A theory of dormant opers on pointed stable curves 9782856299562

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432

ASTÉRISQUE 2022

A THEORY OF DORMANT OPERS ON POINTED STABLE CURVES Yasuhiro WAKABAYASHI

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Astérisque est un périodique de la Société Mathématique de France. Numéro 432, 2022

Comité de rédaction Marie-Claude Arnaud Alexandru Oancea Christophe Breuil Nicolas Ressayre Philippe Eyssidieux Rémi Rhodes Colin Guillarmou Sylvia Serfaty Fanny Kassel Sug Woo Shin Eric Moulines Nicolas Burq (dir.) Diffusion Maison de la SMF Case 916 - Luminy 13288 Marseille Cedex 9 France [email protected]

AMS P.O. Box 6248 Providence RI 02940 USA http://www.ams.org

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ISSN: 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-956-2 doi:10.24033/ast.1171 Directeur de la publication : Fabien Durand

432

ASTÉRISQUE 2022

A THEORY OF DORMANT OPERS ON POINTED STABLE CURVES Yasuhiro WAKABAYASHI

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Yasuhiro Wakabayashi Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan [email protected]

Texte reçu le 22 décembre 2017 ; révisé le 24 avril 2021 ; accepté le 21 octobre 2021. Mathematical Subject Classification (2010). — 14H10, 14H60. Keywords. — p-adic Teichmüller theory, pointed stable curve, moduli space, positive characteristic, algebraic group, Lie algebra, principal bundle, connection, differential operator, oper, p-curvature, fusion ring, Quot-scheme. Mots-clefs. — Théorie de Teichmüller p-adique, courbe stable pointée, espace de modules, caractéristique positive, groupe algébrique, algèbre de Lie, fibré principal, connexion, opérateur différentiel, oper, p-courbure, anneau de fusion, schéma Quot.

A THEORY OF DORMANT OPERS ON POINTED STABLE CURVES by Yasuhiro WAKABAYASHI

Abstract. — This manuscript presents a detailed and original account of the theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The theory of dormant opers (or more generally, opers in positive characteristic) on pointed stable curves, which has proved to be rather rich and deep, was born in the work of S. Mochizuki, who developed the theory for sl2 -opers and used it to establish p-adic Teichmüller theory. Some parts of Mochizuki’s work were later extended in the case of proper smooth curves by K. Joshi, C. Pauly, and other mathematicians. This manuscript represents an advance in the theory of opers that takes the subject beyond the work of Mochizuki, Joshi, and Pauly. In particular, we provide general unified formulations and the basics of principal bundles and connections defined on families of pointed stable curves. The notion of an oper is accordingly introduced in the context of logarithmic algebraic geometry. Some of the results can be regarded as generalizations of results obtained in the fundamental work on the geometric Langlands program developed by A. Beilinson and V. Drinfeld. We also describe various properties and assertions about (dormant) opers, such as duality, comparison with differential operators, and compactification of the moduli space. Our goal is to give an explicit formula, conjectured by Joshi, for the generic number of dormant sln -opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariants for Quot schemes in characteristic zero. This formula reveals an interaction between studies in p-adic Teichmüller theory and certain areas of mathematics, including GromovWitten theory. Résumé. (Une théorie des opers dormants sur des courbes stables pointées) — Ce manuscrit expose de manière détaillée et originale la théorie des opers définis sur des courbes stables pointées en caractéristique arbitraire et leurs modules. En particulier, il développe l’étude des opers dormants, qui sont des opers d’un certain type en caractéristique positive. La théorie des opers dormants (ou plus généralement des opers en caractéristique positive) sur des courbes stables pointées, qui s’est avérée assez riche et profonde, est née des travaux de S. Mochizuki, qui a développé la théorie des sl2 -opers et l’a utilisé pour établir la théorie de Teichmüller p-adique.

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Certains parties des recherches de Mochizuki ont ensuite été approfondies dans le cas des courbes lisses et propres par K. Joshi, C. Pauly et d’autres mathématiciens. Ce manuscrit représente une avancée dans la théorie des opers qui est poussée au-delà des travaux de Mochizuki, Joshi et Pauly. Nous fournissons notamment des formules générales unifiées et les bases des principaux fibrés et connexions définis sur des familles de courbes stables pointées. La notion d’opers est donc introduite dans le contexte de la géométrie algébrique logarithmique. Certains des résultats peuvent être considérés comme des généralisations des résultats obtenus dans les travaux fondamentaux sur le programme géométrique de Langlands développé par A. Beilinson et V. Drinfeld. Nous décrivons également diverses propriétés et assertions sur les opers (dormants), telles que la dualité, la comparaison avec les opérateurs différentiels et la compactification de l’espace des modules. Notre but est de donner une formule explicite, conjecturée par Joshi, pour le nombre générique des sln -opers dormants. Nous y parvenons en acquérant une compréhension détaillée de l’espace des modules des opers dormants et en calculant les invariants de Gromov-Witten pour les schémas Quot en caractéristique zéro. Cette formule révèle une interaction entre les études de la théorie de Teichmüller p-adique et certains domaines des mathématiques, notamment la théorie de Gromov-Witten.

ASTÉRISQUE 432

CONTENTS

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1. Review of opers on complex algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . 0.2. What about opers in positive characteristic? . . . . . . . . . . . . . . . . . . . . . . . . 0.3. Counting problem of dormant opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4. Comparison with arguments due to Mochizuki, Joshi, and Pauly . . . 0.5. Structure and main theorems of the manuscript . . . . . . . . . . . . . . . . . . . . . 0.6. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.7. Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 4 5 6 15 15

1. Logarithmic connections over log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The adjoint representation and the Maurer-Cartan form . . . . . . . . . . . . 1.2. Principal G-bundles and their associated Lie algebroids . . . . . . . . . . . . . 1.3. Logarithmic connections and curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Explicit descriptions of gauge transformations . . . . . . . . . . . . . . . . . . . . . . 1.5. The moduli stack of pointed stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Local pointed curves and monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 22 27 33 38 42

2. Opers on a family of pointed stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. (g, ℏ)-opers on a log curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Local description and automorphisms of a (g, ℏ)-oper . . . . . . . . . . . . . . . 2.3. The case of g = sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The G-bundle † EG and the vector bundle † Vg . . . . . . . . . . . . . . . . . . . . . . . 2.5. ↙ q1 -normality and a torsor structure on the sheaf of (g, ℏ)-opers . . . . . 2.6. (g, ℏ)-opers on pointed stable curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. The adjoint quotient of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Radii of (g, ℏ)-opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. The moduli space of (g, ℏ)-opers of prescribed radii . . . . . . . . . . . . . . . . . 2.10. The universal moduli stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50 55 56 61 65 69 71 74 77

3. Opers in positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Frobenius twists and relative Frobenius morphisms . . . . . . . . . . . . . . . . . 3.2. Lie algebroids in positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. p-curvature on an ℏ-flat G-bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Dormant/p-nilpotent (g, ℏ)-opers and their moduli spaces . . . . . . . . . . . 3.5. The Hitchin-Mochizuki morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Varying the parameter ℏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. The finiteness of the Hitchin-Mochizuki morphism . . . . . . . . . . . . . . . . . . 3.8. The universal moduli stacks of dormant/p-nilpotent (g, ℏ)-opers . . . .

79 79 82 85 89 93 100 103 106

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4. Flat vector bundles and differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Logarithmic ℏ-connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Logarithmic differential operators with parameter ℏ . . . . . . . . . . . . . . . . 4.3. Comparison of log connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. (GLn , ℏ)-opers and comparison with (sln , ℏ)-opers . . . . . . . . . . . . . . . . . . 4.5. (n, ℏ, Θ)-projective connections and (GLn , ℏ, Θ)-opers . . . . . . . . . . . . . . 4.6. (n, ℏ)-theta characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Radii of (GLn , ℏ, ϑ)-opers and (n, ℏ, ϑ)-projective connections . . . . . . . 4.8. Change of (n, ℏ)-theta characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. A canonical PGLn -bundle underlying (sln , ℏ)-opers . . . . . . . . . . . . . . . . . 4.10. Dormant (GLn , ℏ)-opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11. Comparison of the moduli functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12. Hypergeometric differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 115 117 122 128 132 138 141 144 149 154 158

5. Duality of opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. (so2l+1 , ℏ)-opers and (sp2m , ℏ)-opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. (GO2l+1 , ℏ, ϑ)-opers and (GSp2m , ℏ, ϑ)-opers . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Dual opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Isomorphisms of moduli spaces induced by duality . . . . . . . . . . . . . . . . . .

163 163 165 170 173

6. Local deformation of opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Deformation spaces of a curve and an ℏ-flat G-bundle . . . . . . . . . . . . . . 6.2. Cohomology of complexes associated to a (g, ℏ)-oper . . . . . . . . . . . . . . . . 6.3. Infinitesimal deformations of a (g, ℏ)-oper . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Flat vector bundles in positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Infinitesimal deformations of a dormant (g, ℏ)-oper . . . . . . . . . . . . . . . . .

177 177 184 188 193 200

7. The pseudo-fusion ring for dormant opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Semi-graphs and clutching data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Gluing ℏ-flat G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Factorization of (g, ℏ)-opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Pseudo-fusion rules and pseudo-fusion rings . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. The ring structure of the fusion ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Factorization property of a fusion rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7. The pseudo-fusion ring associated to dormant opers . . . . . . . . . . . . . . . . 7.8. The pseudo-fusion rule for sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 210 214 222 224 232 237 240

8. Generic étaleness of the moduli space of dormant opers . . . . . . . . . . . . . . . . . . . 8.1. Formally local description of a flat bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Deformation of a flat bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Deformation of a dormant GLn -oper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Dormant opers on a 3-pointed projective line . . . . . . . . . . . . . . . . . . . . . . . 8.5. The generic étaleness of the moduli stack of dormant opers . . . . . . . . .

247 248 251 254 259 264

9. Comparison with Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Comparison with the moduli space of dormant (GLn , ϑ)-opers . . . . . . 9.3. The relation between the Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 268 273

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9.4. Counting maximal subbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Joshi’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 279

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 0 INTRODUCTION

Mon prince, il y a plus de cinquante ans que j’ay ouï dire à mon pere qu’il y avoit dans ce chasteau une princesse, la plus belle du monde; qu’elle y devoit dormir cent ans, et qu’elle serait réveillée par le fils d’un roy, à qui elle estoit reservée. — Charles Perrault, La Belle au bois dormant (cf. [72]).

This manuscript presents the foundation for a theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The final goal of this manuscript is to prove the following conjectural formula for a general X: Joshi’s conjecture (Cf. [42], Conjecture 8.1). — We have Qn X Zzz... ( i=1 ζi )(n−1)(g−1) p(n−1)(g−1)−1 Q · . (1) deg(Opsln ,X /k) = g−1 n! i̸=j (ζi − ζj ) ×n (ζ1 ,...,ζn )∈C p ζ =1, ζi ̸=ζj (i̸=j) i

We shall explain what is displayed above. Let k be an algebraically closed field of characteristic p > 0 and X a proper smooth curve of genus g > 1 over k. Assume that p > C, where C is a specific constant depending on both g and a fixed positive integer n. It is known (cf. [43], Theorem 5.4.1 and Corollary 6.1.6) that the moduli Zzz... functor Opsln ,X classifying dormant sln -opers on X may be represented by a nonempty finite scheme over k. Toward a further understanding of this moduli space, Kirti Joshi proposed, in [42], the conjectural formula (1) displayed above, computing the Zzz... Zzz... degree deg(Opsln ,X /k) of Opsln ,X over k. Here, the sum on the right-hand side of (1) is taken over the set of n-tuples (ζ1 , . . . , ζn ) ∈ C×n of p-th roots of unity in C (:= the field of complex numbers) that satisfy ζi ̸= ζj if i ̸= j.

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CHAPTER 0. INTRODUCTION

0.1. Review of opers on complex algebraic curves Before continuing our explanation of the present manuscript, let us briefly review what an oper is. Recall (cf. Definition 2.1) that an oper, or more precisely a g-oper for a semisimple Lie algebra g, is defined as a principal bundle over an algebraic curve equipped with a flat connection satisfying certain conditions. Opers on an algebraic curve, or a formal disk, over C play a central role in some areas of mathematics. For example, such objects were studied in the context of the geometric Langlands program (cf. [12]) to construct Hecke eigensheaves on the moduli space of principal bundles by associating them with the base space of Hitchin’s integrable system. Also, as discussed in [23], the moduli space of opers on a formal disk is realized as the phase space of Drinfeld-Sokolov hierarchies, i.e., certain infinite families of nonlinear PDEs; these hierarchies are certainly among the most studied examples of integrable systems because of their tau-symmetries. Many mathematicians have studied various equivalent realizations of opers, including certain kinds of differential operators related to Schwarzian equations. For example, sl2 -opers may be interpreted as a projective structure. A projective structure on a Riemann surface XC is a maximal system of local coordinates on XC in the analytic topology modeled on the complex projective line P1C (= C ⊔ {∞}) such that on any two overlapping coordinate patches, the change of coordinates may be described as a Möbius transformation. On a Riemann surface equipped with a projective structure, there is a projective geometry, in the spirit of F. Klein’s Erlangen program, that locally agrees with the geometry of P1C . Projective structures arise in many areas of mathematics and are still actively being investigated. Indeed, such additional structures on Riemann surfaces play a major role in understanding the framework of uniformization theorem. An important consequence of the uniformization theorem for Riemann surfaces is that any closed Riemann surface is
isomorphic either to P1C , or to the quotient of the complex affine line A1C = C ⊆ P1C by a discrete group of translations (i.e., a 1-dimensional complex
torus), or to the quotient of the 1-dimensional complex hyperbolic space H1C ⊆ P1C by a torsion-free discrete subgroup of SU(1, 1) ∼ = SL(2, R). This implies that by collecting various local inverses of a universal covering map, we have a canonical projective structure on each closed Riemann surface. Moreover, sl2 -opers were introduced and studied, under the name of indigenous bundles, in the work of R. C. Gunning (cf. [31], § 2). One may think of an indigenous bundle as an algebraic object encoding analytic (i.e., non-algebraic) uniformization data for Riemann surfaces. Also, in the work of C. Teleman (cf. [90]), one may find the objects corresponding to sln -opers, under the name of homographic structures. In the case of characteristic zero, we refer the reader to [15], [9], and [82] for further studies of sln -opers, and to [28], [25], and [27] for reviews and expositions concerning g-opers for an arbitrary g over C.

