridifferential holomorphic forms. We will see in Section 4 that for pluricanonical ⊗i ∗ ⊗i forms, it is not true that a section of Ω⊗i X , which is a section of f ΩY ⊂ ΩX along −1 a dense Zariski open set XU = f (U ) ⊂ X is the pull-back of a section of Ω⊗i Y . However Lemma 1.2, (iii) is also true for pluridifferential forms. Remark 1.4. – The exact sequence (1) is a particular case of the conormal bundle exact sequence (2)

∗ 0 → NX → ΩX|Xs → ΩXs → 0, s /X

where in this case NXs /X = f ∗ TY,s . Note also the following standard fact which will apply more generally to a covering family of X by d-dimensional varieties φs : X s → X, that can be seen as a diagram (3)

X

φ

/X

f

 Y,

where f is a fibration and φ is dominant generically finite:

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Lemma 1.5. – In case of a fibration f : X → Y , one has KX|Xs = KXs . For a covering family ( X s )s∈Y , and for a general member X s of the covering family, one has K X s = φ∗s KX + D, where D is effective on X s . Proof. – The first statement follows from (1) by taking determinants, using the fact that f ∗ ΩY is trivial along the fiber X s . Only the second point needs to be proved. However we have by the first property K X s = K X | X s and on the other hand K X = φ∗ KX + R, where R ⊂ X is the ramification divisor of φ. Thus the statement holds once X s is not contained in R. We end this section with a standard lemma that will be used (sometimes implicitly) throughout the paper: Lemma 1.6. – The holomorphic pluridifferential forms, that is sections of Ω⊗k X , k ≥ 0, are bimeromorphic invariants of the compact complex manifold X. Proof. – Let φ : X 99K Y be a bimeromorphic map, with X and Y compact. There exists a Zariski-analytic open set U ⊂ X such that codim X \ U ≥ 2 and φ is well defined on U . We can thus define (because k ≥ 0) ⊗k 0 φ∗ : H 0 (Y, Ω⊗k Y ) → H (U, ΩU ).

This morphism is injective because φ is generically of maximal rank. By Hartogs’ ⊗k 0 theorem, we have H 0 (U, Ω⊗k U ) = H (X, ΩX ). We thus constructed an injective mor⊗k ⊗k ∗ 0 0 phism φ : H (Y, ΩY /K ) → H (X, ΩX/K ), which admits as inverse (φ−1 )∗ . 1.2. Iitaka fibration. – Let X be a smooth projective variety and let L be a line bundle on X. The subset M (L) ⊂ N defined as M (L) = {k ∈ N, H 0 (X, kL) 6= 0} is a submonoid of N hence except for small values of k, it agrees with the set of positive multiples of an integer k0 . We assume from now on that k0 6= 0. The integer k0 has the property that for any sufficiently large number m divisible by k0 , there are nonzero sections of mL. Typical examples where k0 is actually needed, that is, powers of L of order nondivisible by k0 have no nonzero sections, are given by X = E × Y , where E is an elliptic curve, and L = L0  LY , where L0 is a torsion line bundle of order k0 on E and LY is an ample line bundle on Y . The Iitaka dimension κ(X, L) of (X, L) is defined as −∞ if M (L) = {0} and as the maximal dimension of the images of the maps φkL : X 99K PN induced by the linear systems |kL| for k ∈ M (L). For example κ(X, L) = 0 is equivalent to the fact that M (L) 6= {0} and h0 (X, kL) = 1 for any k ∈ M (L). When L = KX , the Iitaka dimension of L is the Kodaira dimension of X.

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Theorem 1.7 (Iitaka). – There exists a rational fibration (4)

φL : X 99K Y,

which is well-defined up to birational modifications of Y , and is characterized by the following conditions: 1. dim Y = κ(X, L). e → Y, τ : X e → X be a desingularization of φL . Then the restriction 2. Let φ˜L : X ∗ of τ L to the general fibers of φ˜L has Iitaka dimension 0. In the case where L = KX , the statement is particularly interesting because first ⊗r ⊗r of all, by Lemma 1.6, the sections of KX agree with the sections of KX e for all r ≥ 0, e which implies that the map φKX : X → Y identifies with the Iitaka fibration of KXe , eb of φ is and secondly, by Lemma 1.5, the restriction of KXe to the smooth fibers X e nothing but the canonical bundle of Xb . Thus we get in this case: Corollary 1.8. – Any variety of nonnegative Kodaira dimension κ has a canonical fibration over a base of dimension κ with general fibers of Kodaira dimension 0. Proof of Theorem 1.7. – We refer to [38] for a more detailed proof. First of all, constructing the rational fibration is immediate since for any k ∈ M (L), choosing a nonzero section σ ∈ H 0 (X, k0 L), there is an inclusion σ : H 0 (X, kL) → H 0 (X, (k + k0 )L) and thus a commutative diagram (5)

X

φ(k+k0 )L

=

 X

φkL

/ PN 0 

π

/ PN ,

where the horizontal map are rational maps, and the second vertical map is a linear projection. It follows that if we take k large enough so that dim Im φkL = κ(X, L), the map π induces a generically finite rational map Im φ(k+k0 )L 99K Im φΦkL , and as all these varieties are dominated by X, these rational maps must be birational when k is large enough. Let Y be a smooth projective model of the varieties Im φkL for k e → Y be a desingularization of φkL (seen large enough divisible by k0 . Let φ : X as rational map to Y ). What remains to be proved is the fact that the map φ has irreducible (or equivalently connected) general fibers, and that the Iitaka dimension e of L to X, e restricted to the general fiber of φ is 0. Note that it of the pull-back L e is the pull-back of a line bundle on Y . However is not true that the line bundle L we observe that by definition, there is a nonzero section of L⊗k ⊗ φ∗kL O Yk (−1) on X, where Yk = Im φkL . It follows that denoting O Y (1) the pull-back to Y of O Yk (1), there e ⊗k ⊗ φ∗ O Y (−1) on X. e The line bundle O Y (1) being big, is a nonzero section α of L e ⊗r ) ⊗ O Y (s) is generically for any r ≥ 0, there is an s > 0 such that the sheaf (R0 φ∗ L e ⊗r globally generated on Y . Hence, if there are two independent sections σ1 , σ2 of L

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e ⊗r ⊗ φ∗ O Y (s) which restrict on the general fiber of φ, there are two sections τ1 , τ2 of L to σ1 , σ2 on the general fiber. Multiplying by αs , we produce two sections τ10 , τ20 e ⊗r+sk with independent restrictions on the general fiber of φ, which contradicts of L e ⊗r has at most one the construction of the fibration φ. Hence we conclude that L section on the general fiber of φ, which implies in particular that it is connected. Remark 1.9. – In the case where L = KX , note that the line bundle O Y (1) and its multiples have in general nothing to do with the canonical bundle of Y , which in some sense stops the analysis. We will come back to this in Section 4 describing results of Campana to analyze further the situation. 1.3. Castelnuovo-de Franchis and Bogomolov theorems. – Castelnuovo-de Franchis Lemma is the following statement: Lemma 1.10. – Let X be a smooth projective (or compact Kähler) manifold and let α, β ∈ H 0 (X, ΩX ) be two independent 1-forms on X such that α ∧ β = 0 in H 0 (X, Ω2X ). Then there exist a morphism φ : X → C, where C is a smooth curve of genus ≥ 2, and two holomorphic 1-forms α0 , β0 on C such that α = φ∗ α0 , β = φ∗ β0 . More generally, if there are g independent (1, 0)-forms αi on X such that αi ∧ αj = 0 in H 0 (X, Ω2X ), X admits a morphism to a curve C of genus ≥ g and the αi ’s are pulled-back from C. Proof. – This immediately follows from the fact that holomorphic forms on X are closed. The two 1-forms α, β are pointwise proportional. Writing α = f β, where f is a rational (or meromorphic) function on X, we thus get df ∧ β = 0 and thus df is also proportional to α and β. The two 1-forms α and β thus vanish along the fibers e → P1 be a desingularization of f and let F : X e → C of f : X 99K P1 . Let f˜ : X be the Stein factorization of f , where C is a smooth projective curve admitting a morphism r : C → P1 such that r ◦ F = f˜. The respective pull-backs α ˜ , β˜ of α, β to X are closed holomorphic 1-forms which are sections of the bundle F ∗ ΩC ⊂ ΩXe , e where F has maximal rank. This implies that there at least on the open set U ⊂ X exists holomorphic 1-forms α0 , β0 on C such that α ˜ = F ∗ α0 , β˜ = F ∗ β0 (we can take d−1 e which has integral 1 on the for α0 the form F∗ ω ∧α ˜ , where ω is a Kähler form on X fibers of F , and similarly for β0 ). It follows that g(C) has genus at least 2, so that the morphism F is already defined on X (this follows from the fact that the irreducible e → X are rationally connected (see components of the fibers of the birational map X Section 3.1 for the definition), hence contracted by F ) since a rational curve cannot have a nonconstant morphism to a curve of positive genus. The general case is treated similarly. Catanese [20] gave a topological version of Castenuovo-de Franchis result, showing that the existence of a dominating map to a curve of genus ≥ 2 is a topological property of compact Kähler manifolds :

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Theorem 1.11. – Let X be a compact Kähler manifold, and assume there are two independent degree 1 cohomology classes β, β 0 ∈ H 1 (X, C) such that β ^ β 0 = 0 in H 2 (X, C). Then there is a morphism X → C to a curve of genus ≥ 2 such that β, β 0 are pulled-back from degree 1 cohomology classes on C. More generally, the existence of a dominant morphism to a curve of genus ≥ g is equivalent of V2 to the existence a vector subspace V ⊂ H 1 (X, C) of dimension g such that V → H 2 (X, C) is identically 0. Proof. – We will just prove the case g = 2. Write the Hodge decomposition of β, β 0 : β = β 1,0 + β 0,1 , β 0 = β 0 where β 1,0 , β 0,1 , β 0 reasons,

1,0

β 1,0 ^ β 0

1,0

(6)

1,0

+ β0

0,1

,

, β 0 0,1 are (classes of) holomorphic 1-forms on X. Then by type = 0 in H 2 (X, C), β 0,1 ^ β 0

0,1

= 0 in H 2 (X, C).

As exact holomorphic forms vanish identically, these equalities also say that (7)

β 1,0 ∧ β 0

1,0

= 0 in H 0 (X, Ω2X ), β 0,1 ∧ β 0 0,1 = 0 in H 0 (X, Ω2X ). 1,0

Thus we can apply Lemma 1.10 if either β 1,0 and β 0 are not proportional or β 0,1 1,0 0,1 and β 0 0,1 are not proportional. Otherwise, we have β 0 = λβ 1,0 , β 0 = µβ 0,1 with λ 6= µ since β and β 0 are not proportional. Next β ^ β 0 = 0 also says that µβ 1,0 ^ β 0,1 + λβ 0,1 ^ β 1,0 = 0 in H 1,1 (X) and as µ 6= λ, this implies that (8)

β 1,0 ^ β 0,1 = 0 in H 1,1 (X).