ASTÉRISQUE 432

0.2. WHAT ABOUT OPERS IN POSITIVE CHARACTERISTIC?

3

0.2. What about opers in positive characteristic? In the present manuscript, we study g-opers in arbitrary characteristic, i.e., ones including the case of positive characteristic. Just as in the complex case mentioned above, one may define the notion of a g-oper in characteristic p > 0 because of the algebraic nature of its formulation. For g = sl2 , various properties of such objects and their moduli space were discussed in the context of the p-adic Teichmüller theory developed by S. Mochizuki (cf. [59], [60]). From a different point of view, Y. Ihara developed, in, e.g., [39], [38], a theory of Schwarzian equations in an arithmetic context. For the case where g is more general (but the underlying curve is assumed to be smooth over an algebraically closed field), g-opers in positive characteristic were studied by K. Joshi, S. Ramanan, E. Z. Xia, J. K. Yu, C. Pauly, T. H. Chen, X. Zhu, et al. (cf., e.g., [44], [43], [42]). One of the common key ingredients in the development of these works is the study of the p-curvature of a connection. Recall that the p-curvature of a connection is an invariant that measures the obstruction to the compatibility of p-power structures appearing in certain associated spaces of infinitesimal (i.e., “Lie”) symmetries. It is well-known that a flat connection on a vector bundle has vanishing p-curvature if and only if the corresponding homogeneous linear differential equation locally has a full set of solutions in the Zariski topology. We say that a g-oper is dormant (cf. Definition 3.15) if the p-curvature of the connection defining this oper vanishes identically. This class of opers enables us to develop geometry in positive characteristic from the same viewpoint as the theory of projective structures on Riemann surfaces. One thing to keep in mind is that due to its analytic formulation, the definition of a projective structure cannot be adopted in positive characteristic as it is. Indeed, if we had defined it in a naive fashion, it would be nothing other than the trivial example of a curve having such a structure. Regarding this matter, Y. Hoshi (cf. [36]) introduced Frobenius-projective structures (of finite level) to deal with appropriate replacements. It was shown that giving a Frobenius-projective structure of level 1 is equivalent to giving a dormant sl2 -oper (cf. [36], Theorem A); this equivalence is an analogue of the relation, mentioned before, between projective structures on a Riemann surface and sl2 -opers on it. We refer the reader to, e.g., [95] and [94] for studies related to this topic. There are some other aspects of dormant opers. For example, dormant sl2 -opers on a proper smooth curve correspond, via Cartier descent, to a certain type of Frobenius-destabilized vector bundles of rank 2 (cf. [69], § 4, Proposition 4.2). This correspondence gives us an approach to understanding the generalized Verschiebung map on the moduli space of rank 2 semistable bundles. Indeed, in the case of genus 2 curves, we can compute the degree of this rational map by using the fact that the set of points classifying such bundles in this moduli space coincides with the undefined locus (cf. [68], § 1, Theorem 1.2). Also, results by S. Mochizuki, F. Liu, and B. Osserman (cf. [60], [54], [93]) provide an interesting connection with the combinatorics of

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CHAPTER 0. INTRODUCTION

convex rational polytopes, as well as edge-colored graphs, and the geometry of moduli spaces classifying dormant sl2 -opers. As another example (cf. [74], Remark 4.1.2), the sheaf of locally exact 1-forms on X may be obtained as the sheaf of horizontal sections of a certain dormant slp−1 -oper twisted by a flat line bundle, i.e., a certain GLp−1 -oper (cf. Definition 4.17, (i)) with vanishing p-curvature. This specific object is of importance because the theta divisor associated to this bundle gives information on the fundamental group of the underlying curve (cf. [74], Theorem 4.1.1). It leads us to the sort of phenomena studied in anabelian geometry for hyperbolic curves in positive characteristic (cf. [73], [75], [88], [89]). (But, we should note that the phenomena studied in the cited references are mostly related to the associated theta divisor and not based on properties of the oper structure on it.) On the other hand, in the context of p-adic Teichmüller theory, it is also worth studying g-opers with nilpotent p-curvature, which will be called p-nilpotent g-opers (cf. Definition 3.15). I. I. Bouw and S. Wewers set up an equivalence between p-nilpotent sl2 -opers and certain deformation data on the underlying curve (cf. [16]). This equivalence allows us to translate the problem of the existence of such deformation data into one on the existence of polynomial solutions of certain differential equations with additional properties. Y. Hoshi advanced the understanding of indigenous bundles with nilpotent p-curvature in characteristic three by creating a complete list of them for the case where the underlying curve is of genus 2 (cf. [35], Theorem 6.1). However, in the present manuscript, we will focus more on dormant g-opers rather than p-nilpotent ones. 0.3. Counting problem of dormant opers With the above-mentioned background in mind, the study of dormant g-opers and their moduli could be placed within developments of certain areas of mathematics, and hence, the following question regarding such objects may arise naturally from various points of view: Can one explicitly calculate how many dormant g-opers exist on a fixed (general) curve? In what follows, we shall review previous results concerning the answer to the above question. Let X/k be as introduced at the beginning of this Introduction. A theorem in p-adic Teichmüller theory due to S. Mochizuki (cf. [60], Chap. II, § 2.3, Theorem 2.8) implies that if X is sufficiently general in the moduli stack Mg of proper Zzz... smooth curves of genus g, then Opsl2 ,X is finite (as we mentioned above) and étale over k. Hence, the task of resolving the above question for the case where g = sl2 Zzz... Zzz... may be reduced to the explicit computation of the degree deg(Opsl2 ,X /k) of Opsl2 ,X over k, that is, to proving Joshi’s conjecture for g = sl2 . In the case of g = 2, S. Mochizuki (cf. [60], Chap. V, § 3.2, Corollary 3.7), H. LangeC. Pauly (cf. [53], Theorem 2), and B. Osserman (cf. [70], Theorem 1.2) verified, by

ASTÉRISQUE 432

0.4. COMPARISON WITH ARGUMENTS DUE TO MOCHIZUKI, JOSHI, AND PAULY

5

applying different methods, the following equality: Zzz... 1 (2) deg(Opsl2 ,X /k) = · (p3 − p). 24 Moreover, by extending the relevant formulations to the case where X admits marked points and nodal singularities, S. Mochizuki also gave an explicit description of dormant sl2 -opers on each totally degenerate curve (cf. Definition 7.15) in terms of the radii of atoms (cf. [60], Introduction, Theorem 1.3). This is essentially equivalent to a previous result obtained by Y. Ihara (cf. [39], or § 4.12 in this manuscript), who investigated the situation when a given hypergeometric differential equation in characteristic p had a full set of solutions. As a result of the explicit description, we obtained Zzz... a combinatorial procedure for explicitly computing the value deg(Opsl2 ,X /k) according to a sort of fusion rule. (These fusion rules will be generalized, in Chapter 7 of the present manuscript, to the case of a more general g). The explicit description also leads to a work by F. Liu and B. Osserman (cf. [54], Theorem 2.1), who have shown Zzz... that the value deg(Opsl2 ,X /k) may be expressed as a degree 3g − 3 polynomial with respect to “p”. They did so by applying Ehrhart’s theory, which concerns computing the cardinality of the set of lattice points inside a polytope. For an arbitrary g and g = sln , K. Joshi conjectured, with amazing insight, an Zzz... explicit description of the value deg(Opsln ,X /k), as displayed in (1). In [92], the author proved, by grace of that idea and discussion due to K. Joshi et al. (cf. [43], [42]), the following equality: (3)

Zzz...

deg(Opsl2 ,X /k) =

p−1 pg−1 X 1 · . 2g−2 2g−1 2 sin ( πθ p ) θ=1

(Notice that we assumed the inequality p > 2(g − 1) in the statement of [92], Corollary 5.4. But, by combining this result with [54], Theorem 2.1, one may remove this assumption; i.e., the asserted equality holds even if g (> 1) is arbitrary. See also [93], Corollary 8.11.) After easy calculations, one verifies that (2) and (3) are consistent with (1). That is to say, the conjecture of Joshi has been proved affirmatively for the case where n = 2 and g are arbitrary. A goal of the present manuscript is to prove Joshi’s conjecture (cf. Theorem H asserted below), i.e., to give an affirmative answer to the question above, for the case where n is an arbitrary integer > 1. In [42], the conjectural formula was formulated under the condition that p > C, where C = n(n−1)(n−2)(g −1). But, the left-hand side of this formula actually makes sense for every n, and we shall give its proof under the assumption that p > n·max{g−1, 2}.

0.4. Comparison with arguments due to Mochizuki, Joshi, and Pauly The proof of Joshi’s conjecture given in the present manuscript may be regarded as a simple generalization of the discussion of the n = 2 case. As in [92], our discussion follows, to a substantial extent, the ideas discussed in [43], and in [42]. Indeed, some of the results are mild generalizations of those obtained in [43] concerning sln -opers to the

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case of families of curves over quite general base schemes. (Such relative formulations allow us to discuss deformations of various types of data.) For example, Proposition 9.1 in the present manuscript corresponds to [43], Theorem 3.1.6 (or [44], § 5.3; [86], § 2, Lemma 2.1); Proposition 9.2 corresponds to [43], Theorem 5.4.1; and Proposition 9.4 corresponds to [43], Proposition 5.4.2, which assoZzz... ciates the moduli space Opsln ,X with a certain Quot scheme in positive characteristic. Also, as in [92], we prove the vanishing of obstruction to lifting the Quot scheme to characteristic 0 (cf. the proof of Theorem 9.8). Thus, in order to complete the proof of Zzz... Joshi’s conjecture, we relate the value deg(Opsln ,X /k) to the degree of the resulting Quot scheme over C and directly apply a formula of Holla (cf. Theorem 9.7), which is a special case of the Vafa-Intriligator formula. Note that this idea of connecting with the formula of Y. Holla is due to K. Joshi (cf. [42]). Zzz... Our discussion is founded on the generic (relative to X) étaleness of Opsln ,X over k. As mentioned above, S. Mochizuki already proved this fact for n = 2. Unfortunately, however, the remaining case (i.e., n > 2) remains unproven. We will prove the generic Zzz... étaleness of Opsln ,X /k for an arbitrary n, which is one of the main results of the present manuscript (cf. Theorem 8.19). To this end, following the discussion in [60] (for the case of n = 2), we start by formulating the notion of an sln -oper (or, more generally, a g-oper for a semisimple Lie algebra g) on a family of pointed stable curves, and then develop a theory of such objects. Moreover, the steps along the path to our goal (e.g., showing the nonemptiness of the moduli space of opers of prescribed radii and the finiteness of the moduli of dormant opers) will require us to study a sort of generalization of g-opers (cf. Definition 2.1, (i); [12], § 3.1.14; [13], § 5.2), which is called a (g, ℏ)-oper (where ℏ is a parameter).

0.5. Structure and main theorems of the manuscript Chapter 1: Logarithmic connections over log schemes. — In the first half of Chapter 1, we discuss the general theory of principal G-bundles and ℏ-connections, where G is a smooth affine algebraic group with Lie algebra g and ℏ is a parameter. The notion of an ℏ-connection was introduced by P. Deligne and used to realize Higgs bundles as a degeneration of flat vector bundles, i.e., vector bundles equipped with a flat connection. Also, C. T. Simpson studied extensively relevant moduli spaces in a series of papers [80], [81], and [82]. Here, we generalize the situation by dealing with ℏ-flat G-bundles (which means principal G-bundles equipped with a flat ℏ-connection) defined on a family of smooth log schemes. Related notions are reviewed in the context of logarithmic algebraic geometry, e.g., the Maurer-Cartan form, Lie algebroids, and curvature, etc. One of the important steps for the subsequent discussion is to describe gauge transformations of an ℏ-connection using the exponential and logarithm maps. (For example, the resulting description will be used to prove that each (g, ℏ)-oper has no nontrivial automorphisms. This fact cannot be ignored from the viewpoint of

ASTÉRISQUE 432

0.5. STRUCTURE AND MAIN THEOREMS OF THE MANUSCRIPT

7

representability of the relevant moduli categories.) If the base field is of characteristic p > 0, then the prime p has to be sufficiently large relative to (a certain invariant associated to) G in order to make that description possible. In the second half of Chapter 1, we briefly recall the definition of a pointed stable curve and the stack Mg,r over a field k (where g and r are nonnegative integers with 2g − 2 + r > 0), i.e., the moduli stack classifying r-pointed stable curves of genus g over k. By considering the natural log structure on a pointed stable curve, we can associate a certain g-valued invariant to each ℏ-flat G-bundle and a choice of marked points. According to the wording in [59], [60], we call this invariant the “monodromy (operator) ”. It determines the sort of boundary condition to glue together ℏ-flat G-bundles on distinct pointed curves along the fibers over the points of attachment. Chapter 2: Opers on a family of pointed stable curves. — In Chapter 2, we study the general theory of opers and their moduli. Given a semisimple algebraic k-group G of adjoint type (equipped with a split maximal torus T and a Borel subgroup B containing T) and ℏ ∈ k, we introduce the notion of a (g, ℏ)-oper, where g denotes the Lie algebra of G, defined on a family of pointed stable curves (or more generally, a family of log curves) in arbitrary characteristic. Compared with the previous studies on opers, this would be a formulation in a more general situation because our discussion especially includes the case of nodal curves. The goal of Chapter 2 is to prove the representability of the moduli category classifying pointed stable curves equipped with a (g, ℏ)-oper. R. C. Gunning (cf. [31], § 4, Corollary 2) proved that the set of projective structures (or equivalently, sl2 -opers) on a smooth curve over C forms an affine space modeled on the space of quadratic differentials. The generalization of this result for a general g was given by A. Beilinson and V. Drinfeld (cf. [13] § 3.4, Theorem; [7], Corollary 3.9). On the other hand, S. Mochizuki (cf. [59], Chap. I, Corollary 2.9) constructed the moduli stack of sl2 -opers on the universal family of pointed curves. Also, in [43], Proposition 3.2.3, we can find the construction of SLn -opers in positive characteristic. There is one thing to keep in mind when generalizing the construction of the moduli space. That is, if the underlying curve has a nodal point or a marked point, there may be a situation where the moduli problem of (g, ℏ)-opers cannot be solved in terms of flat vector bundles or differential operators because of the lack of theta characteristics. Taking this into consideration, we introduce a local version of (g, ℏ)-opers (cf. Definition 2.4) and glue together their moduli spaces defined locally on the underlying curve. We apply this argument to the universal family of pointed curves Cg,r parametrized by Mg,r . Thus, we obtain the moduli stack (4)

Opg,ℏ,g,r

(cf. (303)) representing the category of r-pointed stable curves of genus g over k together with a (g, ℏ)-oper on it. It may be regarded as a stack over Mg,r via the projection Opg,ℏ,g,r → Mg,r induced by forgetting the data of (g, ℏ)-opers. Notice