Wee claim that (8) implies the vanishing (9)

β 1,0 ∧ β 0,1 = 0 in H 2,0 (X).

Indeed, let η = β 1,0 ∧β 0,1 ∈ H 2,0 (X). Then from (8), we deduce η ^ η = 0 in H 2,2 (X). The second Hodge-Riemann bilinear relations [50, 6.3.2] then imply that η = 0, which proves the claim. Note finally that β 0,1 cannot be proportional to β 1,0 as otherwise (8) would contradict the the Hodge-Riemann relations. Using the vanishing (9), we can now apply Lemma 1.10 which concludes the proof. An important consequence of Catanese’s theorem is the following result due independently to Beauville [4] and Siu [45] : Corollary 1.12. – Let X be compact Kähler. The existence of a nonconstant map F : X → C, where C is a smooth projective curve of genus at least 2, is equivalent to the existence of a group morphism α : π1 (X) → π1 (C) whose image has finite index. Proof. – Indeed, as C is a K(π, 1), such a group morphism α is induced by a continuous map f : X → C. As Im α has finite index, the induced map f∗ : H1 (X, Z) →

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H1 (C, Z) has finite cokernel, hence the pull-back map f ∗ : H 1 (C, C) → H 1 (X, C) is injective, and we have a commutative diagram (10)

H 1 (C, C) ⊗ H 1 (C, C) f ∗ ⊗f ∗

 H 1 (X, C) ⊗ H 1 (X, C)

/ H 2 (C, C) f∗

 / H 2 (X, C),

where the horizontal maps are given by cup-product. Consider the subspace V = f ∗ H 1,0 (C) ⊂ H 1 (X, C). It has dimension g(C) because f ∗ is injective. Moreover, one has α ^ α0 = 0 in H 2 (C, C) for any α, α0 ∈ H 1,0 (C), hence by (10), f ∗ α ^ f ∗ α0 = 0 in H 2 (X, C). One can then apply Catanese’s Theorem 1.11. In the other direction, we already proved that a surjective map between compact Kähler manifolds induces a map between their fundamental groups which has finite index image (see Lemma 1.2, (i)). Lemma 1.10 can be given many different generalizations: The following higher dimensional generalization is due to Bogomolov [7], subsequently generalized by Campana [18]. Theorem 1.13. – Let X be a compact Kähler manifold, and let L ⊂ ΩkX be a line bundle (not necessarily saturated). Assume κ(L) ≥ k. Then there exists a rational map φ : X 99K B, where B is smooth projective of dimension k, such that L coincide with φ∗ KB ⊂ ΩkX on a Zariski open set of X. Remark 1.14. – It is not true that B has to be of general type but φ has to be of general type in the sense of Campana. The problem is that φ∗ KB can be nonsaturated and its saturation L can have a different Iitaka dimension than φ∗ KB . A typical such example is described in Section 4 (see Lemma 4.1). Remark 1.15. – Note that Theorem 1.13 implies in particular that L is generically Vk decomposable, that is, generically equal to F for some rank k subsheaf F of ΩX . Proof of Theorem 1.13. – Assume first that the Iitaka fibration of L is given by sections of L. Let 0 6= s0 , . . . , sN be sections generating L generically and giving a dominating rational map φ : X 99K B, with dim B = k. Using the inclusion L ⊂ ΩkX the si ’s give holomorphic k-forms α0 , . . . αN on X. We have αi = φi α0 for some rational function φi . Using the fact that dαi = 0, dα0 = 0, we get that dφi ∧ α0 = 0. At the generic point of X, the dφi generate a rank k subbundle of ΩX , namely φ∗ ΩB . We then apply the following easy linear algebra fact: Vk Lemma 1.16. – Let W be a vector space and 0 6= u ∈ W . Then the vector space V := {v ∈ W, v ∧ u = 0} has dimension ≤ k, with equality if and only if u is a genVk V (in particular u is decomposable). erator of

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Vk ∗ This lemma implies that at the generic point of X, L coincides with (φ ΩB ) = φ∗ ΩkB , which proves the theorem in this case. In general, (and this is the case considered by Campana), we use the well-known but important fact that given a line bundle L on X and a section s of L⊗N , there exist a generically finite dominating map r : X 0 → X and a section s0 of r∗ L such that r∗ (div s0 ) = div s. X 0 is obtained by desingularizing the cyclic cover of X of order N ramified along div s. Iterating this process, it follows that we can find a generically finite cover r : X 0 → X with the property that r∗ L ⊂ ΩkX 0 and the Iitaka fibration of r∗ L is given by sections of r∗ L. We then conclude by the previous argument that there exists a rational fibration φ0 : X 0 99K Y 0 , with Y 0 smooth of dimension k such that r∗ L identifies with φ∗ ΩkY generically on X 0 . The previous proof also shows that the map φ0 = φr∗ L is in fact given by the Iitaka fibration associated with the line bundle r∗ L. On the other hand, r∗ L has Iitaka dimension 0 on the irreducible components of the varieties r−1 (Xs ) where Xs is the general fiber of the Iitaka fibration of X associated with L. It follows that we have a commutative diagram of Iitaka fibrations X0

(11)

φ0

r

 X

φL

/ Y0  /Y

,

r0

with r0 generically finite, showing that L identifies generically on X with φ∗L ΩkY . Definition 1.17. – Following Campana, we will call a rank 1 subsheaf L ⊂ ΩkX a Bogomolov subsheaf if κ(X, L ) = k, with k > 0. Remark 1.18. – By Lemma 1.6, the existence of a Bogomolov sheaf on X is a bimeromorphically or birationally invariant property of X (assumed to be compact or projective). More precisely, for any rank 1 saturated subsheaf L ⊂ ΩkX , and any birational map φ : X 0 99K X, the saturation of φ∗ L ⊂ ΩkX 0 has the same Iitaka dimension as L .

2. Fibrations from families of cycles The results developed in the previous section say nothing in the case of varieties of Kodaira dimension −∞ or with no nonzero pluridifferential forms. We are going to present in this and the following sections various constructions of geometric fibrations explaining, at least conjecturally, the shape of pluridifferential forms. The main conjecture of the field says that the MRC fibration, a fibration with rationally connected fibers that we are going to construct in Section 3.2), is nontrivial (in the sense that its base has dimension < dim X) for varieties X with Kodaira dimension −∞. In a different direction, we will see in Section 4 that the classification of varieties by Kodaira dimension is not (at least conjecturally) the right one from the viewpoint of

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hyperbolicity. Campana introduced in [16] the notion of “special” variety, and constructed on any projective variety X a natural rational fibration with “special” fibers, that is conjectured to explain the degeneracy of the Kobayashi pseudometric of X. A posteriori, this fibration, called the “core” fibration, can be recovered from Bogomolov sheaves on X. We will also describe the Gamma-fibration, which is a crucial tool to study the fundamental group of a variety. 2.1. Generalities about Hilbert schemes and Chow varieties. – In what follows, a very general point in a complex algebraic variety (or complex manifold) X is a point taken away from a specified countable union of closed algebraic (or analytic) subsets in an algebraic (or analytic) variety. Note that the bad set removed depends on the specific considered problem. A typical example is as follows: Let X be a complex projective variety. The theory of the Hilbert scheme tells us the following: Theorem 2.1. – There are countably many projective schemes Z equiped with a flat morphism f to a projective scheme Y and a morphism φ to X (12)

Z

 Y

φ

/X

,

f

which is an embedding on the fibers of f , and such that any subscheme Z ⊂ X is f ( Z y ) for such a family and for some point y ∈ Y . A similar result holds if X is compact Kähler (with closed algebraic subsets or subschemes replaced with closed analytic subsets or ringed subspaces), except that the basis Y will be then only a compact analytic space with Kähler desingularization (see [2]). The proof of Theorem 2.1 is much more difficult in the analytic context. In the algebraic situation, we observe that it suffices to construct the Hilbert schemes for PN itself, since it is clear that the Hilbert schemes of X ⊂ PN are closed algebraic subsets of Hilb PN defined by the conditions that the defining equations for X vanish on the considered subscheme. For PN , we first have to show that subschemes of bounded degree form a bounded family. This means that they are finitely many families parameterized by quasiprojective varieties exhausting all subschemes of a given degree in projective space. This can be proved by showing that closed algebraic subsets of degree d in PN are schematically defined by equations of degree d0 , where d0 depends only on d (we can take in fact d0 = d). This boundedness allows to get uniform vanishing for the cohomology groups H i (PN , I Z/PN (k)) = 0 for i > 0, k ≥ k0 for all subschemes Z of PN of degree d, with k0 depending only on d. It then follows that for k ≥ k0 , h0 (PN , I Z/PN (k)) = P I Z (k), where P I Z is the Hilbert polynomial of I Z/PN . There are countably many Hilbert polynomials, and fixing one, we can then imbed the Hilbert scheme parameterizing subschemes Z of PN with Hilbert polynomial P I Z = P

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into a Grassmannian G(rk , M ) where rk = P I Z (k), and M = h0 (PN , O PN (k)), by the map [Z] 7→ IZ (k) ⊂ H 0 (PN , O PN (k)). The last step in the construction of the Hilbert scheme is the proof that for k large, the image of this map is a closed algebraic subset of the Grassmannian. The algebraic equations for this algebraic subset of the Grassmannian describe the fact that the image of the multiplication map IZ (k) ⊗ H 0 ( O PN (1)) → H 0 ( O PN (k + 1)) has image contained in IZ (k + 1) hence rank ≤ P I Z (k + 1). A key point in Theorem 2.1 is the projectivity or compactness of Y , which is where the projectivity or Kählerness of X is needed. The local existence of analytic spaces parameterizing deformations of closed subvarieties (resp. closed analytic subspaces) works without these assumptions (see [2]). The compactness due to Fujiki uses Bishop compactness [5]. Let us describe an example where closed analytic subspaces (in fact, rational curves) do not have a compact deformation space, even allowing of course degenerations, in a compact complex submanifold which is the twistor family of a K3 surface: Example 2.2. – Choose a positive closed (1, 1) form ω which is the Kähler form of a Kähler-Einstein metric h on a K3 surface S. Then if σS is the holomorphic 2-form on S, σS , ω and the operator I of almost complex structure on S are parallel with respect to the Levi-Civita connection associated to the metric g = Re h. Then consider the conic in P2 with coordinates α, β, γ defined by Z (13) (αω + βσS + γσS )2 = 0. S