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that, in order to construct Opg,ℏ,g,r , we apply in advance some of the results proved in Chapter 4. A key ingredient in the study of opers on a pointed curve is an invariant called radius (cf. Definition 2.29). The radius of a (g, ℏ)-oper at a marked point is an element of the GIT quotient c := cg of the adjoint G-action on g induced by the monodromy. It defines a sort of boundary condition to glue together (g, ℏ)-opers according to the attachment of underlying curves at this point. Let ρ := (ρi )ri=1 be an element of c×r (k) (= c(k)×r ), i.e., a k-rational point of the r-th power product of c over k, where c×0 (k) := {∅}. Each such element ρ induces the substack Opg,ℏ,ρ,g,r

(5)

of Opg,ℏ,g,r classifying (g, ℏ)-oper of radii ρ, i.e., such that the radius of this oper at the i-th marked point coincides with ρi for every i. In order to ensure the nonemptiness of this stack, we observe the deformation family of the moduli stacks Opg,ℏ,g,r parametrized by the value “ℏ” and the degeneration at ℏ = 0. Summarizing the results in Chapter 2, we obtain the following theorem (cf. Theorem 2.39). (The case of g = sln with 1 < n < p will be proved in Chapter 4, Theorem 4.66.) Theorem A (Structure of Opg,ℏ,g,r ). — Let G be a semisimple algebraic k-group of adjoint type (equipped with a split maximal torus T and a Borel subgroup B defined over k containing T.) In the case of p := char(k) > 0, suppose further that G satisfies either “ G = PGLn with 1 < n < p” or “ 2hG < p” (cf. (104) for the definition of hG ). Denote by g the Lie algebra of G. Then, for any ℏ ∈ k and ρ ∈ c×r (k), Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) may be represented by a relative affine space over Mg,r , and the relative dimension is given by the value 2g − 2 + r r (6) ℵ(g) := · dim(g) + · rk(g) 2 2   r 2g − 2 + r c · dim(g) − · rk(g) . resp., ℵ(g) := 2 2 In particular, Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) is a nonempty, geometrically connected, and smooth Deligne-Mumford stack over k of dimension 3g − 3 + r + ℵ(g) (resp., 3g − 3 + r + c ℵ(g)). Chapter 3: Opers in positive characteristic. — In Chapter 3, we study (g, ℏ)-opers in the case of p := char(k) > 0. To begin with, we introduce the logarithmic version of a restricted Lie algebroid and recall the p-curvature associated to a flat G-bundle on a log scheme. For each ρ ∈ c×r (k), denote by   Zzz... Zzz... (7) Opg,ℏ,g,r resp., Opg,ℏ,ρ,g,r the closed substack of Opg,ℏ,g,r classifying dormant (g, ℏ)-opers (resp., dormant (g, ℏ)-opers of radii ρ). One of the important consequences of this chapter is the  Zzz...

properness of Opg,ℏ,g,r

ASTÉRISQUE 432

Zzz...

resp., Opg,ℏ,ρ,g,r over k. As an intermediate step in proving

9

0.5. STRUCTURE AND MAIN THEOREMS OF THE MANUSCRIPT

κ

κ

this fact, we consider a generalization of the Hitchin-Mochizuki morphism  L N∇  L N∇ ∇ resp., ∇ (8) (1) (1) ℏ,univ : Opg,ℏ,g,r → ℏ,ρ,univ : Opg,ℏ,ρ,g,r → C Ψ (ρ),C g,r

ℏ

g,r

(cf. (401)). This morphism is defined as the restriction to the moduli stack of (g, ℏ)-opers of a certain p-curvature analogue of the Hitchin fibration, studied in [43]. When L N∇ specialized at ℏ = 0, it becomes the relative Frobenius morphism of (1) (resp., Cg,r L N∇ (1) ) over Mg,r (cf. Proposition 3.25). This observation helps us to underΨ (ρ),C ℏ

g,r

κ

κ κ

κ

∇ stand the structure of ∇ ℏ,univ (resp., ℏ,ρ,univ ) for a general ℏ, as asserted below. ∇ Here, we denote the inverse image via ℏ (resp., ∇ ℏ,ρ ) of the zero section by   p-nilp p-nilp (9) Opg,ℏ,g,r resp., Opg,ℏ,ρ,g,r .

Each (g, ℏ)-oper classified by this stack is called p-nilpotent. The main results of Chapter 3 are the following Theorems B and C (cf. Theorems 3.33, 3.38, and 3.39 for the full statements). Theorem B generalizes results proved by S. Mochizuki (cf. [59], Chap. II, Theorem 2.3) and T.-H. Chen-X. Zhu (cf. an unpublished version (1) Zzz... of [19], § 1, Theorem 1.1). Also, note that the desired properness of Opg,ℏ,g,r (resp., Zzz...

p-nilp

p-nilp

Opg,ℏ,ρ,g,r ) follows directly from the finiteness of Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) asserted Zzz...

Zzz...

in Theorem B; this is because Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) may be regarded as a closed p-nilp

p-nilp

substack of Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ). p-nilp

κ

κ

Theorem B (Structure of Opg,ℏ,g,r ). — Let G be as in Theorem A, and suppose that p := char(k) > 2hG . Let us fix an element ℏ of k × and an element ρ of c×r (k) with Ψℏ (ρ) = [0]×r (where ρ = ∅ if r = 0). Then, the morphism ∇ ℏ,univ (resp., c ∇ ℵ(g) ℵ(g) ). In particu(resp., p ℏ,ρ,univ ) is finite and faithfully flat of degree p p-nilp

p-nilp

lar, Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) may be represented by a nonempty and proper Deligne-Mumford stack over k of dimension 3g − 3 + r, and the natural projecp-nilp p-nilp tion Opg,ℏ,g,r → Mg,r (resp., Opg,ℏ,ρ,g,r → Mg,r ) is finite and faithfully flat of c degree pℵ(g) (resp., p ℵ(g) ). Zzz...

Theorem C (Structure of Opg,ℏ,g,r ). — Let G be as in Theorem A, and suppose that p := char(k) > 2hG . Also, let us fix an element ℏ of k × and an element ρ Zzz... of c×r (k) (where ρ = ∅ if r = 0). Then, Opg,ℏ,g,r may be represented by a nonempty Zzz...

proper Deligne-Mumford stack over k of dimension 3g − 3 + r. Also, Opg,ℏ,g,r is finite over Mg,r and has an irreducible component that dominates Mg,r . Finally, we have a Zzz... Zzz... (10) Opg,ℏ,ℏ⋆ϱ,g,r = Opg,ℏ,g,r . ϱ∈c×r (Fp )

1. This version is available at: https://arxiv.org/pdf/1306.0299v1.pdf.

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Zzz...

Remark 0.1. — The nonemptiness of Opg,ℏ,g,r asserted in Theorem C can be verZzz...

ified by recalling the nonemptiness of Opsl2 ,ℏ,g,r and constructing a closed immerZzz...

Zzz...

sion Opsl2 ,ℏ,g,r ,→ Opg,ℏ,g,r by using a suitable sl2 -triple in g. In particular, since Zzz...

Opsl2 ,ℏ,g,r is smooth and of dimension 3g − 3 + r, as proved by S. Mochizuki, the Zzz...

existence of such a closed immersion implies that the dimension of Opg,ℏ,g,r is exactly 3g − 3 + r.

Chapters 4-5: Comparison and duality. — In Chapter 4, we consider the description of (sln , ℏ)-opers in terms of differential operators and vector bundles. For a smooth curve X and a theta characteristic Θ on X (i.e., a square root of the canonical bundle), a projective connection on X in the sense of [26], § 4.1.1, is defined as a second-order differential operator Θ⊗(−1) → Θ⊗3 such that the principal symbol is 1 and the subprincipal symbol is 0. In the usual manner, it gives a rank 2 flat vector bundle equipped with a line subbundle such that the Kodaira-Spencer map is an isomorphism. Such an object is called a GL2 -oper and moreover induces an sl2 -oper via projectivization. The resulting correspondence from projective connections to GL2 -opers, as well as to sl2 -opers, is the simplest case of the main theorem of Chapter 4. We refer the reader to [13], § 2, for higher-order differential operators. To generalize the situation (e.g., to families of not necessarily smooth stable curves), we introduce a generalized theta characteristic, i.e., an (n, ℏ)-theta characteristic (where n is a positive integer and ℏ is a parameter). After choosing an (n, ℏ)-theta characteristic ϑ, we define the notions of an (n, ℏ, ϑ)-projective connection and a (GLn , ℏ, ϑ)-oper (cf. Definitions 4.36, 4.37). For example, an (n, ℏ, ϑ)-projective connection is defined as a certain n-th order differential operator such that the principal symbol is 1 and the subprincipal symbol is characterized by ϑ. The main result of Chapter 4 is as follows (cf. Theorem 4.66 for the full statement): Theorem D (Comparison of moduli functors). — Let X be an r-pointed stable curve of genus g over a k-scheme S, ℏ an element of Γ(S, OS ), n an integer > 1, ϑ := (Θ, ∇ϑ ) an (n, ℏ)-theta characteristic of X , and ρ an element of c×r sln (S) (where ρ = ∅ if r = 0). Denote by Sch/S the category of S-schemes and by □ either the absence or presence of “ρ”. In the case of p := char(k) > 0, we suppose further that n < p. Then, there exists a canonical sequence of isomorphisms between functors Diff♣ n,ℏ,ϑ,□ (11)

where

ASTÉRISQUE 432

Op♡ 9 n,ℏ,□ ∼ Λ♣⇒♦ X ,ϑ,□

Λ♦⇒♡ X ,ϑ,□ ∼

& Op♢ n,ℏ,ϑ,□

∼ Λ♡⇒♠ X ,□

% Op♠ n,ℏ,□ ,

11

0.5. STRUCTURE AND MAIN THEOREMS OF THE MANUSCRIPT

♠ — Op♠ n,ℏ (resp., Opn,ℏ,ρ ) := the moduli functor on Sch/S classifying (sln , ℏ)-opers (resp., (sln , ℏ)-opers of radii ρ) on X , i.e.,   (12) Op♠ resp., Op♠ n,ℏ := S ×s,Mg,r Opsln ,ℏ,g,r n,ℏ,ρ := S ×s,Mg,r Opsln ,ℏ,ρ,g,r ,

where s : S → Mg,r denotes the classifying morphism of X ; ♡ — Op♡ n,ℏ (resp., Opn,ℏ,ρ ) := the moduli functor on Sch/S classifying equivalence classes of (GLn , ℏ)-opers (resp., (GLn , ℏ)-opers of radii ρ) on X ; ♢ — Op♢ n,ℏ,ϑ (resp., Opn,ℏ,ϑ,ρ ) := the moduli functor on Sch/S classifying isomorphism classes of (GLn , ℏ, ϑ)-opers (resp., (GLn , ℏ, ϑ)-opers of radii ρ) on X ; ♣ — Diff♣ n,ℏ,ϑ (resp., Diffn,ℏ,ϑ,ρ ) := the moduli functor on Sch/S classifying (n, ℏ, ϑ)-projective connections (resp., (n, ℏ, ϑ)-projective connections of radii ρ) on X .

In particular, all the functors appearing in this sequence may be represented by relative affine spaces over S. We would like to make two comments on the above theorem in characteristic p > 0. The first is that if (n >) p := char(k) > 0, then there always exists an (n, ℏ)-theta characteristic even when it has a nodal point or a marked point (cf. § 4.6.4). In particular, we can construct the moduli functors and the sequence of isomorphisms appearing in (11) for the universal pointed curves Cg,r . Since Diff♣ n,ℏ,ϑ has a natural torsor structure modeled on a vector bundle, it deduces the representability of Opsln ,g,r (hence also Opsln ,g,r,ρ for every ρ) asserted in Theorem A. The second comment concerns the equivalence of the dormancy condition on opers. ×r Denote by p Op♠ n,ℏ,□ , where □ is either the absence or presence of ρ ∈ csln (S), the subZzz...

♠ p scheme of Op♠ n,ℏ,□ classifying dormant (sln , ℏ)-opers, i.e., Opn,ℏ,□ := Opsln ,ℏ,□,X ♣full in the notation of (357) and (358). Also, denote by Diffn,1,ϑ,□ the subscheme of Diff♣ n,ℏ,ϑ,□ classifying (n, 1, ϑ)-projective connections having a full set of root functions in the sense of Definition 4.64. Then, the composite isomorphism ∼ ♠ Diff♣ n,1,ϑ,□ → Opn,1,□ resulting from the above theorem is restricted to an isomorphism of S-schemes

(13)

∼

♠ p Diff♣full n,1,ϑ,□ → Opn,1,□

(cf. Proposition 4.71). At the end of Chapter 4, we review the previous study on hypergeometric differential operators having a full set of root functions, or equivalently, dormant sl2 -oper on a 3-pointed projective line. This observation allows us to enumerate explicitly such objects in terms of radii. In Chapter 5, we discuss (g, ℏ)-opers for some classical types g other than sln , i.e., so2l+1 and sp2m . One important concept related to these opers is the duality of (GLn , ℏ, ϑ)-opers; it may be interpreted as taking the transposes of the corresponding differential operators. In the cases of g = so2l+1 with 2l +1 = n and g = sp2m with 2m = n, we describe (g, ℏ)-opers as certain self-dual (GLn , ℏ, ϑ)-opers equipped with

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a nondegenerate bilinear form. This description gives rise to a canonical closed im♠,Sp ♠,O ♠,Sp ♠ ♠ mersion Op♠,O n,ℏ,ρ ,→ Opn,ℏ,ρ (resp., Opn,ℏ,ρ ,→ Opn,ℏ,ρ ), where Opn,ℏ,ρ (resp., Opn,ℏ,ρ ) denotes the moduli scheme classifying (so2l+1 , ℏ)-opers (resp., (sp2m , ℏ)-opers) on X ×r of radii ρ ∈ c×r so2l+1 (S) (resp., ρ ∈ csp2m (S)). If p = char(k) > 0, it is restricted to a ♠,Sp ♠,O ♠ ♠ p p p p closed immersion p Op♠,O n,ℏ,ρ ,→ Opn,ℏ,ρ (resp., Opn,ℏ,ρ ,→ Opn,ℏ,ρ ), where Opn,ℏ,ρ ♠,O ♠,Sp (resp., p Op♠,Sp n,ℏ,ρ ) denotes the closed subscheme of Opn,ℏ,ρ (resp., Opn,ℏ,ρ ) classifying dormant ones. The existence of such a closed immersion deduces the generic unramZzz... Zzz... ifiedness of Opg,g,r (for g = so2l+1 , sp2m ) by applying that of Opsln ,g,r . Moreover, the duality yields a bijection between (dormant) (g, ℏ)-opers of radii ρ and (dormant) (g, ℏ)-opers of radii (−1) ⋆ ρ for g = sln , so2l+1 , sp2m . These observations are essential ingredients for constructing the pseudo-fusion ring associated to dormant (g, ℏ)-opers for such g’s, as discussed in Chapter 7. In the following, we summarize the main assertions proved in Chapter 5 (cf. Theorems 5.15 and 5.17 for the full statements).

Theorem E (Duality of (dormant) opers). — Let X , ℏ, and n be as in Theorem D. Then, the following assertions hold: (i) For each ρ ∈ c×r sln (S), there exists a canonical S-isomorphism D ♠ ρ

D

D

p

(15)

= id. Moreover, if p := char(k) > 0, then ♠ ρ

D

♠ ◦ satisfying (−1)⋆ρ to an isomorphism

∼

♠ : Op♠ n,ℏ,ρ → Opn,ℏ,(−1)⋆ρ ♠ ρ

D

♠ ρ

(14)

restricts

∼

♠ p : p Op♠ n,ℏ,ρ → Opn,ℏ,(−1)⋆ρ .

(ii) Suppose that there exists an integer l (resp., m) with 2l + 1 = n (resp., 2m = n) ×r and ρ lies in c×r so2l+1 (S) (resp., csp2m (S)). (In particular, we have ρ = (−1) ⋆ ρ.) ♠,O (resp., Op♠,Sp Then, ρ♠ restricts to the identity morphism of Opn,ℏ,ρ n,ℏ,ρ ). If, p ♠ moreover, p := char(k) > 0, then ρ restricts to the identity morphism ♠,Sp p of p Op♠,O (resp., Op ). n,ℏ,ρ n,ℏ,ρ

D

D

Chapters 6-7: Deformation and degeneration of (dormant) opers. — In Chapter 6, we study local deformations of (dormant) (g, ℏ)-opers, where g is as before. First, we review classical deformation theory of pointed stable curves and flat vector bundles in the context of logarithmic algebraic geometry. After that, we describe the deformation space of a (g, ℏ)-oper (of prescribed radii) using the de Rham cohomology of a certain complex associated to that oper (cf. Propositions 6.7, 6.9). A key observation concerning flat bundles in characteristic p > 0 is that the Cartier operator may be interpreted as describing deformation of p-curvature (cf. Proposition 6.11). By taking this into account, we achieve a detailed understanding of the tangent space of the moduli stack classifying dormant (g, ℏ)-opers (cf. Proposition 6.19, Corollary 6.20). One of the results in Chapter 6 is a criterion for the étaleness of that stack (cf. Corollary 6.21), which plays a crucial role in the proof of Joshi’s conjecture.

ASTÉRISQUE 432

13

0.5. STRUCTURE AND MAIN THEOREMS OF THE MANUSCRIPT

In Chapter 7, we examine the behavior, according to decomposition arising from various clutching morphisms, of the generic degree Zzz...

deg(Opg,ℏ,ℏ⋆ρ,g,r /Mg,r )

(16) Zzz...

of Opg,ℏ,ℏ⋆ρ,g,r /Mg,r for each ρ := (ρi )ri=1 ∈ c×r (Fp ) and ℏ ∈ k × . This value depends neither on the choice of ℏ ∈ k × nor on the ordering of ρ1 , . . . , ρr ; this is why there exists a well-defined function Pr Zzz... c(Fp ) p Ng,g : N≥3−2g → Z; ρ (:= i=1 ρi ) 7→ deg(Opg,ℏ,ℏ⋆ρ,g,r /Mg,r ) (17) c(F )

p (cf. (999)) on a certain subset N≥3−2g of the free commutative monoid with basis Under some assumptions, p Ng,g (ρ) coincides precisely with the number of dormant (g, ℏ)-opers on a sufficiently general r-pointed stable curve of genus g. Also, as proved in (cf. Proposition 7.33), the function p Ng,0 satisfies a certain factorization property, which will be referred to as a “pseudo-fusion rule” on c(Fp ) (cf. Definition 7.20). The notion of a pseudo-fusion rule may be regarded as a somewhat weaker variant of a fusion rule. (For reviews and expositions concerning fusion rules, we refer the reader to [8] or [29].) The latter half of Chapter 7 is mainly devoted to developing a general theory of pseudo-fusion rules. By applying the results proved there, we obtain an effective way to calculate the values p Ng,g (ρ) (for various g’s and ρ’s) combinatorially from the case of (g, r) = (0, 3). Moreover, the function p Ng,0 induces a commutative ring

p

Y

(18)

g,

which will be referred to as the pseudo-fusion ring for dormant g-opers (cf. Definition 7.34). At the end of Chapter 7, we explicitly describe the structure of p sl2 . In general, the explicit description of (the characters of) p g allows us to perform a computation for the various values p Ng,g (ρ), as asserted in the following assertion (cf. Theorem 7.36, (ii)): Y

Y

Theorem F (Factorization property). — Suppose that p := char(k) > 0 and g is either sln with 2n < p, so2l+1 with 4l+2 < p, or sp2m with 4m < p. Let ρ = (ρi )ri=1 ∈ c×r (Fp ) (where ρ = ∅ if r = 0) and ℏ P ∈ k × . Write p Sg for the set of ring homomorphisms p p ▼ p g → C and write Casg := g ). Then, we have the following λ∈c(Fp ) λ ∗ λ (∈ equality: ! r r X X Y Zzz... p (19) deg(Opg,ℏ,ℏ⋆ρ,g,r /Mg,r ) = Ng,g ( ρi ) = χ(p Casg )g−1 · χ(ρi ). Y

Y

i=1

χ∈p Sg

i=1

If, moreover, r = 0 (which implies g > 1), this equality means that   X Zzz... (20) deg(Opg,ℏ,g,0 /Mg,0 ) = p Ng,g (0) = p Ng,g (∅) = χ(p Casg )g−1 . χ∈p Sg

Chapters 8-9: A proof of Joshi’s conjecture. — Chapter 8 is devoted to proving the Zzz... generic étaleness of Opg,ℏ,ℏ⋆ρ,g,r over Mg,r for certain classical g’s. For g = sl2 , this

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fact was already proved by S. Mochizuki (cf. [60], Chap. II, § 2.8, Theorem 2.8). Thanks to the results in the previous chapters, it suffices to prove that there are no nontrivial deformations of any dormant sln -oper on the 3-pointed projective line P. We give a proof of this claim (cf. Corollary 8.17) based on the idea that infinitesimal deformations of a dormant oper on P are completely determined by their radii and the sheaves of their horizontal sections (as shown in [60]). Consequently, we obtain the following theorem (cf. Theorem 8.19): Theorem G (Generic étaleness). — Let k be a perfect field of characteristic p > 0, ℏ an element of k × , and ρ an element of c×r g (Fp ) (where ρ = ∅ if r = 0). Suppose that g is either sln with 2n < p, so2l+1 with 4l + 2 < p, or sp2m with 4m < p. Then, Zzz... Opg,ℏ,ℏ⋆ρ,g,r is étale over the points of Mg,r classifying totally degenerate curves. In Zzz...

particular, Opg,ℏ,ℏ⋆ρ,g,r is generically étale over Mg,r ; i.e., any irreducible component that dominates Mg,r admits a dense open subscheme which is étale over Mg,r . In Chapter 9, we give a proof of Joshi’s conjecture by investigating the relation Zzz... between (the fiber over a point in Mg,0 of) the moduli stack Opsln ,1,g,0 and certain n,O Quot schemes in positive characteristic, denoted by Quotn,0 Θ and QuotΘ . These Quot schemes are associated to a smooth curve X/S equipped with a theta characteristic Θ, and both classify rank n subbundles of a rank p vector bundle on X induced from Θ. The discussion in Chapter 9 is parallel to the one in [92], §§ 4-5, where the case of g = sl2 was studied. Just as in loc. cit., we construct an isomorphism between the O moduli space of dormant opers and Quotn, Θ via Cartier descent of flat connections with vanishing p-curvature (cf. Proposition 9.4). In particular, (under the assumption that X/S is sufficiently general in Mg,0 ) the problem can be reduced to a computation O n,0 of the degree of Quotn, Θ , and moreover, to that of the other Quot scheme QuotΘ . By Zzz... the generic étaleness of Opsln ,1,g,0 resulting from Theorem G, this value coincides with the degree of a certain Quot scheme over C obtained by deforming the underlying curve and the rank p vector bundle on it. On the other hand, there is a formula given by Y. Holla for computing the degree of this Quot scheme, which may be regarded as a special case of the Vafa-Intriligator formula for explicitly computing the GromovWitten invariants. By applying this formula, we thus prove the following theorem (cf. Theorem 9.10). That is to say, Joshi’s conjecture holds for sufficiently general curves.

Theorem H (Joshi’s conjecture). — Suppose that p > n · max{g − 1, 2} and ℏ ∈ k × . Zzz... Zzz... Then, the generic degree deg(Opsln ,ℏ,g,0 /Mg,0 ) of Opsln ,ℏ,g,0 over Mg,0 can be calculated by the following formula: Qn X Zzz... ( i=1 ζi )(n−1)(g−1) p(n−1)(g−1)−1 Q (21) deg(Opsln ,ℏ,g,0 /Mg,0 ) = · . g−1 n! i̸=j (ζi − ζj ) ×n (ζ1 ,...,ζn )∈C p ζ =1, ζi ̸=ζj (i̸=j) i

ASTÉRISQUE 432

0.7. NOTATION AND CONVENTIONS

15

0.6. Acknowledgements We cannot express enough our sincere and deep gratitude to Professors Shinichi Mochizuki and Kirti Joshi. Without their philosophical viewpoints, theoretical insights, and endless creativity, my study of mathematics would have remained “dormant”. We deeply appreciate Professor Yuichiro Hoshi for his helpful suggestions, as well as for reading preliminary versions of the present manuscript. However, responsibility for any errors or misconceptions in the present manuscript ultimately remains with the author. Also, we would like to express our sincere appreciation to the referee for a careful reading of the manuscript and for various useful comments and suggestions. Finally, we would like to thank Professors Go Yamashita and Akio Tamagawa (for their constructive comments concerning the discussion in Chapter 7), and the various individuals (including pointed stable curves and their moduli space!) with whom the author became acquainted in Kyoto. I wrote the present manuscript as a gratitude letter to them. This work was partially supported by Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385, 21K13770). 0.7. Notation and Conventions Categories and Schemes. — Denote by Set the category of (small) sets. Throughout this manuscript, all schemes (and Deligne-Mumford stacks) are assumed to be locally noetherian. For a scheme S, we denote by Sch/S the category of schemes over S and by Et/S the small étale site on S. If S = Spec(k) for a field k, then we write Sch/k := Sch/Spec(k) . Fields. — Let k be a field. Then, char(k) denotes the characteristic of k. For a k-vector space h, we denote by Sk (h) the symmetric algebra on h over k. Occasionally, we will regard h as the affine k-scheme Spec(Sk (h∨ )), where h∨ is the dual space of h. Sheaves. — Let S be a scheme. Given a sheaf V on S, we use the notation “ v ∈ V ” for a local section v of V . If V is an OS -module, then we denote by V ∨ the dual of V , i.e., V ∨ := Hom OS (V , OS ). Next, by a vector bundle on S, we mean a locally free OS -module of finite rank. For a vector bundle V on S and an integer i ≥ 0, we Vi denote by V the i-th exterior power of V . Also, SOY (V ) denotes the symmetric L algebra of V over OS ; it decomposes into the direct sum SOS (V ) = j≥0 SjOS (V ), where SjOS (V ) is the subsheaf of SOS (V ) consisting of sections of homogenous degree j. If X is a scheme (or more generally, a Deligne-Mumford stack) over S, then we shall write TX/S (resp., ΩX/S ) for the tangent bundle (resp., the sheaf of 1-forms) on X relative to S. Relative affine schemes. — Let s : S ′ → S be an affine morphism of schemes. Then, we shall write OS ⟨S ′ ⟩ for the OS -algebra corresponding to S ′ , i.e., OS ⟨S ′ ⟩ := s∗ (OS ′ ). In particular, we have S pec(OS ⟨S ′ ⟩) ∼ = S ′ . Let S be a scheme, or more generally a

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CHAPTER 0. INTRODUCTION

Deligne-Mumford stack, and V a vector bundle on S. Denote by V(V ) the relative affine scheme over S associated to V , i.e., the spectrum S pec(SOT (V ∨ )) of SOT (V ∨ ). If S is defined over a field k and v : Spec(R) → S is
a k-morphism, where R denotes a ring over k, then we denote by Tv S := v ∗ (TS/k ) the R-module defined to be the tangent space of S at v. Complexes. — Let ∇ : K 0 → K 1 be a morphism of sheaves of abelian groups on a scheme S. It may be regarded as a complex concentrated at degrees 0 and 1; we denote this complex by K • [∇]. Next, let f : X → S be a morphism of schemes and i an integer ≥ 0. Then, one may define the sheaf Ri f∗ (K • [∇]) on S obtained from K • [∇] by applying the i-th hyper-derived functor Ri f∗ (−) of f∗ (−) (cf. [47], (2.0)). In particular, R0 f∗ (K • [∇]) = f∗ (Ker(∇)). If S is affine, then Γ(S, Ri f∗ (K • [∇])) may be identified with the i-th hypercohomology group Hi (X, K • [∇]). Given an integer n and a sheaf F on S, we define the complex F [n] to be F (considered as a complex concentrated at degree 0) shifted down by n, so that F [n]−n = F and F [n]i = 0 (i ̸= −n). Algebraic groups. — Suppose that we are working over a field k. Then, denote
by Gm := Spec(k[z ±1 ]) the multiplicative group over k. For a positive integer n, we shall write GLn (resp., PGLn ) for the general (resp., projective) linear group of rank n over k. The Lie algebra gln of GLn consists of n × n matrices over k. Denote by sln the Lie subalgebra of gln consisting of n × n matrices with vanishing trace. If n is invertible in k, then the Lie algebra pgln of PGLn is isomorphic to sln via the natural composite sln ,→ gln ↠ pgln ; we will occasionally identify sln with pgln via this composite isomorphism. Next, let G be an affine algebraic k-group and h a k-vector space equipped with a G-action. Then, the dual h∨ of h is equipped with the G-action given by (h · f )(v) := f (h−1 · v) for h ∈ G, f ∈ h∨ , and v ∈ h. In particular, this G-action induces a G-action on the coordinate ring Sk (h∨ ) of h.