For a point 0 6= t = (α, β, γ) in this conic, the 2-form ωt := αω + βσS + γσS is parallel on S, hence so is its square. Equation (13) then implies that ωt is everywhere degenerate on S. On the other hand, it is nowhere 0, hence it defines a rank 2 vector subbundle F ⊂ TS,C which is in fact transverse to TS,R . To see this last point, observe that if ωt 6= 0 then ωt ∧ ω t > 0 (this is true pointwise as forms on S as one easily checks). We then have an almost complex structure It such that F is the sheaf of (1, 0)-tangent vectors for It and it is easy to show, using the flatness for the LeviCivita connection, that It is integrable, giving rise to a deformation St of the complex structure on S. Working a little more, this construction provides a compact complex threefold X with a morphism f : X → P1 = C whose fibers are the K3 surfaces St . X is diffeomorphic to S × C and the curves x × C, for x ∈ S are holomorphic P1 ’s contained in X. The family of these P1 ’s is not holomorphic however; it is the real part of a 4-dimensional holomorphic family M of deformations of these P1 ’s. The fact that there is no compact family M of such curves and their degenerations follows from the observation (used by Campana in [12]) that the deformation space M x of such P1 passing through one point x ∈ X is 2-dimensional. Assume M is compact, then we have accordingly a compact 2-dimensional family M x of rational curves in X

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passing through x. This implies by Mori’s argument [22] that there is a reducible P1 passing through this point. As the degree of these P1 ’s over C is 1, this is possible only if the limiting rational curve has an irreducible component which is a rational curve in S passing through x. Choosing for x a point which does not lie on a rational curve contained in S, we get a contradiction. Remark 2.3. – Campana used in [12] these rational curves and their Brody reparametrization to show that in the twistor family of any hyper-Kähler manifold, at least one fiber is not Kobayashi hyperbolic. This result is now essentially subsumed by Verbitsky’s work [48]. The limits are thus entire curves and not closed analytic subsets. An immediate consequence of Theorem 2.1 is the following result that will be crucial for the construction of fibrations: Theorem 2.4. – Let X be a smooth projective variety (resp. a compact Kähler manifold). Let x ∈ X be a very general point of X. Then if Z ⊂ X is a closed subvariety of X passing through x, there exists a covering family (14)

X

φ

/X

f

 Y, where f is a fibration, Z and Y are smooth projective (resp. compact Kähler manifolds) and φ is surjective, such that for some point s ∈ Y , φ| X s identifies to the e → X, where Ze is a desingularization of Z in X. natural map Z Proof. – We apply Theorem 2.1. We then have countably families of subvarieties or subschemes in X parameterized as in (12). For each of them, the morphism φ is projective, hence has Zariski closed image. Let us declare that the bad set is the countable union of the Im φ which are not the whole of X. Let now x ∈ X be taken away from this set, and let Z ⊂ X be a subvariety (or subscheme!) of X passing through x. We know by Theorem 2.1 that there exists a family (15)

Z

φ

/X

f

 Y,

such that Z identifies with one fiber Z y ,→ X. We can assume that Y is smooth by desingularization. By definition of the bad set, the morphism φ has to be surjective. The conclusion follows by taking for X a desingularization of Z , using the fact that Z is reduced, hence generically smooth, so that Z is generically smooth along Z y .

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Remark 2.5. – If the original variety Z is smooth, the variety Z above is smooth along Z y , hence we can assume that X ∼ = Z along Z y , which is then isomorphic to Z. Remark 2.6. – As the general fiber Z is reduced, we can also if needed assume, by restricting and desingularizing Y , that the morphism Z → X is generically finite. Indeed, let z ∈ Z be a smooth point such that Y is smooth at y = f (z) and the fiber Z y of f passing through z is smooth and φ is a submersion at z. Then as φ| Z y is an immersion, for a general vector subspace V ⊂ TY,y of dimension k := codim Z y ⊂ X, −1 the differential φ∗ : f∗,z (V ) → TX,φ(z) is an isomorphism, hence for any subvariety Y 0 of Y of dimension k with tangent space V at y, the map φ| Z Y 0 is generically finite onto X. 2.2. First application: fibrations associated with a dominant self-map. – Let X be smooth projective and let f : X → X be a dominant rational selfmap. (Some nontrivial examples with X a projective hyper-Kähler or Calabi-Yau manifold with Picard number 1 have been constructed in [51].) The following appears in [1]. Theorem 2.7. – Either for the very general point x ∈ X, the orbit {x, f (x), . . ., f k (x) . . .} is Zariski dense in X, or there exists a nontrivial fibration φ : X 99K B such that f acts on X over B: φ ◦ f = φ. Proof. – Let us assume for simplicity that f is a morphism. For any x ∈ X, we can consider the closed algebraic subset Ox which is defined as the Zariski closure of {x, f (x), . . . , f k (x) . . .}. Ox is stable under f , hence its irreducible components Γ of a given dimension are preperiodic, that is f l (Γ) is stable under f k for some l ≥ 0, k > 0 which can be taken independent of x, by a countability argument and because x is very general. Let Γx be an irreducible component of Ox passing through x. If Γx = X, then we are done. Assume thus Γx is different from X. Then f l (Γx ) is stable under f k and passes through f l (x). Note that, as x is very general and f is dominant, the point f l (x) is also a very general point of X. We thus know that through the general point f l (x) of X, there is a variety f l (Γx ) which is invariant under f k for some k > 0. We can thus apply Theorem 2.4, and conclude that these varieties form a family, that is, X is swept-out by a family of varieties invariant under a power f k . We now observe that the varieties of minimal dimension, irreducible and invariant under some power f k passing through the general point of X form a meromorphic fibration of X. This means that there is only one such variety passing through the general point of X. This is obvious since each irreducible component of Γ1 ∩ Γ2 passing through x satisfies these properties if Γ1 and Γ2 do (maybe up to changing k). We thus constructed a fibration of X which is invariant under some power f k , for some k > 0. It is then immediate that f acts meromorphically on the base B of this fibration, providing g : B 99K B such that g k = IdB . Then the composite rational map X 99K B 99K B/ satisfies the desired properties.

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3. MRC and Γ-fibrations 3.1. Rationally connected varieties. – Rationally connected varieties were introduced by Kollár, Miyaoka and Mori in [36]. There are in fact several notions which are equivalent for a smooth projective variety over C. The basic one is chain rational connectedness. Definition 3.1. – A projective variety X is chain rationally connected if given any two points x, y ∈ X, there exists a chain of rational curves in X, that is rational maps fi : P1 → X, i = 1, . . . , N , such that f1 (0) = x, fN (∞) = y and for i = 1, . . . , N − 1, fi (∞) = fi+1 (0). X is said to be rationally connected if one can join any two general points by a single rational curve. For example, the projective cone CX over any projective variety X is chain rationally connected, but if X does not contain any rational curve, the only way to join x to y by a chain in CX is to use the lines passing through x and y and meeting at the vertex O of the cone. This shows that one cannot in general assume that a single rational curve contains both x and y. Example 3.2. – The example of the cone also shows that a projective variety can be chain rationally connected without having any chain rationally connected smooth projective model. In particular, this is not a birationally invariant property. To the contrary, rational connectedness is clearly a birationally invariant property. However, when X is smooth, one has the following: Theorem 3.3. – (See [36]) Let X be smooth projective chain rationally connected over C. Then for x, y two general points of X, there exists an irreducible rational curve f : P1 → X passing through x and y, that is f (0) = x, f (∞) = y, so X is rationally connected. Equivalently, there exists a rational curve f : P1 → X such that f ∗ TX is a positive vector bundle on P1 . Thus a projective variety is rationally connected if and only if one (or any) of its desingularizations is chain rationally connected. Here we use Grothendieck’s theorem saying that any vector bundle E on P1 is a direct sum of O P1 (ai ). E is said to be positive if all ai are positive. The fact that the second statement is equivalent to the first follows from the observation that the first order deformations of P1 → X with the condition f (0) = x for fixed x are parameterized by H 0 (P1 , f ∗ TX (−0)) and that the differential of the evaluation map at (f, ∞) identifies then to the evaluation map (16)

H 0 (P1 , f ∗ TX (−0)) → (f ∗ TX (−0))|∞ .

Thus if this differential is surjective, which by Sard’s theorem will be the case for a general f if the general rational curve through x which is a deformation of f passes through the general point of X, the evaluation map (16) is surjective, which says that f ∗ TX (−0) is generically generated by sections, hence f ∗ TX is positive. The first

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statement in Theorem 3.3 (passing from chains to reducible curves) is more tricky as one needs to deform the chain of P1 ’s to a morphism from a smooth P1 . One ingredient is the following lemma: Lemma 3.4. – Let X be smooth projective, and let x ∈ X be a very general point. Let f : P1 → X be a rational curve passing through x. Then f ∗ TX is semi-positive. Proof. – Indeed, by Theorem 2.4, there is a family of rational curves F : B × P1 → X which dominates X, and such that Fb = f for some b ∈ B. As x is general and F is dominating, the differential of F can furthermore be assumed to be surjective at (b, 0), where Fb (0) = x. This differential factors as before through an evaluation map TB,b → H 0 (P1 , Fb∗ TX ) → (Fb∗ TX )|∞ and the surjectivity of the composed map thus implies that f ∗ TX is generically generated by sections, that is, semipositive. Remark 3.5. – A rational curve f : P1 → X satisfying the conclusion of Lemma 3.4 is said to be free. Free rational curves have a very good deformation theory, since they are unobstructed. The difficulty left in proving Theorem 3.3 is the following: We have for x, y general points of X a chain of rational curves from x to y. By Lemma 3.4, the first curve f1 : P1 → X and the last curve fN : P1 → X both satisfy the property that fi∗ TX is semipositive. Unfortunately, it could be very well that the intermediate curves do not sweep-out X and in fact have some negative summands in fi∗ TX . What is done in [36] is to attach to these intermediate curves more “legs”, coming for example from the deformations of C1 , CN . This process of attaching free rational curves increases the positivity of the normal bundle. One can then deform progressively C1 ∪ C2 , C1 ∪ C2 ∪ C3 etc. to a morphism from a smooth curve. 3.2. The MRC fibration . – The maximal rationally connected fibration of a smooth projective variety X is constructed by Campana [13]. The sketch of proof given here uses further ingredients coming from [36], [28]. Theorem 3.6. – Given any smooth projective variety X, there exists a rational map φ : X 99K B which has the following properties: (i) For a very general point x ∈ X, any rational curve C ⊂ X passing through a general point x0 of the fiber Xx of φ through x is contained in Xx . (ii) The fibers of φ are rationally connected. (iii) The rational map φ is almost holomorphic, which means that it is well-defined along the general fiber. Remark 3.7. – Using property (iii), the general fiber of φ is smooth. Remark 3.8. – Point (iii) is essential and will be obtained as a consequence of a stronger version of (i), namely: (i)’ For a very general point x ∈ X, any rational curve C ⊂ X intersecting Xx is contained in Xx .