ASTÉRISQUE 432

CHAPTER 1 LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

In the first half of this chapter, we discuss a general theory of principal bundles and ℏ-connections, where ℏ is a parameter, in the context of logarithmic algebraic geometry. We shall refer the reader to, e.g., [80], [81], [82], [5], [4], and [91] for descriptions of ℏ-connections. One of the important results in this chapter is the description of gauge transformations using the exponential and logarithm maps. This fact will be used to prove, e.g., that a (g, ℏ)-oper has no nontrivial automorphisms. In the second half of this chapter, we briefly recall pointed stable curves and the moduli stack classifying them. By considering the natural log structure on a pointed stable curve, we can associate a certain invariant, called the “monodromy (operator),” to each ℏ-flat principal bundle on the curve and a choice of a marked point. As we will discuss in Chapter 7, the notion of monodromy determines the sort of boundary condition to glue together ℏ-flat principal bundles on pointed curves along the fibers over the points of attachment. Notation and Conventions. — Let us fix a field k and a connected smooth affine algebraic k-group G. Denote by e the identity element of G and by g the Lie algebra of G, i.e., the tangent space Te G of G at e. Occasionally, we regard g as a k-scheme.

1.1. The adjoint representation and the Maurer-Cartan form 1.1.1. Derivations. — Let T be a k-scheme and f : Y → T a T -scheme. Each OY -module V defines End f −1 (OT ) (V ) (resp., End OY (V )), the sheaf of f −1 (OT )-linear (resp., OY -linear) endomorphisms of V . Denote by (22)

∆ : OY → End f −1 (OT ) (V )

the diagonal embedding, i.e., the f −1 (OT )-linear morphism which, to any local section a ∈ OY , assigns the locally defined endomorphism of V given by multiplication by a. If V is a vector bundle, then the local freeness of V implies that ∆ is injective. The sheaf End f −1 (OT ) (V ) may be regarded as an OY -module in such a way that a · D := ∆(a) ◦ D for any a ∈ OY and D ∈ End f −1 (OT ) (V ). Also, End f −1 (OT ) (V ) has a structure of OT -Lie algebra on Y (in the sense of [52], § 2.1), where the Lie bracket

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

operation is given by [D1 , D2 ] := D1 ◦ D2 − D2 ◦ D1 . The sheaf End OY (V ) is the OT -Lie subalgebra of End f −1 (OT ) (V ) consisting of local sections that commute with Im(∆). Recall that a derivation on OY relative to T is an f −1 (OT )-linear endomorphism D of OY (i.e., a global section of End f −1 (OT ) (OY )) that satisfies D(a · b) = a · D(b) + D(a) · b

(23)

for any local sections a, b ∈ OY . In particular, we have D(a) = 0 for any a ∈ f −1 (OT ).  of Y /T may be identified, in the usual manner, The tangent bundle TY /T := Ω∨ Y /T with the sheaf of locally defined derivations on OY relative to T ; this sheaf specifies an OT -Lie subalgebra of End f −1 (OT ) (OY ), meaning that TY /T is closed under both the OY -action and the Lie bracket operation on End f −1 (OT ) (OY ). 1.1.2. Right/Left-translations. — Let T be a scheme over k and h : T → G a morphism of k-schemes. Denote the graph of h by Γh (:= (idT , h)) : T → T ×k G. We write Rh (resp., Lh )

(24)

for the right-translation (resp., the left-translation) by h, i.e., the composite isomorphism (25)

(idT ×sw)◦(Γh ×idG )

id ×mult

Rh : T ×k G −−−−−−−−−−−−−→ T ×k G ×k G −−T−−−−→ T ×k G   Γ ×idG id ×mult resp., Lh : T ×k G −−h−−−→ T ×k G ×k G −−T−−−−→ T ×k G ,

where “mult” denotes the multiplication G ×k G → G in G and “sw” denotes the automorphism of G ×k G given by switching the two factors (h1 , h2 ) 7→ (h2 , h1 ). Note that (26)

Rh1 ◦ Rh2 = Rh2 h1 , Lh1 ◦ Lh2 = Lh1 h2 , and Rh1 ◦ Lh2 = Lh2 ◦ Rh1

for any h1 , h2 ∈ G(T ). By considering the left G-action on G defined by L(−) , we often identify g with the space of left-invariant vector fields on G. That is to say, given any element v ∈ g = Te G, we can use left-translation to produce a global vector field vG with vG | = dLh | (v) for every h ∈ G, where dLh | denotes the differential Te G → Th G h e e of Lh at e. 1.1.3. The adjoint action. — Given a morphism of k-schemes h : T → G, we shall write (27)

Innh := Lh ◦ Rh−1 (= Rh−1 ◦ Lh ) : T ×k G → T ×k G,

which specifies an automorphism of the T -scheme T ×k G. Denote by GL(g) the automorphism group of the k-vector space g and by eT : T → G the identity in G(T ), e i.e., the composite T → Spec(k) − → G. Then, the adjoint representation of G is, by definition, the morphism (28)

ASTÉRISQUE 432

AdG : G → GL(g)

19

1.1. THE ADJOINT REPRESENTATION AND THE MAURER-CARTAN FORM

of algebraic k-groups which, to any k-morphism h : T → G, assigns the automorphism dInnh | of T ×k g defined as the differential of Innh at eT . We occasionally identify eT
AdG (h) with the corresponding OT -linear automorphism of OT ⊗k g = Γ∗eT (TT ×k G/T ) . Write
(29) ad := dAdG | : g → End(g) = Tidg GL(g) , e

i.e., ad(v)(v ′ ) := [v, v ′ ] for v, v ′ ∈ g. The Lie algebra g has a left G-action determined by AdG . Also, the composition of AdG and AdGL(g) : GL(g) → GL(End(g)) determines the left G-action on End(g), which is given by (h, a) 7→ AdG (h) ◦ a ◦ AdG (h−1 ) for every h ∈ G, a ∈ End(g). This G-action is compatible with the adjoint G-action on g via ad : g → End(g). 1.1.4. The Maurer-Cartan form. — Here, let us recall the Maurer-Cartan form on the algebraic group G (cf. [79], Chap. 3, § 1, Definition 1.3, or [30], § 8.6). Definition 1.1. — The (left-invariant) Maurer-Cartan form on G is the global section ΘG ∈ Γ(G, ΩG/k ⊗k g)

(30)

corresponding to the differential
(31) dLinv | : TG/k = Γ∗idG (TG×k G/G ) → Γ∗eG (TG×k G/G ) (= OG ⊗k g) Γid G

∼

∼

of the left-multiplication Linv : G ×k G → G ×k G by the inversion inv : G → G in G, where TG×k G/G denotes the tangent bundle of G ×k G relative to the first projection G×k G → G. (In particular, ΘG is a g-valued 1-form on G in the terminology below.) Occasionally, it will be convenient to identify ΘG with the corresponding OG -linear morphism TG/k → OG ⊗k g. Remark 1.2. — The Maurer-Cartan form ΘG can be characterized by the condition that, for any k-morphism h : T → G, the pull-back
(32) h∗ (ΘG ) ∈ Γ(T, h∗ (ΩG/k ) ⊗k g) ∼ = HomO (h∗ (TG/k ), OT ⊗ g) T

corresponds to the differential (33)

dLh−1 |

Γh


: h∗ (TG/k ) = Γ∗h (TT ×k G/T ) → Γ∗eT (TT ×k G/T ) (= OT ⊗k g)

of Lh−1 at Γh ∈ (T ×k G)(T ). This implies that if ∂ ∈ Γ(G, TG/k ) is a left-invariant vector field, then the equality ΘG (∂) = 1 ⊗ ∂ | holds. e

Let T be a smooth k-scheme (hence, TT /k is a vector bundle). For an integer q ≥ 0, Vq a g-valued q-form on T is a global section of the vector bundle ( ΩT /k ) ⊗k g. Let q1 , q2 be nonnegative integersVsuch that both q1 ! and Vq q2 ! are invertible in k. q Also, let us take local sections δ1 ∈ ( 1 ΩT /k ) ⊗k g, δ2 ∈ ( 2 ΩT /k ) ⊗k g defined on a common open subset of T . Then, we define (34)

q1^ +q2

[δ1 , δ2 ] ∈ (

ΩT /k ) ⊗k g

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to be the g-valued (q1 + q2 )-form given by (35)

[δ1 , δ2 ](∂1 , . . . , ∂q1 +q2 ) X 1 (−1)sgn(σ) [δ1 (∂σ(1) , . . . , ∂σ(q1 ) ), δ2 (∂σ(q1 +1) , . . . , ∂σ(q1 +q2 ) )], · := q1 !q2 ! σ∈Sq1 +q2

where Sq1 +q2 denotes the symmetric group of q1 +q2 letters. For example, for g-valued 1-forms δ1 , δ2 , we have (36)

[δ1 , δ2 ](∂1 , ∂2 ) = [δ1 (∂1 ), δ2 (∂2 )] − [δ1 (∂2 ), δ2 (∂1 )].

In particular, any g-valued 1-form δ satisfies (37)

[δ, δ](∂1 , ∂2 ) = 2 · [δ(∂1 ), δ(∂2 )].

Finally, the exterior differential is the morphism q q+1 ^ ^ d : ( ΩT /k ) ⊗k g → ( ΩT /k ) ⊗k g Vq (q ≥ 0) determined by d(δ ⊗ v) = dδ ⊗ v for any δ ∈ ΩT /k , v ∈ g. One may verify immediately that

(38)

(39)

d[δ1 , δ2 ] = [dδ1 , δ2 ] + (−1)q1 · [δ1 , dδ2 ]

Vq for every δi ∈ ( i ΩT /k ) ⊗k g (i = 1, 2). Proposition 1.3 (The Maurer-Cartan equation). — Suppose that 2 is invertible in k. Then, the Maurer-Cartan form ΘG satisfies an equality 1 (40) dΘG + · [ΘG , ΘG ] = 0 2 V2 of elements in Γ(G, ( ΩG/k ) ⊗k g). Proof. — Let us take two sections ∂1 , ∂2 ∈ Γ(G, TG/k ). Since ΘG is a vector-valued 1-form on the smooth k-scheme G, the equality (41)

dΘG (∂1 , ∂2 ) = ∂1 (ΘG (∂2 )) − ∂2 (ΘG (∂1 )) − ΘG ([∂1 , ∂2 ])

holds because of an argument similar to the proof of [79], Chap. 1, § 5, Lemma 5.15. If both ∂1 and ∂2 are left-invariant (i.e., invariant under the natural left G-action on Γ(G, TG/k )), then [∂1 , ∂2 ] is left-invariant. Moreover, in accordance with the final comment in Remark 1.2, we obtain the equalities (42)

∂1 (ΘG (∂2 )) = ∂2 (ΘG (∂1 )) = 0

and obtain the following sequence of equalities: (43)

ΘG ([∂1 , ∂2 ]) = 1 ⊗ [∂1 , ∂2 ]|

e

= 1 ⊗ [∂1 | , ∂2 | ] e

e

= [1 ⊗ ∂1 | , 1 ⊗ ∂2 | ] e

e

= [ΘG (∂1 ), ΘG (∂2 )]

ASTÉRISQUE 432

1.1. THE ADJOINT REPRESENTATION AND THE MAURER-CARTAN FORM

(37)

=

21

1 · [ΘG , ΘG ](∂1 , ∂2 ). 2

By (41), (42), and (43), the equality   1 (44) dΘG + · [ΘG , ΘG ] (∂1 , ∂2 ) = 0 2 holds for any left-invariant vector fields ∂1 , ∂2 on G. Since the left-invariant vector fields on G generates TG/k , (44) implies the desired equality (40). 1.1.5. Change of structure group. — Let G′ be another connected smooth affine algebraic k-group with Lie algebra g′ and w : G → G′ a k-morphism of algebraic groups. The differential dw : TG/k → w∗ (TG′ /k ) of w restricts to a k-linear morphism dw| : g → g′ . e Proposition 1.4. — The Maurer-Cartan form ΘG′ on G′ makes the following square diagram commute:

(45)

/ OG ⊗k g

ΘG

TG/k

id⊗dw|

dw

e



w∗ (TG′ /k )

 / OG ⊗k g′ .

w∗ (ΘG′ )

Proof. — The assertion follows from the definitions of ΘG and ΘG′ . Example 1.5. — Let us consider the case where G = GLn for a positive integer n. By identifying GLn with the open subscheme {R ∈ gln | det(R) ∈ k × } of gln (regarded as a k-scheme), we have an open immersion A : GLn ,→ gln . The differential of A defines an OGLn -linear morphism (46)

dA : TGLn → (A∗ (Tgln ) = A∗ (Ogln ⊗k gln )) = OGLn ⊗k g,

where we also use the notation dA to denote the corresponding gln -valued 1-form on GLn . Then, for any k-morphism h : T → GLn and a local section ∂ ∈ TT /k , the Maurer-Cartan form ΘGLn on GLn satisfies (47)

h∗ (ΘGLn )(∂) = dLh−1 |

Γh

(∂)

= dLh−1 (dh(∂)) = (A ◦ h)−1 (d(A ◦ h)(∂)) = h∗ (A−1 dA)(∂). This implies an equality (48)

ΘGLn = A−1 dA

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of gln -valued 1-forms on GLn . Moreover, if G is an algebraic subgroup of GLn , then it follows from Proposition 1.4 that (49)

ΘG = (A−1 dA)| . G

1.2. Principal G-bundles and their associated Lie algebroids Next, we discuss principal bundles on log schemes. We refer the reader to [46], [40], and [45] for the basic properties on log schemes. 1.2.1. Principal G-bundles. — To begin with, recall the definition of a principal G-bundle. Let Y be a k-scheme. Definition 1.6. — (i) A (principal) G-bundle (in the étale topology) on Y is a scheme E over Y equipped with a right G-action E ×k G → E which is étale locally trivial in the following sense: there exist an étale surjective k-morphism ∼ Y ′ → Y and a G-equivariant isomorphism Y ′ ×Y E → Y ′ ×k G over Y ′ , where ′ ′ both Y ×Y E and Y ×k G are equipped with the right G-action induced from the G-action on the second factor of the product. (ii) Let E , E ′ be G-bundles on Y . Then, an isomorphism of G-bundles from E to E ′ ∼ is defined as a G-equivariant isomorphism E → E ′ over Y . Example 1.7 (Trivial bundle). — The trivial G-bundle on Y is the product Y ×k G, where G acts only on the second factor, considered as a scheme over Y by the first projection Y ×k G → Y . Any G-bundle is, by definition, étale locally isomorphic to the trivial G-bundle. The left-translation Lh by each element h ∈ G(Y ) specifies an automorphism of the trivial G-bundle Y ×k G, i.e., preserves the right G-action. Conversely, each automorphism η of the trivial G-bundle Y ×k G can be expressed as a left-multiplication. (idY ,eY )

η

pr

2 Indeed, let us take h to be the composite Y −−−−−→ Y ×k G − → Y ×k G −−→ G. The two automorphisms η, Lh of Y ×k G coincide when they are restricted to the closed subscheme Im(ΓeY ) (⊆ Y ×k G). Hence, for any y ∈ Y and h ∈ G, we have

(50)

η(y, h) = η(y, e) · h = Lh (y, e) · h = Lh (y, h).