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Note that we cannot impose a better condition than (i)’, removing the generality assumption on x. For example, it is not possible in general to construct a morphism or rational map with rationally connected general fiber and contracting all rational curves. Indeed, consider the case of an elliptic surface S with Kodaira dimension ≥ 0. Then S is not ruled. On the other hand, many such surfaces contain countably many rational curves. It suffices for this to have one rational curve, whose intersection with the general fiber St is not proportional in Pic St to h|St , where h is a given ample line bundle on S, of degree d on fibers St . Then we can use the rational maps φk : S 99K S, St 3 x 7→ kh|St − (kd − 1)x to construct infinitely many rational curves in S. No nonconstant rational map can contract all of them. Indeed, a generically finite rational map S 99K S 0 contracts only finitely many curves. For a rational map S 99K B to a curve, either there are finitely many rational curves contained in fibers, or all fibers are rational, since there are finitely many singular fibers. As our surface is not ruled, the second possibility is also excluded. Sketch of the proof of Theorem 3.6. – We first use Lemma 3.4 which says that if x ∈ X is a very general point, then any rational curve f : P1 → X passing through x is free. We also observe that similarly, a very general point x0 of the curve Cx = f (P1 ) is a very general point of X, and thus Lemma 3.4 applies to x0 . Assume there is now a rational curve Cx0 0 , C 0 = Im f 0 , f 0 : P1 → X, passing through S x0 , which is different from C. Then we will get at least a surface S = x0 ∈P1 Cx0 0 passing through x. This surface is clearly chain rationally connected since any general point in it is joined to x by a chain of two rational curves, a curve Cx0 and the curve C), but it is not clear that it is rationally connected: as it has been constructed, it could be singular along the curve C and after normalization becomes a ruled surface over a nonrational covering of C. To circumvent this problem, one has to use the fact that both C and the curves Cx0 0 are free. Note that one can also assume that the two curves C and Cx0 0 are smooth and transverse at x0 . The following deformation lemma is needed (see [36]): Lemma 3.9. – Let f : P1 → X, f 0 : P1 → X be two free rational curves such that f (0) = f 0 (00 ). Then there is a smoothification of (f, f 0 ) : C 00 → X, where C 00 is the union of C and C 0 glued via 0 = 00 , that is a morphism g : C → X, where C → B is a smooth surface over the smooth curve B, with smooth fiber isomorphic to P1 over b 6= b0 , and with fiber C b0 ∼ = C 00 , g| C b0 = (f, f 0 ). Furthermore, the morphism gb is free for general b. This deformation lemma is called “weak gluing lemma” in [42] where it is even mentioned that one can assume that the deformations gb all pass through the given point f (0) = f 0 (00 ). Applying this lemma to the general chain C ∪ C 0 of two free rational curves, we conclude that there is for very general x ∈ X at least a surface Sx in X consisting of points joined to x by a free rational curve. We can then continue arguing with Cx replaced by Sx , until the process stops. When the process stops, one gets a variety Xx passing through the very general point x ∈ X, with the property (i).

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One then proves that these varieties give a fibration of X. The very general point x0 of a variety Xx is a very general point of X and Xx = Xx0 has the property that a general point y of Xx is joined to x0 by a rational curve, hence Xx is rationally connected. Hence (i) and (ii) are proved. Point (iii) can be obtained as an application of Theorem 3.10 below. Indeed, the fibration whose construction is sketched above e → B, where τ : X e → X is constructed by a sequence of gives a morphism φ˜ : X blow-ups starting from X. The irreducible components of the positive dimensional fibers of τ are rationally connected. Hence if E is an exceptional divisor of τ , either E does not dominate B or the morphism φ˜ : E → B factors through τ . One thus concludes that generically over B, the morphism φ is well-defined. An essential property satisfied by the MRC fibration map, and already used above to prove (iii) in Theorem 3.6 and (i)’ in Remark 3.8, has been established by GraberHarris-Starr [28]. Theorem 3.10. – Let X be smooth and projective and let φ : X 99K B be the MRC fibration of X. Then the variety B (which is defined up to birational equivalence) is not uniruled. Remark 3.11. – Item (iii) had been given a direct proof (see [13]) before Theorem 3.10 was proved. Theorem 3.10 is obtained as a consequence of the general properties (i), (ii) of the MRC fibrations and of the following fundamental result: Theorem 3.12. – Let f : Y → C be a projective morphism with rationally connected general fiber, where C is a smooth curve. Then there is a section C → Y of f . Furthermore, there is a section of f passing through the general point of the general fiber of f . Proof of the implication Theorem 3.12 ⇒ Theorem 3.10. – Let φ : X 99K B be the MRC fibration of X and let b be a general point of B. Up to a birational transformation, we may assume f is a morphism. Assume there is a rational curve, that is a nonconstant map α : P1 → B passing through b, hence a point 0 ∈ P1 such that α(0) = b. Then as b is general, Xα := X ×B P1 has only one irreducible component Xα,d dominating P1 . Furthermore Xα,d is smooth along the fiber over 0 of the induced morphism φα : Xα,d → P1 which identifies with the fiber Xb , hence is rationally con1 ] nected. Thus, even after desingularization of Xα,d , the morphism φ˜α : X α,d → P has ] rationally connected fibers and we can apply to it Theorem 3.12. The variety X α,d thus contains rational curves not contained in the fiber over 0 and passing through ] the general point of the fiber over 0. The morphism X α,d → X is finite onto its image near the fiber over 0, hence we conclude that X contains rational curves not contained in the fiber Xb and passing through the general point of Xb . This contradicts property (i) in Theorem 3.6.

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3.2.1. Mumford’s conjecture. – Let us start with the following observation: Lemma 3.13. – Let X be a rationally connected variety. Then we have H 0 (X, Ω⊗k X )=0 for k > 0. 1 Proof. – Indeed, L by Theorem 3.3, there exists a morphism f : P → X such ∗ that f TX = i O P1 (ai ), where all ai ’s are > 0. The deformations of this morphism on the other hand sweep-out X since they are unobstructed and the first order evaluation map f : P1 × B → X has surjective differential at any point of P1 . Note that the generic deformation L ft : P1 → X also satisfies the property that ft∗ TX = i O P1 (a0i ), where all ai ’s are > 0, since this is L a Zariski open property on families of vector bundles on P1 . Next, ∗ 0 1 ∗ ⊗k as f TX = i O P1 (ai ), with ai > 0 for any i, we get that H (P , f ΩX ) = 0 0 1 ∗ ⊗k for k > 0 and similarly H (P , ft ΩX ) = 0 for k > 0 for a general deformation ft 1 of f . Thus H 0 (X, Ω⊗k X ) = 0 for k > 0 since the curves ft (P ) sweep-out a dense Zariski open set of X.

The converse statement is the following conjecture attributed to Mumford. Conjecture 3.14. – Let X be smooth projective such that H 0 (X, Ω⊗k X ) = 0 for k > 0. Then X is rationally connected. Using the MRC fibration, this conjecture has been reduced in [28] to the following conjecture, which is known to hold true in dimension ≤ 3 (see [42]): Conjecture 3.15. – Let X be a variety with Kodaira dimension −∞. Then X is uniruled. The reduction to Conjecture 3.15 goes as follows: Assume Conjecture 3.15 and let X be such that H 0 (X, Ω⊗k X ) = 0 for k > 0. Let X 99K B be the MRC fibration ⊗l ⊗ldim B ⊗l of X. Then H 0 (B, KB ) injects into H 0 (X, ΩX ) for l > 0, hence H 0 (B, KB )=0 for k > 0. As Conjecture 3.15 is assumed to hold, B is uniruled unless it is a point. But by Theorem 3.10, the basis of the MRC fibration of any variety is never uniruled. So B is a point and X is rationally connected. A line bundle L on a projective variety X is pseudoeffective if it is a “limit” of effective line bundles, which means concretely that there are sequences of positive 0 0 integers (kn , kn0 ) with limm→∞ km /km = 0 such that km L + km H is effective for all m, where H is a given ample line bundle on X. Pseudoeffective line bundles or divisors D have a nonnegative intersection with moving curves, i.e., curves whose deformations sweep-out the ambient variety. Indeed, we have [D] = lim [D] + m→∞

0 km

0 km [H] in H 2 (X, R), km

0 km

where [D]+ km [H] = km [Em ], with Em effective. Thus [D]·C = limm→∞ [Em ]·C ≥ 0 if C is moving. In particular, a pseudoeffective divisor on a smooth projective variety X

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has nonnegative intersection with the class c1 (L)n−1 , where L is any ample line bundle on X and dim X = n. Working a little more, this intersection number is in fact positive if [D] 6= 0. An important recent progress on Conjecture 3.15 has been obtained in [10] which proves: Theorem 3.16. – Let X be a smooth projective variety such that KX is not pseudoeffective. Then X is uniruled. The question left is thus whether Kodaira dimension −∞ implies that the canonical bundle is not pseudoeffective, which is one instance of the abundance conjecture. 3.2.2. An application to Calabi-Yau manifolds. – Recall first from [3] that a smooth projective variety with trivial canonical bundle has a finite étale cover which is a product of irreducible hyper-Kähler manifolds, simply connected Calabi-Yau manifolds, and abelian varieties. The Ricci flat Kähler metric existing on such variety is the product metric, and on the Calabi-Yau factors, say of dimension k, the holonomy is SU (k), while on a hyper-Kähler factor of dimension 2k, the holonomy is Sp(k). A K-trivial manifold will be said irreducible Calabi-Yau if, for the decomposition above, it has only one factor. The following theorem is proved in [39] (for Lagrangian fibrations on hyper-Kähler manifolds, it follows from [54] combined with [41]). Theorem 3.17. – Let X be a projective irreducible simply connected Calabi-Yau manifold. Then for any dominant rational map φ : X 99K Y , where Y is smooth projective and dim Y < dim X, Y is rationally connected (or a point). Proof. – We first observe that it suffices to prove that under the assumptions above, Y is uniruled. Indeed, starting from φ : X 99K Y , we have the dominating composite rational map φ0 : X 99K Y 99K B, where the second map is the MRC fibration of Y . We know by Theorem 3.10 that if B is not a point, then B is not uniruled, hence this produces a dominating rational map from X to a not uniruled variety. Once this is excluded, we find that B has to be a point hance Y is rationally connected. Next by Theorem 3.16, Y is uniruled if and only if its canonical bundle is not pseudoeffective. So we only have to prove that given a dominant rational map φ : X 99K Y , KY is not pseudoeffective. Assume the contrary: The map φ provides a rank 1 subsheaf φ∗ ΩkY , k = dim Y , of ΩkX on the Zariski open set U ⊂ Y where φ is defined. There is thus a line bundle D ⊂ ΩkX which is the saturation of φ∗ ΩkY . If we choose an ample line bundle H on X, we know by the existence of Kähler-Einstein metric on X with Kähler class c1 (H) that ΩX is H-stable and that Ωk is H-polystable (see [34], [40]). More precisely, in the case of SU (n) holonomy, the vector bundle ΩkX is stable, while in the case of Sp(n) holonomy, there is a decomposition M r (17) ΩkX = σX ∧ Ωk−2r X,0 , 2r≤k

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∧σ n−k+2r+1

k−2r where 2n = dim X and Ωk−2r −−X−−−−−→ Ω2n−k+2r+2 ). Here σX deX,0 := Ker (ΩX,0 − X notes the holomorphic 2-form of X, and the summands Ωk−2r are stable. The line X,0 k bundle D ⊂ ΩX being pseudoeffective has nonnegative H-degree. This is possible only if D is one of the summands in the decomposition (17) of ΩkX as a direct sum k/2 of H-stable bundles. However the only rank 1 such summand is σX when k is even and it does not correspond to a locally decomposable subsheaf of ΩkX when k < 2n. This is a contradiction which concludes the proof.

3.3. Shafarevich maps (or Γ-reductions) and Shafarevich conjecture. – Recall that a complex manifold M is said to be holomorphically convex if the holomorphic functions on M give a proper holomorphic map M → U where U is Stein. Let now X be a smooth projective variety (or compact Kähler manifold). The Shafarevich conjecture states the following: e is holomorphically convex. Conjecture 3.18. – The universal cover X Note that the conjecture is wrong for intermediate covers, as shows Example 4.5 in [43], attributed to Narasimhan: For a general lattice Z4 ∼ = Λ ⊂ C2 , let X be the 2 complex torus C /Λ. Then for a general rank 3 submodule Λ0 ⊂ Λ, the quotient variety C2 /Λ0 is a cover of X which is not holomorphically convex: it is not compact, but has no nonconstant holomorphic function. e is holomorphically convex, and let F : X e → U be the proper map Assume X e The fibers of F should thus be exactly the given by holomorphic functions on X. e But via the natural holomorphic map X e → X, compact closed analytic subsets of X. the later correspond to closed analytic subsets Z of X having the property that π1 (Z 0 ) → π1 (X) has finite image where Z 0 is a desingularization of Z. Note here that there is a delicate and crucial issue on whether we consider Z or one of its desingularizations Z 0 . If the map π1 (Z) → π1 (X) has finite image, one concludes e is a countable disjoint union of compact closed that the inverse image of Z in X analytic subsets. If we only know that the map π1 (Z 0 ) → π1 (X) has finite image, e is a countable where Z 0 is a desingularization of Z, then the inverse image of Z in X but a priori not disjoint union of compact closed analytic subsets. This problem has been suggested as a possible reason for a negative answer to Shafarevich conjecture (cf. [8]). More generally, given a group morphism ρ : π1 (X)→Γ, one considers closed ρ analytic subsets Z of X having the property that π1 (Z) → π1 (X) → Γ has finite image. The Γ-reduction was invented independently by Campana in [14] and Kollár in [35] (under the alternative name of Shafarevich map) in order to understand closed analytic subsets Z satisfying this property. Theorem 3.19. – Let X be a smooth projective variety or compact Kähler manifold and let ρ : π1 (X) → Γ be a group morphism. There exists a rational fibration φρ : X 99K Y which is well defined up to birational transformations of Y , and satisfies the following properties:

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(i) For a very general point x ∈ X, the fiber Xx of φρ passing through ex ) → Γ has finite image, where X ex property that the natural morphism π1 (X desingularization of Xρ . (ii) For a very general point x ∈ X and any subvariety Z ⊂ X passing e such the map π1 (Z) e → Γ has finite image, Z is with desingularization Z, in Xx . (iii) The rational map φρ is almost holomorphic.

x has the is a (any) through x contained

Remark 3.20. – It follows from (iii) that the general fiber Xs of φρ is smooth, so that ex by Xx . in (i), we can replace X Sketch of proof of Theorem 3.19. – Point (iii) is an immediate consequence of (ii). e → X be a smooth model of X on which φρ is a morphism. If φρ is not almost Let τ : X e which dominates Y and such that holomorphic, there is an exceptional divisor E ⊂ X the fibers of τ|E are not contracted by φρ . The fibers of τ|E are rationally connected e such that its image P1 ∼ and it follows that there is a rational curve P1 ∼ = =R→X eR R → Y passes through the very general point of Y . Then any desingularization X e has a dominant map p1 to R by the first projection (note of the variety R ×Y X e has only one irreducible component that, φρ having irreducible general fiber, R ×Y X eR will be in fact a desingularization of this component). Let dominating R, and X e R → R ×Y X e → R. We have R0 ⊂ R be the regular locus of the composite map p1 : X the exact sequence of π1 ’s et ) → π1 (X eR0 ) → π1 (R0 ) → 1, t ∈ R e0 . π1 (X e so On the other hand, there is a natural copy of R which is contained in R ×Y X, 0 e by desingularization, we get a curve R ⊂ XR which maps to R (or rather its image) e In particular it is contracted by τ and thus the map π1 (R0 0 ) → π1 (X) is trivial. in X. 0 On the other hand, the natural map π1 (R0 ) → π1 (R0 ) has image of finite index. et ) → Γ has finite image, so does the map π1 (X eR0 ) → Γ, As the natural map π1 (X which contradicts the maximality of the fibers (property (ii)), using the fact that e R 0 ) → π1 ( X eR ) is surjective since X eR is smooth. π1 ( X In order to construct the desired fibration, we will need the following Lemma 3.21 (which has been in fact implicitly used in the previous proof). Let X, ρ : π1 (X) → Γ be as in the theorem. In what follows, Ye will denote a smooth model of the variety Y . Lemma 3.21. – Let Z ⊂ X be a subvariety and let φW : W → X, π : W → B be a family of closed subvarieties W b of X parameterized by a quasiprojective basis B such that: (i) For a general b ∈ B, the map ρ ◦ ib∗ : π1 ( f W b ) → Γ has finite image, where ib : f W b → X is the natural map. (ii) For any b ∈ B, W b intersects Z. e → Γ, where j : Ze → X is the natural morphism, has (iii) The map ρ ◦ j∗ : π1 (Z) finite image.

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f of the Zariski closure W in X of the Then for any smooth projective model W S f union b∈B W b , the natural map π1 (W ) → Γ has finite image. f Proof. – Let φW : W → X be a smooth and projective model of W mapping to X. f is a desingularization of the image φW ( f Then by definition W W ) ⊂ X and we may f assume by further blowing-up W that the morphism φW factors through a dominant f f . By Lemma 1.2, (i), the map ψW ∗ : π1 ( f f ) has morphism ψW : W → W W ) → π1 (W f finite index, so it suffices to show that ρ ◦ φW ∗ : π1 ( W ) → Γ has finite image. 0 f f The variety W contains a Zariski open set W which is a desingularization of W , 0 f and in particular maps to B. Shrinking B, we can even assume that W → B is a 0 f f smooth proper fibration. The map π1 ( W ) → π1 ( W ) is surjective so we can restrict 0 f to W =: f W . We now observe that by assumption (ii), the variety f W contains a subvariety Σ which dominates B via the morphism π, can be assumed to be smooth and maps to Z via φW . Up to further blow-ups and shrinking, we can also assume e The that the morphism φΣ : Σ → Z factors through a morphism ψΣ : Σ → Z. map π∗ : π1 (Σ) → π1 (B) has finite index image. On the other hand, by (iii), the e of Γ which by assumption is map ρ ◦ φW ∗ maps π1 (Σ) to the subgroup ρ ◦ j∗ (π1 (Z)) finite. Writing then the homotopy exact sequence 0 π1 ( f W b ) → π1 ( f W ) → π1 (B ) → 1

and using the fact that ρ ◦ ib∗ (π1 ( f W b )) has finite image, we conclude that Im (ρ ◦ f φW ∗ : π1 ( W b ) → Γ) has finite index in Im (ρ ◦ φW ∗ : π1 ( f W ) → Γ). By property (i), f Im (ρ ◦ ib∗ = Im (ρ ◦ φW∗ )|π ( f : π1 ( W b ) → Γ) is finite and we are done. W ) 1

b

The proof of Theorem 3.19 is now quite easy. For a very general point x ∈ X, let k be the maximal integer such that there exists an irreducible closed subvariety Zx ⊂ X passing through x and such that the natural map π1 (Zex ) → Γ has finite image. We claim that Zx is unique and that it is the fiber of a fibration X 99K B satisfying the desired assumptions. First of all, we apply Theorem 2.4 which provides (18)

Z

φ

/X

f

 Y,

where f is proper, φ is dominant and may be assumed to be finite, and Zx corresponds to the inclusion in X of one fiber Z y of f . Note that we can assume that f ( Z y ) 6= f ( Z y0 ) for y general and y 0 6= y, as otherwise there is a map Y → Y 0 such that the family of varieties f ( Z y ) factors through a family parameterized by Y 0 . We now desingularize Z to e Z and restrict to the Zariski open set Y 0 of Y over which f is