This implies η = Lh . One may verify that such an element h is uniquely determined. 1.2.2. Pull-back and base-change. — Let E be a G-bundle on Y and u : Y ′ → Y a scheme over Y . Then, the fiber product (51)

u∗ (E ) := Y ′ ×Y E

forms a G-bundle on Y ′ by considering the right G-action induced from the G-action on the second factor E of the product. We shall refer to u∗ (E ) as the pull-back of E to Y ′ .

ASTÉRISQUE 432

1.2. PRINCIPAL G-BUNDLES AND THEIR ASSOCIATED LIE ALGEBROIDS

23

Suppose that Y ′ comes from a base-change via a k-morphism t : T ′ → T , i.e., Y ′ = T ′ ×T Y . Then, we denote the pull-back of E to Y ′ by (52)

t∗ (E ) (:= Y ′ ×Y E = T ′ ×T E )

and refer to it as the base-change of E to T ′ . 1.2.3. Change of structure group. — Let G′ be another connected smooth affine algebraic k-group and w : G → G′ a morphism of algebraic k-groups. Then, each G-bundle E on Y induces a G′ -bundle via change of structure group by w; we denote this G′ -bundle by
G ′ G,w E × G or E × (53) G′ . To be precise, let us set E ×G G′ to be the quotient of E ×k G′ by the right G-action given by (v, h′ ) · h := (v · h, w(h)−1 · h′ ), where v ∈ E , h ∈ G, and h′ ∈ G′ . The natural right G′ -action on E ×k G′ induces a well-defined right G′ -action on E ×G G′ , by which E ×G G′ forms a G′ -bundle on Y . Definition 1.8. — Let B be a subgroup of G and E a G-bundle on Y . By a B-reduction ∼ of E , we mean a B-bundle EB on Y together with an isomorphism EB ×B G → E of G-bundles. Let Y be a k-scheme and E a G-bundle on Y . Given a k-vector space h equipped with a left G-action, we write (54)

hE

for the vector bundle on Y associated to the affine space E ×G h (:= (E ×k h)/G) over Y (hence V(hE ) ∼ = E ×G h). This vector bundle in the case of h = g, i.e., gE , is called the adjoint (vector) bundle associated to E . It is the vector bundle corresponding to
∼ the GL(g) = GLdim(g) -bundle E ×G,AdG GL(g) and equipped with the OY -Lie algebra structure induced from the Lie algebra structure on g. 1.2.4. Logarithmic derivations. — Given a log scheme (or more generally, a log stack) indicated, say, by Y log , we shall write Y for the underlying scheme (stack) of Y log , and αY : MY → OY for the morphism of sheaves of monoids defining the log structure of Y log . Let us recall the notion of a logarithmic derivation. Let T log := (T, αT : MT → OT ), log Y := (Y, αY : MY → OY ) be fine log schemes over k (cf. [46], (2.3)) and f log : Y log → T log a k-morphism of log schemes. By definition, a logarithmic derivation ∂ on (OY , MY ) relative to (OT , MT ) (cf. [65], § 1, Definition 1.1.2) is given as a pair (55)

∂ = (D : OY → OY , δ : MY → OY ),

where D denotes a (non-logarithmic) derivation on OY over T and δ denotes a monoid homomorphism from MY to the underlying additive monoid of OY satisfying the following two conditions:

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— D(αY (m)) = αY (m) · δ(m) for any local section m ∈ MY ; — δ(f ∗ (m)) = 0 for any local section m ∈ MT . Let us write (56)

TY log /T log

for the sheaf of locally defined logarithmic derivations of (OY , MY ) relative to (OT , MT ). This sheaf forms an OT -Lie algebras on Y with Lie bracket operation defined by (57)

[(D1 , δ1 ), (D2 , δ2 )] := ([D1 , D2 ], D1 ◦ δ2 − D2 ◦ δ1 )

(cf. [65], § 1, Proposition 1.1.7). Also, write (58)

ΩY log /T log

for the sheaf of logarithmic 1-forms on Y log relative to T log , as defined in [46], (1.7). In the usual manner, the sheaf TY log /T log may be identified with the dual of ΩY log /T log . (The formations of such sheaves may be extended naturally to the case where both T log and Y log are log stacks.) If Y log /T log is log smooth (i.e., “smooth” in the sense of [46], (3.3)), then both TY log /T log and ΩY log /T log are vector bundles (cf [46], Proposition (3.10)). The universal logarithmic derivation on Y log (relative to T log ) is denoted by (59)

d : OY → ΩY log /T log .

Example 1.9 (Trivial log structure). — Suppose that f log : Y log → T log is strict in the sense of [40], § 1.2. Then, the assignment given by ∂ 7→ D for every ∂ as in (55) defines ∼ an isomorphism TY log /T log → TY /T . Indeed, if D is a local section of TY /T , then the morphism OY× → OY determined by m 7→ D(m) extends to a well-defined monoid m homomorphism δD : MY → OY . The pair (D, δD ) specifies a section of TY log /T log and the assignment D 7→ (D, δD ) gives the inverse to the assignment ∂ 7→ D.

Example 1.10 (Normal crossing divisor). — Let T be a k-scheme, Y a smooth T -scheme, and D the relative reduced divisor on Y (relative to T ) with normal crossings. If j : Y \ D ,→ Y denotes the open immersion, then the inclusion MY = OY ∩ j∗ (OY×\D ) ,→ OY defines a fine log structure on Y . Choose a system of local coordinates (x1 , . . . , xn ) on Y (relative to T ) such that D is defined by the Qr equation i=1 xai i = 0 (where 0 ≤ r ≤ n and a1 , . . . , ar ≥ 1). Then, the sheaf of Qr Z monoids MY can be expressed locally as MY = OY× · i=1 xi ≥0 . ∂ For each i = 1, . . . , n, let ∂i denote the non-logarithmic derivation ∂x on OY . i In addition, for each i = 1, . . . , r, denote by ∂ilog the logarithmic derivation Qr Z (xi · ∂i , δi ), where δi is the monoid homomorphism OY× · i=1 xi ≥0 → OY given

ASTÉRISQUE 432

1.2. PRINCIPAL G-BUNDLES AND THEIR ASSOCIATED LIE ALGEBROIDS

by u ·

Qr

i=1

xlii 7→ xi ·

25

∂i u u

+ li . Then, TY log /T decomposes locally into the direct sum ! ! r n M M log TY log /T = OY · ∂i ⊕ OY · ∂i .

(60)

i=1

i=r+1

1.2.5. Lie algebroids. — Now, let us discuss the logarithmic version of a Lie algebroid (cf. [52], § 2.1, Definition 2.1 for the non-logarithmic version) and the Lie algebroid associated to a G-bundle. Definition 1.11. — A Lie algebroid on Y log /T log is a pair (V , α)

(61)

consisting of an OT -Lie algebras V on Y and an OY -linear morphism α : V → TY log /T log , which preserves the Lie bracket operation and satisfies the equality [∂1 , a · ∂2 ] = α(∂1 )(a) · ∂2 + a · [∂1 , ∂2 ]

(62)

for any local sections a ∈ OY and ∂1 , ∂2 ∈ V . The morphism α is called the anchor (map) of V . When there is no fear of confusion, we will only use the symbol V to indicate the Lie algebroid (V , α). In what follows, let us construct a Lie algebroid associated to a G-bundle on Y in the context of logarithmic algebraic geometry. Let E be a G-bundle on Y , and denote by π : E → Y its structure morphism. By pulling-back the log structure of Y log via π, one may obtain a log structure on E ; we denote the resulting log scheme by E log . The right G-action on E induces a right G-action on the direct image π∗ (TE log /T log ) (resp., π∗ (TE log /Y log )) of the sheaf of logarithmic derivations TE log /T log (resp., TE log /Y log ). We shall set f log log := π (T log log )G , f log log := π (T log log )G , T (63) T E

/T

∗

E

/T

E

/Y

∗

E

/Y

f log log and T f log log are the subsheaves of G-invariant sections of π (T log log ) i.e., T ∗ E /T E /Y E /T f and π (T log log ) respectively. The sheaf T log log may be identified with the adjoint ∗

E

/Y

E

/Y

vector bundle gE associated to E (cf. [6], § 2). Indeed, the identification of g with the space of left-invariant vector fields on G gives V(TE log /Y log ) ∼ = E ×k g, where the G-action on the right-hand side is given by (q, v) · h = (q · h, AdG (h−1 )(v)) for q ∈ E , v ∈ g, and h ∈ G. The quotient of E ×k g by this G-action is naturally isomorphic to the relative affine space V(gE ) associated to gE , so we have (64)

∼

f log log ) ∼ V(T = V(TE log /Y log )/G ∼ = (E ×k g)/G → V(gE ). E /Y

f log log = ∼ gE . This implies T E /Y Differentiating π yields a sequence of OE -modules (65)

0 −→ TE log /Y log −→ TE log /T log −→ π ∗ (TY log /T log ) −→ 0

(i.e., the dual of the sequence displayed in [46], Proposition (3.12)), which is exact because of the local triviality of the G-bundle E . By applying the functor π∗ (−)G to

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

this sequence, we obtain a short exact sequence of OY -modules dlog

E f log log −→ TY log /T log −→ 0 0 −→ gE −→ T E /T

(66)

f log log via the (cf. [6], § 2, Theorem 1). We shall regard gE as an OY -submodule of T E /T f log log admits a Lie bracket operation f log log . The sheaf T second arrow g ,→ T E

E

/T

E

/T

[−, −] arising from that of TE log /T log , and dlog preserves the Lie bracket operation. E Moreover, by means of the local description of [−, −] (cf. Example 1.12 below) the equality [∂1 , a · ∂2 ] = dlog (∂1 )(a) · ∂2 + a · [∂1 , ∂2 ] E

(67)

f log log . That is to say, can be verified for any local sections a ∈ OY and ∂1 , ∂2 ∈ T E /T the pair f log log , dlog ) (68) (T E /T E specifies a Lie algebroid on Y log /T log . We call it the Lie algebroid (on Y log /T log ) associated to E . Example 1.12 (Trivial bundle). — Let us consider the Lie algebroid associated to the trivial G-bundle. Denote by pr1 (resp., pr2 ) the first projection Y ×k G → Y (resp., the second projection Y ×k G → G). In particular, pr1 is nothing but the structure morphism of the trivial G-bundle Y ×k G on Y . The direct sum of the differentials of pr1 and pr2 gives an isomorphism
∼ ∗ ∗ (69) T(Y ×k G)log /T log = TY log ×k G/T log → pr1 (TY log /T log ) ⊕ pr2 (TG/k ). By applying the functor pr1∗ (−)G to this isomorphism, we obtain a decomposition ∼

f log τ :T Y ×k G/T log → TY log /T log ⊕ OY ⊗k g

(70)

f log of T Y ×k G/T log , where the second factor OY ⊗k g of the codomain is naturally isomorphic to gY ×k G . The morphism dlog Y ×k G coincides, via τ , with the first projection TY log /T log ⊕ OY ⊗k g → TY log /T log . Under the decomposition τ , the Lie bracket f log operation on T log is given by Y

×k G/T

(71) [(∂1 , a1 ⊗ v1 ), (∂2 , a2 ⊗ v2 )] = ([∂1 , ∂2 ], a1 a2 ⊗ [v1 , v2 ] + ∂1 (a2 ) ⊗ v2 − ∂2 (a1 ) ⊗ v1 ) for any local sections ∂i ∈ TY log /T log , ai ∈ OY , and any vi ∈ g (i = 1, 2). By means of this description, (67) can be immediately verified for the trivial G-bundle. 1.2.6. Morphisms of Lie algebroids. — Next, we define a morphism of Lie algebroids and exhibit two examples arising from G-bundles. Definition 1.13. — Let V , V ′ be Lie algebroids on Y log /T log . A morphism of Lie algebroids from V to V ′ is defined as an OY -linear morphism V → V ′ that preserves both the Lie bracket operation and the anchor map.

ASTÉRISQUE 432

1.3. LOGARITHMIC CONNECTIONS AND CURVATURE

27

Example 1.14. — An automorphism of G-bundles induces an automorphism between their associated Lie algebroids. Indeed, let π : E → Y and π ′ : E ′ → Y be G-bundles ∼ on Y and η : E → E ′ an isomorphism of G-bundles. The differential of η gives a ∼ G-equivariant isomorphism dη : π∗ (TE log /T log ) → π∗′ (TE ′ log /T log ) of OY -modules. Then, the restriction ∼ f f log log → (72) dη G : T T ′ log log E

/T

/T

E

of this isomorphism specifies an isomorphism of Lie algebroids. Example 1.15. — The change of structure group for a G-bundle induces a morphism of Lie algebroids. In fact, let E be a G-bundle on Y , and let G′ and w be as in § 1.2.3. Then, the differential of the natural morphism E → E ×G G′ (i.e., the morphism given by v 7→ (v, e) for every v ∈ E ) yields a morphism of Lie algebroids f log log → T f G ′ log log . (dw)E : T E (E × G ) /T /T

(73)

f log log can be described by means of a section Remark 1.16. — The Lie algebroid T E /T of E . To see this, suppose that there exists a global section σ : Y → E of the G-bundle ∼ π : E → Y . This section induces an isomorphism of G-bundles ησ : Y ×k G → E given by (y, g) 7→ σ(y) · g for y ∈ Y and g ∈ G. Hence, we have the following composite isomorphism: (74)

TY log /T log

∼

⊕ OY ⊗ g → Γ∗eY (pr∗1 (TT log /T log )) ⊕ Γ∗eY (pr∗2 (TG/k )) ∼

→ Γ∗eY (TY log ×k G/T log ) ∼

→ Γ∗eY (ησ∗ (TE log /T log )) ∼

→ σ ∗ (TE log /T log ), where the second arrow arises from (69). By composing this isomorphism with τ ◦ d(ησ−1 )G ◦ (cf. (70) for the definition of τ ), we obtain an isomorphism of OY -modules ∼ f log log → (75) T σ ∗ (T log log ). E

/T

E

/T

This description will be used in the proof of Lemma 1.21.