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smooth. As x is very general, we can assume that y ∈ Y 0 , and thus the locally constant map π1 ( e Z y ) → Γ has finite image for any y ∈ Y 0 . We claim that the map f : Z → X is birational. To see this, consider f −1 (f ( Z y )) ⊂ Z . It clearly contains Z y . If f is not birational, f −1 (f ( Z y )) has another irreducible component T dominating Z y , whose image B in Y has positive dimension. We now apply Lemma 3.21 to Z B := φ−1 (B). It satisfies properties (i), (ii) (with Z = Z y ) and (iii), and thus we conclude that its image φ( Z B ) is a closed algebraic subset Z 0 of X, passing through x and such that π1 (Ze0 ) → Γ has finite image. This contradicts the fact that Z had maximal dimension. Remark 3.22. – Exactly as in the case of the Iitaka-Kodaira fibration where it was observed that the basis needs not be of general type and even can have negative Kodaira fibration, it is quite possible that the basis of the Shafarevich map has trivial fundamental group. Here is an example: Let S be a K3 surface with an involution ιS acting freely (so the quotient is an Enriques surface), and let C be a hyperelliptic curve with its hyperelliptic involution ιC . Then let X := S × C/(ιS , ιC ). The fundamental group of X is infinite because X has S ×C as étale cover. On the other hand, X admits a natural map f : X → C/ιC = P1 with general fiber isomorphic to S which is simply connected. Thus the very general fibers St must by definition of the Shafarevich map be the fibers of the Shafarevich map which is thus equal to f . In this case the basis of the Shafarevich map is thus P1 . The Shafarevich fibration is well-understood for certain types of groups: Theorem 3.23 (Campana [15], see also [31]). – If X has nilpotent fundamental group, the Shafarevich fibration is given by the Stein factorization X → YSt of the Albanese map X → Y ⊂ Alb X. Equivalently, let φ : X 0 → Y 0 be a desingularization of X → YSt (so that X 0 and Y 0 are smooth and φ is a morphism), then π1 (X) = π1 (X 0 ) and φ∗ : π1 (X 0 )→π1 (Y 0 ) is surjective with finite kernel. Proof. – We follow Campana’s proof. The fact that φ∗ is surjective is Lemma 1.2, (i) since the fibers of φ are connected. Observe next that the map φ∗ : H1 (X 0 , Z) → H1 (Y 0 , Z) is surjective with finite kernel (indeed, the map alb∗ : H1 (X 0 , Z) → H1 (Alb X 0 , Z) has finite kernel). Furthermore, the map φ∗ : H 2 (Y 0 , Q) → H 2 (X 0 , Q) → is injective by Lemma 1.2, (ii). We claim now that if H = π1 (Y 0 ), the natural map nY 0 : H 2 (H, Q) → H 2 (Y 0 , Q) is injective. This can be proved as follows: Let EH → BH = EH /H be the classifying space f0 × EH )/H of H, with EH contractible. Then Y 0 is homotopically equivalent to (Y 0 0 f where Y is the universal cover of Y , and denoting by uY 0 the continuous map (Ye × EH )/H → BH , the map nY 0 is equal to f0 × EH )/H, Q). u∗Y 0 : H 2 (BH , Q) → H 2 (Y 0 , Q) = H 2 ((Y

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It is clear that the fibers of uY 0 are homeomorphic to Ye , hence are simply connected. The Leray spectral sequence of uY 0 then shows that f0 × EH )/H, Q) u∗Y 0 : H 2 (BH , Q) → H 2 ((Y is injective. We can thus conclude by dualizing and composing the two surjective maps that the map φ∗ : H2 (X, Q) → H2 (H, Q) is surjective. Let G = π1 (X). It follows that the map φ∗ : H2 (G, Q) → H2 (H, Q) is also surjective, using the following commutative diagram: (19)

H2 (X, Q)

/ H2 (Y, Q)

 H2 (G, Q)

 / H2 (H, Q).

The proof of Theorem 3.23 is then concluded with the following lemma due to Stallings [46]. For any group Γ, let us denote by Γn ⊂ Γ the n-th term of the lower central series: Γn = [Γ, Γn−1 ]. Lemma 3.24. – Let α : G → H be a morphism between finitely generated groups. Assume that α induces an isomorphism H1 (G, Q) ∼ = H1 (H, Q) and a surjective map H2 (G, Q) → H2 (H, Q). Then for any n, the natural map G/Gn → H/Hn has finite kernel and cokernel. Corollary 3.25 ([31]). – If π1 (X) has a finite index subgroup which is nilpotent, then the Shafarevich conjecture is true for X. Proof. – Replacing X by a finite étale cover, we immediately reduce to the case where X has nilpotent fundamental group. We use the same notation as before: Y 0 is a smooth model of the Stein factorization of albX : X → Alb X and a : e be the uniX 0 → Y 0 is a desingularization of the rational map X 99K Y 0 . Let A e maps in versal cover of Alb X = Alb Y 0 . Then the cover Y 0 ab := Y 0 ×Alb(X) A e a proper way to the Stein analytic space A. Theorem 3.23 shows that the morphism a∗ : π1 (X 0 ) → π1 (Y 0 ) of fundamental groups is surjective with finite kernel. f0 univ the universal covers of X 0 , Y 0 respectively, it follows that f0 univ , Y Denoting by X f0 univ → Y f0 univ , X f0 univ identifies with a finite étale cover via the natural map auniv : X 0 0 f f0 univ maps in a of X ×Y 0 Y univ . In particular, auniv is proper. On the other hand, Y 0 e We then use proper way to the universal cover of the Stein manifold Y ×Alb(X) A. f0 univ has a the fact that the universal cover of a Stein manifold is Stein [47]. Thus X proper holomorphic map to a Stein manifold. Although Theorem 3.19 is not precise enough for applications to the Shafarevich conjecture, the idea of the Shafarevich map has played a crucial role in the proof of the Shafarevich conjecture under extra assumptions on the fundamental group. The case of linear fundamental groups has been solved in [26].

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Theorem 3.26. – Let X be a smooth projective variety such that π1 (X) admits a faithe is holomorphically convex. ful linear representation. Then X The case where the fundamental group admits a reductive representation had been treated by Eyssidieux in [25]. Let us say that a projective or compact Kähler variety has a large fundamental group if for any positive dimensional closed algebraic sube → π1 (X) has set Z ⊂ X, with desingularization Ze → X, the natural map π1 (Z) infinite image. We will say that the fundamental of X is generically large if the same is true for varieties passing through the very general point of X. Theorem 3.19 can be rephrased saying that a variety has generically large fundamental group if and only if the Shafarevich fibration is the trivial fibration X → X. As we mentioned in Remark 3.22, it is unfortunately not the case that the base of the Shafarevich fibration has generically large fundamental group. Furthermore, only its very general fiber is controled, and it could even be that special fibers have configurations of their components leading to a disproof of the Shafarevich conjecture. The paper [26] constructs in a more controled way, using nonabelian Hodge theory, the Shafarevich fibration associated with a given linear representation of the fundamental group, and introduces an orbifold structure Yorb on the base, such that the given representation of the fundamental group of the original variety X factors through the orbifold fundamental group. This is the starting point of the works [25], [26], the second crucial ingredient being again anabelian Hodge theory related to the given faithful representation of π1 (X). This allows the authors to construct a plurisubharmonic exhaustion function on the universal cover of the orbifold or stack Yorb , in order to show that it is Stein. 4. Special varieties and the core 4.1. Morphisms of general type. – The starting point of Campana’s work [16] is illustrated by the following example: Consider a hyperelliptic curve C of genus g ≥ 2 with hyperelliptic involution ι and an elliptic curve E with nonzero 2-torsion point η. Then the involution σ := (ι, η) acts freely on C × E. The quotient surface Σ : C × E/σ admits a morphism to the elliptic curve E/η, and a morphism f : Σ → P1 = C/ι whose fibers are elliptic curves. Lemma 4.1. – (i) There is no dominant morphism φ : Σ → B, where B is a variety of general type. (ii) The morphism f has multiple fibers over the branch divisor D ⊂ P1 of degree 2g + 2 of C/P1 . (iii) The twisted canonical bundle f ∗ KP1 ( 21 D) pulls-back to pr1∗ KC on the surface C × E. Proof. – (i) As the fibers of f are elliptic, Σ has Kodaira dimension 1 and does not dominate a surface of general type. If φ : Σ → B is dominant with B a curve of genus > 1, the fibers of f cannot dominate B via φ because they are elliptic, hence

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are contracted by φ. Thus φ factors through f , and B is dominated by P1 , which is absurd. (ii) As σ has no fixed points, the local form of the map f is the same as the local form of the map f ◦ q : C × E → P1 , where q is the quotient map C × E → Σ. But the fibers of f ◦ q are clearly multiples fibers 2E × x over each fixed point of the hyperelliptic involution ι. (iii) This follows from the Hurwitz formula which says that KC = r∗ KP1 ( 21 D), where r : C → P1 is the natural map. Note also that the surface Σ behaves as the surface C × E from the viewpoint of hyperbolicity. Indeed, any holomorphic map from a disk or C to Σ lifts, by simple connectedness, to C × E. Hence we conclude that the Kobayashi pseudodistance of Σ, although degenerate, is not 0: it is the pull-back of a certain pseudodistance on P1 which takes into account the divisor D. Campana introduced special varieties in [16]. The idea is that these varieties should not map in any way to a variety of general type and the conjecture is that these varieties are those which have a trivial Kobayashi pseudodistance. However Lemma 4.1 above shows that the surface Σ constructed above should not be considered as special. The crucial corrected Definition 4.8 below was introduced by Campana. We first define a morphism of general type. Definition 4.2. – Let φ : X → Y be a dominant proper morphism with X and Y smooth of dimension k > 0. φ is said to be of general type if the saturated rank 1 subsheaf L φ := (φ∗ KY )sat ⊂ ΩkX has Iitaka dimension k. Remark 4.3. – In the situation above, Bogomolov’s Theorem 1.13 says that the Iitaka dimension of (φ∗ KY )sat is at most k. For a rational map φ : X 99K Y , we will say that φ is of general type if a (in fact e → Y of φ is of general type. As noticed above, the notion any) desingularization φ˜ : X of morphism or rational map of general type does not say much on the base. However, Campana realized that there is an orbifold structure on the base which under extra assumptions allows to interpret the morphism φ being of general type as a certain orbifold or log structure on Y being of general type. Let φ : X → Y be a morphism. The orbifold structure on Y will be determined by the Q-divisor ∆φ on Y of multiple fibers of φ defined as follows: For any irreducible reduced divisor D ⊂ Y , we can P write the effective divisor φ∗ (D) as E mE E + D0 , where the divisor D0 does not dominate D. Let mD := Inf E mE . We then define X mD − 1 ∆φ = (20) D. mD D

The multiplicities appearing in the divisor D are inspired from the case of a morphism between curves. The local structure of the morphism is vl : U → V, z 7→ z l , where

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U, V are open neighborhoods of 0 in C, and the local multiplicity ν is then l. On the other hand, we have locally vl∗ (KV ) = KU (−(l − 1)0), vl∗ (0) = l0, that is vl∗ (KV ( l−1 l 0) = KU . With this construction of the orbifold base, the above computation shows that if φ : X → Y is finite, the divisor KX − (φ∗ KY (∆φ )) is effective. For morphisms of relative dimension > 0, a similar interpretation is also possible but only after birational transformations. In fact, the Kodaira dimension of the pair (Y, ∆φ ) is not a birational invariant of (X, φ) if no assumptions are made on the morphism φ. This problem is addressed by Campana (see [19]) as follows: Definition 4.4. – A morphism φ : X → Y is neat if there exists a birational morphism u : X → X 0 with X 0 smooth, such that divisors D in X contracted by φ (in the sense that their image in Y has codimension ≥ 2) are contracted by u. Lemma 4.5. – Any fibration X → Y has a neat birational model, with X and Y smooth. Proof. – Let (21)

X0

u0

φ0

 Y0

v

0

/X  /Y

φ

be a flattening of φ, that is u0 and v 0 are projective and birational and φ0 is flat. We can also assume Y 0 is smooth by resolving the singularities of Y 0 and base-changing. Let now τ : X 00 → X 0 be a resolution of singularities and let φ00 = φ0 ◦ τ , u = u0 ◦ τ . We claim that φ00 is neat. Indeed, if D is a divisor contracted by φ00 , the image of D under τ is mapped by φ0 to a codimension ≥ 2 closed algebraic subset of Y 0 , and as φ0 is is flat, it follows that τ (D) cannot be of codimension 1 in X 0 , that is D is contracted by τ , and a fortiori by u. Proposition 4.6. – If the morphism φ is neat, the orbifold Kodaira dimension κ(Y, ∆φ ) is equal to the Iitaka dimension κ(X, L φ ), where L φ is the saturation of the rank 1 subsheaf φ∗ KY ⊂ ΩpX , p = dim Y . ⊗m

Proof. – We prove that for m divisible enough, the sections of L φ on X identify via the pull-back with the sections of m(KY + ∆φ ). Here we ask that m is divisible by all multiplicities appearing in ∆φ so that m(KY + ∆φ ) is a honest divisor. As L φ is saturated, we only have to worry about what happens in codimension 1 on X. The sheaves L φ and φ∗ KY differ along divisors contracted by φ and along divisors of P multiple fibers. For a contracted divisor E = n E , as the Ei are contracted i i i by u : X → X 0 , with X 0 smooth, one can use the fact that pluridifferential forms on X are pulled-back from X 0 (see Lemma 1.6) and that u(E) has codimension ≥ 2 in X 0

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⊗N

to conclude that it suffices to compare sections of L φ and φ∗ (KY (∆φ )⊗N ) on X \E, and more generally, over Y \ W for any codimension ≥ 2 closed analytic subset W of Y . For the divisors of multiple fibers, it suffices to look by the above argument at what happens at the generic point of an irreducible component D of ∆φ : If we compare L φ and φ∗ KY (∆φ ) we see that they coincide along at least one component D1 of φ−1 (D), namely the one where the multiplicity ν of fibers is minimal. It then ⊗m follows that the sections m(KY +∆φ ) and L φ coincide because otherwise the divisor φ∗ m0 D − m0 νD1 is effective along the fibers of φ−1 (D) → D. A simple but interesting property of rational fibrations of general type is the following: Lemma 4.7. – Let φ : X 99K Y be of general type. Then φ is almost holomorphic. e → Y by a single blowProof. – Let us assume that φ admits a desingularization φ : X up along a smooth center Z ⊂ X along which φ has indeterminacies. Let E → Z ⊂ X e → X. We have to show that E → Y cannot be be the exceptional divisor of X dominant. Assume the contrary: The saturated sheaf L φ restricted to E then injects into ΩpE and its Iitaka dimension is at least k = dim Y . We use then the fact that E is a projective bundle over Z, so that any Bogomolov sheaf on E comes from Z, using the fact that there are no nonzero pluridifferential forms on the fibers Pk and the relative cotangent exact sequence 0 → τE∗ ΩZ → ΩE → ΩE/Z → 0. Thus the map φ|E factors rationally through Z, which contradicts the fact that the morphism has indeterminacies generically along Z. Definition 4.8 (Campana [16], see also [21]). – A smooth projective variety X is special e of X admits a morphism of general type X e → Y . Equivaif no birational model X lently, X admits no Bogomolov subsheaf (see Definition 1.17). e By Proposition 4.6, X is special if and only if for any birational model X e of X and morphism φ : X → Y , with Y smooth of positive dimension, one has κ(Y, ∆φ ) < dim Y . The surface Σ considered above is not special as the morphism f : Σ → P1 is of general type by Lemma 4.1. Lemma 4.9. – (i) A rationally connected variety is special. (ii) A variety with trivial canonical bundle is special. Proof. – Point (i) follows from Lemma 3.13 while, if a variety X has a birational e admitting a morphism of general type φ : X e → Y , then there exist for N model X ⊗N k ⊗N large nonzero sections of L φ ⊂ (ΩXe ) , hence also nonzero sections of (ΩkX )⊗N by Lemma 1.6. For point (ii), we use Yau’s theorem on the existence of Kähler-Einstein metrics [53] which has as a consequence ([34], [40]) the polystability (with respect to any Kähler

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class) of the tangent bundle of a K-trivial projective manifold. It follows that ΩkX is also polystable and that its effective rank 1 subsheaves have Iitaka dimension ≤ 0.

4.2. The core fibration. – The following result has been obtained by Campana [16]. Theorem 4.10. – For any smooth projective variety X, there exists a canonically defined rational fibration φ : X 99K Y which satisfies the following properties: (i) φ is of general type. In particular, φ is almost holomorphic by Lemma 4.7. (ii) The general fiber of φ is special. (iii) For a very general point x ∈ X, any subvariety X 0 ⊂ X passing through x and with special desingularization is contained in the fiber of φ through x. (iv) φ is universal in the following sense: Any rational map of general type X 99K Y 0 factors through Y . Sketch of the proof. – The construction of φ is given by the choice of a Bogomolov subsheaf L ⊂ ΩpX with p maximal. If p = 0, X is special by definition and there is nothing to prove. Applying Theorem 1.13, one gets a rational map of general type X 99K Y . One then has to prove that it satisfies all the properties (ii) to (iv). The key point is the proof of (ii), which will follow from Theorem 4.11 below by the following argument. Assume the general fiber is not special. Then by induction on dimension, it has a well-defined core fibration, so that one can construct a relative (over Y ) core fibration ψ

χ

ψ : X 99K Y 0 99K Y. Here, over the very general point t ∈ Y , the morphism ψt : Xt 99K Yt0 is the core fibration of Xt . For an adequate birational model of ψ, the orbifold canonical bundle of (Yt0 , ∆ψt ) is then of general type by Proposition 4.6. We then have a fibration χ : Y 0 → Y that we can assume to be neat. Furthermore Y 0 is equipped with an orbifold structure ∆ψ whose general fibers are of general type. Finally, the morphism Y 0 → Y is easily seen to be of general type since X → Y is. It follows from the additivity Theorem 4.11 in the case of a base of general type that (Y 0 , ∆ψ ) is of general type and since dim Y 0 > dim Y , we get a contradiction. Theorem 4.11. – Let (Y 0 , D) be an orbifold, and let Y 0 → Y be a neat morphism of general type. Then κ(Y 0 , D) ≥ κ(Yt0 , Dt ) + dim Y . Here κ(Yt0 , Dt ) is the logarithmic Kodaira dimension of the general fiber Yt0 equipped with the restricted orbifold structure.

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4.3. Conjectures. – In the inspiring paper [37], the subject of Kobayashi hyperbolicity, which had been exposed in [33] from the viewpoint of complex analysis and geometry, is treated from the viewpoint of arithmetics. The case of curves is seen as a model, with Faltings finiteness theorem for points on curves of genus ≥ 2 defined over a number field [27]. There is a beautiful parallel developed by Lang between Brody curves on one hand and points defined over a number field on the other hand. The following conjectures are an example: Conjecture 4.12 (Green-Griffiths [29]). – Let X be a complex projective manifold which is of general type. Then entire curves are not Zariski dense in X. This conjecture is optimal since even when the canonical bundle of X is ample, there might be rational curves in X. Note that it is not even known that surfaces with ample canonical bundle (eg quintic surfaces in P3 ) contain only finitely many rational curves (despite positive results by Bogomolov under extra assumptions on Chern numbers, see [6]). Conjecture 4.12 is also known to hold for subvarieties of abelian varieties by results of Kawamata [32] and Ochiai [44] (cf. [52]). In this case, one can use Brody curves (which are entire curves with bounded derivative) and observe that Brody curves in abelian varieties are projections of an affine line in the universal cover of A. It follows that the Zariski closure of a Brody curve in an abelian variety is a translate of an abelian subvariety. The arithmetic counterpart of the Green-Griffiths conjecture is the following: Conjecture 4.13 (Lang-Bombieri). – Let X be a variety of general type which is defined over a number field K. Then K-points of X are not Zariski dense. This conjecture is Faltings’ theorem in the case of subvarieties of abelian varieties. What should be the generalization of these conjectures to varieties of any Kodaira dimension? Again, it is generally conjectured that for varieties of Kodaira dimension ≤ 0, the Kobayashi pseudodistance is trivial (see [33]). On the arithmetic side, the following analogue is completely open: Conjecture 4.14 (Campana). – Let X be a variety of Kodaira dimension ≤ 0, defined on a field K which is either a number field or the function field of a complex curve. Then rational points are potentially dense on X, meaning that there is a finite extension L of K such that L-points of X are Zariski dense in X. Note that potential density is a property which is invariant under proper étale covers of projective varieties (Chevalley-Weil’s theorem, see [30]). It follows that the Lang-Bombieri conjecture also implies : Conjecture 4.15. – If X is smooth projective and there exists a dominant rational map from an étale cover X 0 of X to a variety of general type, then X is not potentially dense.

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In the situation of Conjecture 4.15, X 0 is not special and thus, as specialness is a property invariant under proper étale covers, X is not special. Campana goes further and proposes that the obstruction to potential density lies in the existence of a rational map of general type, or equivalently, in the fact that X is not special. Conjecture 4.16 (Campana). – A smooth projective variety defined over a number field is potentially dense if and only if it is special. This is the arithmetic counterpart of the following conjecture on hyperbolicity: Conjecture 4.17 (Campana). – A smooth projective variety defined over a number field has vanishing Kobayashi pseudo-distance if and only if it is special. Campana asked in [16] whether a variety which is not special has a finite étale cover admitting a dominant rational map to a variety of general type. The following example, due to Bogomolov and Tschinkel [9] provides a negative answer. In other words, the assumption of Conjecture 4.15 is stronger than being nonspecial. Example 4.18. – The examples constructed by Bogomolov and Tschinkel are elliptic threefolds φ : X → B fibered over an elliptic surface B and satisfying the following conditions: (i) π1 (X) = {1}. (ii) The pair (B, ∆φ ) is of general type. Such an X is constructed as follows: First of all one construct a simply connected elliptic surface E → P1 with one multiple fiber, say of multiplicity 2, over ∞. Then one chooses an elliptic surface S and a rational map f : S 99K P1 given by an ample enough Lefschetz pencil of curves on S and smooth fiber S∞ over ∞. Let B be a blowup of S such that f becomes well-defined on B and let X = B ×P1 E . The divisor ∆φ for the morphism X → B contains the curve B∞ with multiplicity 21 . Let F be the exceptional divisor of τ : B → S. One has B∞ = τ ∗ S∞ (−F ) and KB = τ ∗ KS (F ) hence KB ( 21 B∞ ) = τ ∗ (KS ( 12 S∞ )) + 21 F , hence the logarithmic Kodaira dimension of (B, ∆φ ) is as large as we want, by choosing the curve S∞ positive enough. One then shows that X can be chosen simply connected. Being fibered over B into elliptic curves, with B not a surface of general type and not admitting a morphism to a curve of genus ≥ 2, the variety X does not admit a dominant rational map to a variety of general type, and neither does any of its finite étale covers (since they are all isomorphic to X). On the other hand, X is not special since φ is of general type.