1.3. Logarithmic connections and curvature 1.3.1. Logarithmic connections. — Here, we introduce an ℏ-twisted version of a logarithmic connection defined on a principal bundle. Let Y log /T log be as above. Also, let E be a G-bundle on Y and ℏ an element of Γ(T, OT ). Definition 1.17. — An ℏ-T log -connection on E is an OY -linear morphism f log log , (76) ∇ : TY log /T log → T E /T

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

satisfying dlog ◦ ∇ = ℏ · idTY log /T log . For simplicity, a 1-T log -connection will be referred E to as a T log -connection. Example 1.18 (The case of ℏ = 0). — Note that giving a 0-T log -connection on E is equivalent to giving an OY -linear morphism TY log /log → gE . Indeed, the image of a f log log is contained in Ker(dlog ) = g because 0-T log -connection ∇ : T log log → T Y

/T

E

/T

E

of the equality dlog ◦ ∇ = 0. E

E

Remark 1.19 (The space of connections). — Let ∇ be an ℏ-T log -connection on E . Then, f log log , it defines an ℏ-T log -connection for an OY -linear morphism ∇′ : TY log /T log → T E /T ′ on E if and only if ∇ = ∇ + R for some
∼ R ∈ HomO (TY log /T log , gE ) = Γ(Y, ΩY log /T log ⊗ gE ) . Y

It follows that the set of ℏ-T log -connections on E forms, if it is nonempty, an affine space modeled on the k-vector space Γ(Y, ΩY log /T log ⊗ gE ).

1.3.2. The trivial bundle. — We here consider the case of the trivial G-bundle E = Y ×k G. Given an ℏ-T log -connection ∇ on Y ×k G, we shall write ∇g for the composite (77)

pr2 ∇ f τ ∇g : TY log /T log −→ T → TY log /T log ⊕ OY ⊗k g −−→ OY ⊗k g. Y log ×k G/T log −

This morphism characterizes the connection ∇ because the equality ∇ = (ℏ · idTY log /T log , ∇g ) f log holds under the identification T Y ×k G/T log = TY log /T log ⊕ OY ⊗k g given by τ . In particular, the assignment ∇ 7→ ∇g defines a bijective correspondence between the set of ℏ-T log -connections on Y ×k G and the set of OY -linear morphisms TY log /T log → OY ⊗k g. If G = Gm (hence, g = k), then any ℏ-T log -connection on the trivial Gm -bundle
Y ×k Gm is determined by a global section of ΩY log /T log ∼ = Hom OY (TY log /T log , OY ) . Denote by f log (78) ∇triv : T log log → T log Y

/T

Y

×k G/T

log

the ℏ-T -connection corresponding to the zero map TY log /T log → OY ⊗k g. We will refer to ∇triv as the trivial connection on Y ×k G.

1.3.3. Pull-back. — Let ∇ be an ℏ-T log -connection on E . Also, let Y ′ log be a fine log scheme equipped with a log étale morphism ulog : Y ′ log → Y log . Then, we can obtain an ℏ-T log -connection on the pull-back E ′ := u∗ (E ) (cf. (51)) arising from ∇. Indeed, because of the log étaleness of ulog , the differential of the natural projection uE :

ASTÉRISQUE 432

29

1.3. LOGARITHMIC CONNECTIONS AND CURVATURE

E ′ (=

Y ′ ×Y E ) → E yields the following isomorphism of short exact sequences: / gE ′

0 (79)

/ TY ′ log /T log

≀

≀

/0

≀

 f log log ) / u∗ (T E /T

 / u∗ (gE )

0

dlog E′

f ′ log log /T E /T

u∗ (dlog ) E

 / u∗ (TY log /T log )

/ 0.

f log log ) of ∇ by u specifies, via the above isoThe pull-back u∗ (TY log /T log ) → u∗ (T E /T f ′ log log forming an morphism of sequences, an OY ′ -linear morphism TY ′ log /T log → T E /T log ′ ℏ-T -connection on E . We shall write u∗ (∇) (or u∗E (∇))

(80)

for this connection and refer to it as the pull-back of ∇ to Y ′ log . Conversely, the formation of an ℏ-T log -connection on Y log /T log has descent with respect to the étale topology on Y . 1.3.4. Base-change. — Let T ′ log be a fine log scheme over k equipped with a k-morphism tlog : T ′ log → T log . Write Y ′ log := T ′ log ×T log Y log , where the fiber product is taken in the category of fine log schemes over k. Here, we use the notation t∗ (−) to denote base-change by t, i.e., pull-back by the projection Y ′ → Y (cf. (52)). Note that we can construct the base-change of an ℏ-T log -connection over T ′ log . Indeed, the differential of the natural projection t∗ (E ) → E yields an isomorphism (81)

f∗ log ′ log T t (E ) /T

∼ f log log ) → t∗ (T E /T

(cf. [67], Chap. IV, § 1.2, Proposition 1.2.15) compatible with the respective surjec
tions onto TY ′ log /T ′ log ∼ = t∗ (TY log /T log ) . For an ℏ-T log -connection ∇ on E , its pullf log log ) by the projection Y ′ → Y specifies, via (81), a back t∗ (T log log ) → t∗ (T Y

/T

E

t∗ (ℏ)-T ′ log -connection

/T

f log ′ log t∗ (∇) : TY ′ log /T ′ log → T t(E ) /T

(82)

on t∗ (E ), where t∗ (ℏ) denotes the image of ℏ in Γ(T ′ , OT ′ ). 1.3.5. Gauge transformation. — We shall define the notion of a gauge transformation and examine how the local description of an ℏ-T log -connection changes after applying a gauge transformation. ∼ Let π ′ : E ′ → Y be another G-bundle on Y and η : E → E ′ an isomorphism log of G-bundles. Then, for an ℏ-T -connection ∇ on E , the composite f ′ log log (83) η (∇) := dη G ◦ ∇ : T log log → T ∗

Y

G

/T

(cf. (72) for the definition of dη ) forms an ℏ-T

E

log

/T

-connection on E ′ .

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

Definition 1.20. — Suppose that E ′ = E , i.e., η is an automorphism of E . Then, we shall refer to η∗ (∇) as the gauge transformation of ∇ by η. Let us prove the following lemma. f log Lemma 1.21. — We shall identify T Y ×k G/T log with TY log /T log ⊕ OY ⊗k g by means of τ (cf. (70)). Then, for a morphism of k-schemes h : Y → G, the automorphism dLG h (cf. (72)) of TY log /T log ⊕ OY ⊗k g can be expressed as −1 ∗ dLG ) (ΘG ) + AdG (h)), h = (idTY log /T log , (h

(84)

where ΘG and AdG (h) are regarded as OY -linear morphisms TY log /T log → OY ⊗k g and ∼ OY ⊗k g → OY ⊗k g respectively. log Proof. — The automorphism dLG h is compatible with the surjection dY ×k G , i.e., the first projection TY log /T log ⊕ OY ⊗k g → TY log /T log . This implies that there exist OY -linear morphisms

(85)

D1 : TY log /T log → OY ⊗k g and D2 : OY ⊗k g → OY ⊗k g,

1 2 satisfying dLG h = (idTY log /T log , D + D ). In what follows, we shall find out what D1 and D2 are. First, observe that, for any local section ∂ ∈ h∗ (TG/k ), the following sequence of equalities holds:

D1 (∂) = dLG h ((∂, 0)) = dLh |

(86)

Γh−1

(∂) = (h−1 )∗ (ΘG )(∂),

where the second equality follows from (75) and the last equality follows from Remark 1.2. This implies D1 = (h−1 )∗ (ΘG ). On the other hand, recall that the identification of g with the space of left-invariant vector fields on G gives a canonical identification
f log ∼ (87) V(T log )/G = ((Y × G) × g)/G log ) = V(T log Y

×k G/Y

Y

×k G/Y

k

k

(cf. the discussion following Definition 1.11). Here, the right-hand side is obtained as the quotient of (Y ×k G) ×k g by the right G-action given by ((y, g1 ), v) · g2 = ((y, g1 · g2 ), AdG (g2−1 )(v)) for y ∈ Y , v ∈ g, and g1 , g2 ∈ G. Under this identification, dLG maps each h restricts to an automorphism of ((Y  ×k G) ×k g)/G, which  point ((y, e), v) for y ∈ Y , v ∈ g to ((y, h), v) = ((v, e), AdG (h)(v)) . Hence, since

f log the natural identification Y ×k g (= V(OY ⊗k g)) ∼ = V(T Y ×k G/Y log ) is given, via (87), by the correspondence (y, v) ↔ ((y, e), v), we have D2 = AdG (h). This completes the proof of this lemma. The following proposition follows directly from the above lemma. Proposition 1.22. — Let h be a morphism of k-schemes Y → G. Then, for an ℏ-T log -connection ∇ on the trivial G-bundle Y ×k G, the gauge transformation

ASTÉRISQUE 432

1.3. LOGARITHMIC CONNECTIONS AND CURVATURE

31

Lh∗ (∇) of ∇ by Lh satisfies the following equality of morphisms TY log /T log → OY ⊗k g: (88)

Lh∗ (∇)g = ℏ · (h−1 )∗ (ΘG ) + AdG (h) ◦ ∇g .

(cf. (77) for the definition of ∇g ). 1.3.6. Curvature. — Here, we define the notion of curvature. Roughly speaking, this invariant measures the failure of an ℏ-T log -connection to preserve the Lie bracket f log log be an ℏ-T log -connection on E . For local operation. Let ∇ : TY log /T log → T E /T sections ∂1 , ∂2 ∈ TY log /T log , we write (89)

ψ ∇ (∂1 ∧ ∂2 ) := [∇(∂1 ), ∇(∂2 )] − ℏ · ∇([∂1 , ∂2 ]).

If a is a local section of OY , then the following equalities hold: (90)

ψ ∇ (∂1 ∧ ∂2 ) = −ψ ∇ (∂2 ∧ ∂1 ), ψ ∇ ((a · ∂1 ) ∧ ∂2 ) = ψ ∇ (∂1 ∧ (a · ∂2 )) = a · ψ ∇ (∂1 ∧ ∂2 ),

where the lower sequence of equalities follows from (62). So the assignment ∂1 ∧ ∂2 7→ ψ ∇ (∂1 ∧ ∂2 ) defines an OY -linear morphism (91)

ψ∇ :

2 ^

TY log /T log

f log log . →T E /T

Definition 1.23. — We shall refer to ψ ∇ as the curvature of ∇. Also, we shall say that ∇ is flat (or integrable) if ψ ∇ = 0. Remark 1.24. — Since dlog preserves the Lie bracket operation, we have E (92)

dlog ◦ ψ ∇ (∂1 ∧ ∂2 ) = dlog ([∇(∂1 ), ∇(∂2 )] − ℏ · ∇([∂1 , ∂2 ])) E E = [dlog ◦ ∇(∂1 ), dlog ◦ ∇(∂2 )] − ℏ · dlog ◦ ∇([∂1 , ∂2 ]) E E E = [ℏ · ∂1 , ℏ · ∂2 ] − ℏ · ℏ · [∂1 , ∂2 ] = 0.

It follows that dlog ◦ ψ ∇ = 0, or equivalently, the image of ψ ∇ is contained E  in gE = Ker(dlog ) . E V2 ∇ Hence, ψ may be regarded as an OY -linear morphism TY log /T log → gE . Remark 1.25 (Log curve). — Suppose that Y log /T log is a log curve in the sense of V2 Definition 1.40 described later. Then, TY log /T log is a line bundle, so TY log /T log = 0. It follows that any ℏ-T log -connection is automatically flat. Since the main part of the subsequent discussions in this manuscript deals with connections on log curves, the concept of flatness may not be so important to us. Example 1.26 (Trivial bundle). — Suppose that 2 is invertible in k. Let ∇ be an ℏ-T log -connection on the trivial G-bundle Y ×k G. In the natural manner, we shall regard ∇g (cf. (77)) as an element of Γ(Y, ΩY log /T log ⊗k g) and ψ ∇ as an element

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V2 of Γ(Y, ( ΩY log /T log ) ⊗k g). Then, the following equality holds: ℏ · [∇g , ∇g ], 2 where the operations “[−, −]” and “d” are defined in the
same fashions as (34) and (38) respectively. In particular, if ω ∈ Γ(Y, ΩY log /T log ) is the 1-form corresponding to an ℏ-T log -connection ∇ on the trivial Gm -bundle (cf. § 1.3.2), then ∇ is flat if and only if ω is closed. ψ ∇ = d∇g +

(93)

Proposition 1.27. — Let E be a G-bundle on Y , ∇ an ℏ-T log -connection on E , and η an automorphism of E . Denote by AdG (η) the automorphism of gE induced from η via change of structure group by AdG . Then, the equality ψ η∗ (∇) = AdG (η) ◦ ψ ∇

(94) V2

of morphisms TY log /T log → gE holds. In particular, the flatness of an ℏ-T log -connection is closed under gauge transformations. Proof. — The assertion follows immediately from the fact that the automorf log log and restricts phism dη G is compatible with the Lie bracket operation on T E /T to AdG (η). Definition 1.28. —

(i) An ℏ-flat G-bundle on Y log /T log is a pair (E , ∇)

(95)

consisting of a G-bundle E on Y and a flat ℏ-T log -connection ∇ on E . For simplicity, a 1-flat G-bundle will be referred to as a flat G-bundle. (ii) Let (E , ∇), (E ′ , ∇′ ) be ℏ-flat G-bundles on Y log /T log . Then, an isomorphism of ℏ-flat G-bundles from (E , ∇) to (E ′ , ∇′ ) is defined as an isomorphism ∼ of G-bundles η : E → E ′ with η∗ (∇) = ∇′ . 1.3.7. Change of structure group. — Let E and ℏ be as before. Also, let G′ and w be as in § 1.2.3. If ∇ is an ℏ-T log -connection on E , then the composite (96)

(dw)E f ∇ f w∗ (∇) : TY log /T log −→ T E log /T log −−−→ T(E ×G G′ )log /T log

specifies an ℏ-T log -connection on E ×G G′ . This connection is denoted by ∇ ×G,w G′ or ∇ ×G G′ depending on the situation. Since (dw)E preserves the Lie bracket operation, we have (97)

ψ w∗ (∇) = (dw)E ◦ ψ ∇ .

It follows that the flatness of ψ ∇ implies the flatness of ψ w∗ (∇) , and vice versa if w is injective. In particular, an ℏ-flat G-bundle (E , ∇) yields an ℏ-flat G′ -bundle (E ×G G′ , w∗ (∇)).

ASTÉRISQUE 432

1.4. EXPLICIT DESCRIPTIONS OF GAUGE TRANSFORMATIONS

33

1.3.8. Change of parameter. — Let ∇ be an ℏ-T log -connection on E . Also, let us take c ∈ Γ(T, OT ). Then, the OT -linear morphism f log log , c · ∇ : TY log /T log → T E /T

(98)

i.e., the morphism ∇ multiplied by c, forms a (c · ℏ)-T log -connection on E . The curvature of c · ∇ is given by ψ c·∇ = c2 · ψ ∇ .