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Claire Voisin, CNRS, IMJ-PRG, 4 Place Jussieu, 75005 Paris, France E-mail : [email protected]

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PANORAMAS ET SYNTHÈSES

2021 56. B. CLAUDON, P. CORVAJA, J.-P. DEMAILLY, S. DIVERIO, J. DUVAL, C. GASBARRI, S. KEBEKUS, M. PĂUN, E. ROUSSEAU, N. SIBONY, B. TAJI, C. VOISIN – Hyperbolicity properties of algebraic varieties 55. D. BEN-ZVI, D. CALAQUE, J. GRIVAUX, E. MANN, D. NADLER, T. PANTEV, M. ROBALO, P. SAFRONOV, G. VEZZOSI – Derived algebraic geometry 2019 54. F. ANDREATTA, R. BRASCA, O. BRINON, X. CARUSO, B. CHIARELLOTTO, G. FREIXAS I MONTPLET, S. HATTORI, N. MAZZARI, S. PANOZZO, M. SEVESO, G. YAMASHITA – An excursion into p-adic Hodge theory : from foundations to recent trends 2018 53. F. BENAYCH-GEORGES, C. BORDENAVE, M. CAPITAINE, C. DONATI-MARTIN, A. KNOWLES (edited by F. BENAYCH-GEORGES, D. CHAFAÏ, S. PÉCHÉ, B. DE TILIÈRE) – Advanced Topics in Random Matrices 2017 52. P. HABEGGER, G. RÉMOND, T. SCANLON, E. ULLMO, A. YAFAEV – Autour de la conjecture de Zilber-Pink 51. F. MANGOLTE, J.-P. ROLIN, K. KURDYKA, S. BASU, V. POWERS (edited by K. BEKKA, G. FICHOU, J.-P. MONNIER, R. QUAREZ) – Géométrie algébrique réelle 2016 50. A. NOVOTNÝ, R. DANCHIN, M. PEREPELITSA, edited by D. BRESCH – Topics on Compressible Navier-Stokes Equations 49. T. SAITO, L. CLOZEL, J. WILDESHAUS – Autour des motifs. École d’été franco-asiatique de géométrie algébrique et de théorie de nombres 48. T. T. Q. LÊ, C. LESCOP, R. LIPSHITZ, P. TURNER – Lectures on quantum topology in dimension three 2015 47. M. DEMAZURE, B. EDIXHOVEN, P. GILLE, W. VAN DER KALLEN, T.-Y. LEE, S. P. LEHALLEUR, M. ROMAGNY, J. TONG, J.-K. YU – B. EDIXHOVEN, P. GILLE, G. PRASAD, P. POLO, eds. – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume III 46. B. CALMÈS, P.-H. CHAUDOUARD, B. CONRAD, C. DEMARCHE, J. FASEL – B. EDIXHOVEN, P. GILLE, G. PRASAD, P. POLO, eds. – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume II 45. B. DE TILIÈRE, P. FERRARI – C. BOUTILLIER, N. ENRIQUEZ, eds. – Dimer Models and Random Tilings 44. L. BORCEA, H. KANG, H. LIU, G. UHLMANN – H. AMMARI, J. GARNIER, eds. – Inverse Problems and Imaging 2014 42–43. S. BROCHARD, B. CONRAD, J. OESTERLÉ – Autour des schémas en groupes, École d’été « Schémas en groupes », Volume I

2013 41. M. LEVINE, J. WILDESHAUS, B. KAHN – Asian-French summer school on algebraic geometry and number theory 39–40. P. DEGOND, V. GRANDGIRARD, Y. SARAZIN, S. C. JARDIN, C. VILLANI – N. CROUSEILLES, H. GUILLARD, B. NKONGA, E. SONNENDRÜCKER, eds. – Numerical models for fusion 2012 38. V. BANICA, L. VEGA, C. BARDOS, D. LANNES, J. EGGERS, M. A. FONTELOS, A. MELLET, Y. POMEAU, M. LE BERRE – C. JOSSERAND, L. SAINT-RAYMOND, eds. – Singularities in mechanics : formation, propagation and microscopic description 37. D. CHAFAÏ, O. GUÉDON, G. LECUÉ, A. PAJOR – Interactions between compressed sensing random matrices and high dimensional geometry 36. K. BELABAS, H. W. LENSTRA JR., P. GAUDRY, M. STOLL, M. WATKINS, W. MCCALLUM, B. POONEN, F. BEUKERS, S. SIKSEK – Explicit methods in number theory rational points and Diophantine equations 2011 34-35. M. BRUNELLA, S. DUMITRESCU, P. EYSSIDIEUX, A. GLUTSYUK, L. MEERSSEMAN, M. NICOLAU. S. DUMITRESCU, ed. – Complex manifolds, foliations and uniformization 33. V. P. KOSTOV – Topics on hyperbolic polynomials in one variable 2010 32. J. BARRAL, J. BERESTYCKI, J. BERTOIN, A. H. FAN, B. HAAS, S. JAFFARD, G. MIERMONT, J. PEYRIÈRE – Quelques interactions entre analyse, probabilités et fractals 31. L. BONAVERO, B. HASSETT, J. M. STARR, O. WITTENBERG – Variétés rationnellement connexes : aspects géométriques et arithmétiques 30. S. CANTAT, A. CHAMBERT-LOIR, V. GUEDJ – Quelques aspects des systèmes dynamiques polynomiaux 2009 29. M. KIM, R. SUJATHA, L. LAFFORGUE, A. GENESTIER, NGÔ B. C. – École d’été francoasiatique de géométrie algébrique et de théorie des nombres 28. C. BERTHON, C. BUET, J.-F. COULOMBEL, B. DESPRÈS, J. DUBOIS, T. GOUDON, J. E. MOREL, R. TURPAULT – Mathematical models and numerical methods for radiative transfer 2008 27. P.-L. CURIEN, H. HERBELIN, J.-L. KRIVINE, P.-A. MELLIÈS – Interactive Models of Computation and Program Behaviour 26. Z. DJADLI, C. GUILLARMOU, M. HERZLICH – Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques 25. M. DISERTORI, W. KIRSCH, A. KLEIN, F. KLOPP, V. RIVASSEAU – Random Schrödinger operators 2007 24. M. COSTE, T. FUKUI, K. KURDYKA, C. McCRORY, A. PARUSIŃSKI, L. PAUNESCU – Arc spaces and additive invariants in Real Algebraic and Analytic Geometry 23. R. CERF – On Cramér’s Theory in infinite dimensions 2006 ˜ 22. S. VU NGO . C – Systèmes intégrables semi-classiques : du local au global 21. S. CROVISIER, J. FRANKS, J.-M. GAMBAUDO, P. LE CALVEZ – Dynamique des difféomorphismes conservatifs des surfaces : un point de vue topologique

Panoramas et Synthèses La série Panoramas et Synthèses publie, en français ou en anglais, des textes de 100 à 150 pages environ faisant le point sur l’état présent d’un sujet mathématique. Dans une présentation soignée, les auteurs s’attachent à mettre en évidence les difficultés, à donner un parfum des démonstrations et un aperçu de l’histoire récente du sujet. Les textes, destinés à des mathématiciens professionnels non spécialistes, doivent être utilisables par des étudiants de doctorat.

In the series Panoramas et Synthèses are published texts from 100 to 150 pages, in French or in English, which give an account of the present state of some mathematical area. The authors aim at explaining the main problems, while giving some flavor of the proofs and an overview of the recent developments of their subject. The texts, which are intended to be read by non-specialists, should be accessible to graduate students.

Instructions aux auteurs / Instructions to Authors Le manuscrit doit être envoyé en double exemplaire au secrétariat des publications en précisant le nom de la revue. Le fichier source TEX (un seul fichier par article) peut aussi être envoyé par courrier électronique ou par transfert FTP, sous réserve que sa compilation par le secrétariat SMF soit possible. Contacter le Secrétariat à l’adresse électronique [email protected] pour obtenir des précisions. La SMF recommande vivement l’utilisation d’AMS-LATEX avec sa classe smfart.cls et la feuille de style panoramas.sty, disponibles ainsi que leur documentation sur le serveur http://smf.emath.fr/ ou sur demande au Secrétariat des publications SMF. Les autres formats TEX et les autres types de traitement de texte ne sont pas utilisables par le secrétariat et sont fortement déconseillés. Avant de saisir leur texte, les auteurs sont invités à prendre connaissance du document Recommandations aux auteurs disponible au secrétariat des publications de la SMF ou sur le serveur de la SMF.

Two copies of the original manuscript should be sent to the editorial board of the SMF, indicating to which publication the paper is being submitted. The TEX source file (a single file for each article) may also be sent by electronic mail or by FTP transfer, in a format suitable for typesetting by the Secretary. Please, send an email to [email protected] for precise information. The SMF has a strong preference for AMS-LATEX together with the documentclass smfart.cls and the style file panoramas.sty, available with their User’s Guide at http: // smf. emath. fr/ (Internet) or on request from the editorial board of the SMF. Files prepared with other TEX dialects or other word processors cannot be used by the editorial board and are not encouraged. Before preparing their electronic manuscript, the authors should read the Advice to authors, available on request from the editorial board of the SMF or from the web site of the SMF.

Since its introduction in the 70’s, the notion of Kobayashi hyperbolicity has attracted a lot of attention in the mathematical community. Besides its aspects exclusively belonging to the several complex variables world, an extremely fascinating theme is that of its interactions with the algebraic, arithmetic, and differential geometric properties of algebraic varieties. These interactions are essentially what this book is about. Some of the issues addressed are: distribution and distribution of values of entire curves, algebraic analogues of hyperbolicity, hyperbolicity properties of projective hypersurfaces and of varieties of general type, hyperbolicity of moduli spaces, relationships between hyperbolicity and negative curvature, distribution of rational points on hyperbolic (arithmetic) varieties, and interplay of different kinds of natural fibrations on algebraic varieties and hyperbolicity. The volume has the ambition to make a point of the state of the art, each chapter treating a different aspect of the subject, trying to keep the language friendly enough to encourage in particular PhD students as well as young researchers in complex geometry to get into the most recent advances in the study of hyperbolicity properties of algebraic varieties.

Société Mathématique de France