(99)

In particular, c · ∇ is flat if ∇ is flat, and the inverse is true when c ∈ Γ(T, OT× ). 1.3.9. Products. — Let G′ be another connected smooth affine algebraic k-group with Lie algebra g′ . Also, let E (resp., E ′ ) be a G-bundle (resp., a G′ -bundle) on Y . The fiber product (100)

E

⊠ E ′ := E ×Y E ′

admits, in the natural manner, a structure of G ×k G′ -bundle. The Lie algebra of the product G ×k G′ is the direct sum g ⊕ g′ , and there exists a canonical isomorphism ∼ (g ⊕ g′ )E ⊠E ′ → gE ⊕ g′E ′ . Moreover, the following sequence is exact: (dlog ,−dlog ′ ) E

f ′ log log −−−−−−E−→ T log log −→ 0, f log log ⊕ T f ′ log log −→ T (101) 0 −→ T Y /T E /T E /T (E ⊠E ) /T where the second arrow is obtained by differentiating the projections E ⊠ E ′ → E , f ′ log log as an O -subE ⊠ E ′ → E ′ . By using the second arrow, we shall regard T Y (E ⊠E ) /T f ′ log log . f log log ⊕ T module of T E

/T

E

/T

Now, let ∇ (resp., ∇′ ) be an ℏ-T log -connection on E (resp., E ′ ). The image of the f ′ log log . f ′ log log is contained in T f log log ⊕ T morphism (∇, ∇′ ) : TY log /T log → T (E ⊠E ) /T E /T E /T The resulting OY -linear morphism f ′ log log (102) ∇ ⊠ ∇′ : T log log → T Y

/T

(E ⊠E )

/T

′

specifies an ℏ-T log -connection on E ⊠ E ′ . The curvature ψ ∇⊠∇ of ∇ ⊠ ∇′ satisfies an equality (103)

′

′

ψ ∇⊠∇ = (ψ ∇ , ψ ∇ )


of morphisms TY log /T log → (g ⊕ g′ )E ⊠E ′ ∼ = gE ⊕ g′E ′ . In particular, if (E , ∇) and (E ′ , ∇′ ) respectively form an ℏ-flat G-bundle and an ℏ-flat G′ -bundle, then the pair (E ⊠ E ′ , ∇ ⊠ ∇′ ) forms an ℏ-flat G ×k G′ -bundle; we will refer to it as the product of (E , ∇) and (E ′ , ∇′ ).

1.4. Explicit descriptions of gauge transformations In what follows, we describe gauge transformations by using the exponential and logarithm maps, which connect unipotent elements in G with nilpotent elements in g.

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1.4.1. p-unipotent/p-nilpotent elements. — Denote by p the characteristic char(k) of k. In this section, we assume (except for the discussion in Remark 1.35) that k is perfect and that (G, T) is a split semisimple algebraic k-group of adjoint type, where T denotes a maximal torus of G. In particular, the adjoint representation AdG : G → GL(g) becomes a closed immersion; by using this immersion, we regard G as a closed subgroup of GL(g). Fix a Borel subgroup B of G defined over k containing T, and set N := [B, B], i.e., the unipotent radical of B. Denote by n theP Lie algebra of N. Here, note that each root β in G can be expressed uniquely as β = α∈Γ nα · α with P nα ∈ Z, where Γ denotes the set of simple roots in B with respect to T; we call α∈Γ nα the height of β. We shall set (104)

hG := 1 + (the height of a root with maximum height).

If G is simple, then hG coincides with the Coxeter number of G. We assume further that p = 0 or p > 2hG . In particular, we will not deal with the case where p = 2, 3. Remark 1.29. — From the semisimplicity of G and the assumption “p = 0 or p > 2hG ” imposed above, the Killing form on g is nondegenerate; this fact follows from the formula for the discriminant of the Killing form proved in [85], § 4, (4.8) (cf. [64], § 1.1). Hence, since k is a perfect field of characteristic ̸= 2, 3 and (G, T) is split over k, the semisimple Lie algebra g is classical in the sense of [77], Chap. II, § 3. It follows (cf. [77], Chap. III, § 5, Theorem III.5.1) that the image of the adjoint representation AdG coincides with the identity component Aut(g)0 (⊆ GL(g)) of the group of Lie algebra ∼ automorphisms of g. In particular, AdG induces an isomorphism G → Aut(g)0 . Also, according to [77], Chap. I, § 7, corollary, any semisimple Lie algebra with nondegenerate Killing form can be decomposed into a direct sum of simple Lie algebras. Thus, the problem of understanding the structures of such semisimple Lie algebras can be largely reduced to those of simple Lie algebras. For example, by applying [17], Chap. 5, § 5.5, Proposition 5.5.2, we see that if p > 2hG , then the equality ad(v)p = 0 holds for any nilpotent element v of g. Denote by Gunip (resp., gnilp ) the unipotent variety of G (resp., the nilpotent variety of g), which is the Zariski closed subset of G (resp., g) consisting of all unipotent elements of G (resp., all nilpotent elements of g). If p = 0 (resp., p > 0), then we shall write (105)

GL(g)p-unip

for the Zariski closed subset in GL(g) consisting of all elements h with (h − idg )m = 0 for some m ∈ Z>0 (resp., (h − idg )p = 0). Also, write (106)

End(g)p-nilp

for the Zariski closed subset in End(g) consisting of all elements v satisfying that v m = 0 for some m ∈ Z>0 (resp., v p = 0). We endow Gunip (resp., gnilp ; resp., GL(g)p-unip ; resp., End(g)p-nilp ) with the reduced subscheme structure of G (resp.,

ASTÉRISQUE 432

35

1.4. EXPLICIT DESCRIPTIONS OF GAUGE TRANSFORMATIONS

g; resp., GL(g); resp., End(g)). In particular, there exist natural closed immersions N ,→ Gunip , n ,→ gnilp . 1.4.2. Exponentials and logarithms. — For each v ∈ End(g)p-nilp , the exponential of v is defined to be ! p−1 ∞ X X 1 s 1 s (107) Exp(v) := ·v · v , if p > 0 . = s! s! s=0 s=0 Also, for h ∈ GL(g)p-unip , the logarithm of h is defined to be (108) Log(h) :=

∞ X (−1)s−1 s=1

s

· (h − idg )

s

=

p−1 X (−1)s−1

s

s=1

! s

· (h − idg ) , if p > 0 .

The assignments v 7→ Exp(v) and h 7→ log(h) determine isomorphisms of k-schemes (109)

∼

Exp : End(g)p-nilp → GL(g)p-unip ,

∼

Log : GL(g)p-unip → End(g)p-nilp ,

which are inverses of each other. Proposition 1.30. — Let v : T → End(g)p-nilp (resp., h : T → GL(g)p-unip ) be a morphism of k-schemes. Then, the following equality holds:
(110) Exp(−v) = Exp(v)−1 resp., Log(h−1 ) = −Log(h) . Proof. — The non-resp’d assertion follows from the observation that (111)

Exp(v) · Exp(−v) = Exp(v + (−v)) = Exp(0) = idg .

The resp’d assertion is a direct consequence of the non-resp’d assertion together with the fact that Exp and Log are inverses of each other. The morphism of algebraic k-groups AdGL(g) : GL(g) → GL(End(g)) (resp., Inn(−) : GL(g) → Aut(GL(g))) defines a left GL(g)-action on End(g)p-nilp (resp., GL(g)p-unip ). Moreover, it induces a left G-action on End(g)p-nilp (resp., GL(g)p-unip ) via composition with AdG : G → GL(g). Both Exp and Log are compatible with the GL(g)-actions, hence also with the induced G-actions. In particular, if p is sufficiently large relative to hG , then Exp defines a Springer isomorphism for GL(g). According to the final comment in Remark 1.29, the morphisms ad : g ,→ End(g) and AdG : G ,→ GL(g) restrict to closed immersions gnilp ,→ End(g)p-nilp and Gunip ,→ GL(g)p-unip respectively. Moreover, by the definition of Exp, the image Exp(ad(v)) ∈ GL(g)p-unip of each v ∈ gnilp defines a Lie algebra automorphism of g. This implies (since G ∼ = Aut(g)0 as mentioned in Remark 1.29) that Exp restricts to a closed immersion (112)

Exp : gnilp ,→ Gunip .

This morphism in fact is an isomorphism because gnilp and Gunip are irreducible and of the same dimension.

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Proposition 1.31. — The isomorphism Exp restricts to an isomorphism of k-schemes ∼

Exp : n → N.

(113)

Proof. — Let us fix an identification of (the underlying k-vector space of) g with k ⊕l , where l := dim(g). Under this identification, we have GL(g) = GLl and End(g) = gll . Denote by NGLl (resp., ngll ) the subgroup of GLl (resp., gll ) consisting of upper triangular matrices with diagonal entries 1 (resp., 0). The isomorphism ∼ Exp : End(g)p-nilp → GL(g)p-unip restricts, by construction, to an isomorphism ∼ End(g)p-nilp ∩ ngll → GL(g)p-unip ∩ NGLl . On the other hand, N is contained, via AdG , in a maximal closed connected unipotent subgroup of GLl , which is conjugate to NGLl (cf. [55], § 6, Corollary 6.11, (c)). Hence, since Exp is GL(g)-equivariant, we may assume, after possibly replacing the fixed identification g ∼ = k ⊕l with another, unip that N ⊆ NGLl . This assumption implies that N = G ∩ NGLl and n = gnilp ∩ ngll .

∼ Thus, Exp restricts to n = gnilp ∩ ngll → N = Gunip ∩ NGLl (cf. (112)). This completes the proof of the assertion. By the above proposition, we obtain the inverse ∼

Log : N → n

(114) ∼

to Exp : N → n, which makes the following square diagram commute: AdG |

N

N (115)

/ GL(g)p-unip ≀ Log

Log ≀

 n

ad|

 / End(g)p-nilp .

n

The commutativity of this diagram implies the following proposition. Proposition 1.32. — For a k-scheme T and a T -rational point h : T → N of N, we have an equality ! ∞ X 1 s (116) AdG (h) = Exp(ad(Log(h))) = · ad(Log(h)) s! s=0 of OT -linear automorphisms of OT ⊗k g. 1.4.3. Gauge transformation using the logarithm map. — Since the tangent bundle Tg/k of the k-scheme g is canonically isomorphic to Og ⊗k g, the differential ∼ of Log : N → n defines an ON -linear morphism (117)

dLog : TN/k → ON ⊗k n,

or equivalently, an element of Γ(N, ΩN/k ⊗k n).

ASTÉRISQUE 432

1.4. EXPLICIT DESCRIPTIONS OF GAUGE TRANSFORMATIONS

37

Proposition 1.33. — The following equality holds: ΘN = dLog.

(118)

In particular, for a k-scheme T and a T -rational point h : T → N of N, the equality h∗ (ΘN ) = dLog(h) (:= d(Log ◦ h))

(119)

of morphisms TT /k → OT ⊗ n holds. Proof. — After possibly changing the base by a field extension of k, we may assume that k is algebraically closed. Denote by A : GL(g) ,→ End(g) the natural open immersion, and consider N to be a closed subscheme of End(g) via the composite of AdG and A. In particular, (since N ⊆ GL(g)) we have ΘN = ΘGL(g) | = (A−1 dA)|

(120)

N

N

(cf. Example 1.5). Now, let h be a k-rational point of N (⊆ End(g)) and take an element R ∈ Th N, which classifies the element h + ϵ · R of N(k[ϵ]/(ϵ2 )) ⊆ End(g)(k[ϵ]/(ϵ2 )) = End(g)(k) ⊕ ϵ · End(g)(k). The following sequence of equalities holds: ∞ X (−1)s−1 (121) Log(h + ϵ · R) = · (h − idg + ϵ · R)s s s=1 =

∞ X (−1)s−1

s

s=1

· (h − idg )s + ϵ ·

∞ X

(−1)s−1 · (h − idg )s−1 · R

s=1 −1

= Log(h) + ϵ · h

· R.

Hence, we have (122)

(121)

(120)

dLog| (R) = h−1 · R = ΘN | (R). h

h

By applying this fact to various h’s and R’s, we have ΘN = dLog, as desired. Corollary 1.34. — (Let us keep the notation and assumptions described at the beginning of § 1.4.1.) Let Y log /T log be as in § 1.2.4. Also, let ℏ be an element of Γ(T, OT ), ∇ an ℏ-T log -connection on the trivial G-bundle Y ×k G, and h : Y → N a Y -rational point of N. Then, we have an equality ∞ X 1 g (123) Lh∗ (∇) = −ℏ · dLog(h) + · ad(Log(h))s ◦ ∇g s! s=0 of morphisms TY log /T log → OY ⊗k g. Proof. — The assertion follows from Propositions 1.22, 1.30, 1.32, and 1.33. Remark 1.35. — Let n be an integer > 1. If p = 0 or n < p, then we can obtain a description of a gauge transformation (123) for G = GLn (n > 1) although this is not of adjoint type. Indeed, let N (resp., n) denote the subgroup of GLn (resp., gln ) consisting of upper triangular matrices with diagonal entries 1 (resp., 0). In the case of p > 0, it follows from the assumption n < p that v p = (h − 1)p = 0 for any v ∈ n

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and h ∈ N. Hence, the exponential Exp(v) of v ∈ n and the logarithm Log(h) of h ∈ N can be defined as in (107) and (108) respectively. The assignments v 7→ Exp(v) and ∼ ∼ h 7→ Log(h) respectively specify isomorphisms Exp : n → N and Log : N → n, which are inverses of each other and satisfy the same assertion as Proposition 1.30. For each h ∈ N(T ) (⊆ Γ(T, OT ⊗k gln )) and R ∈ Γ(T, OT ⊗k gln ), the equality AdGLn (R) = h · R · h−1 holds. On the other hand, as in the proof of Proposition 1.33, we have ΘN = dLog. Consequently, keeping the notation in Corollary 1.34, we obtain from Proposition 1.22 an equality Lh∗ (∇)gln (∂) = −ℏ · dLog(h) + h · ∇gln (∂) · h−1

(124)

for any local section ∂ ∈ TY log /T log .

1.5. The moduli stack of pointed stable curves Let us recall pointed stable curves and their canonical log structures pulled-back from the moduli stack. Let us fix a pair of nonnegative integers (g, r). 1.5.1. Pointed curves. — Let S be a k-scheme. By a curve over S, we mean a proper flat scheme f : X → S over S all of whose geometric fibers are connected, reduced, and of dimension 1. Definition 1.36. —

(i) An r-pointed curve over S is defined as a collection of data X := (f : X → S, {σi }ri=1 )

(125)

consisting of a curve f : X → S over S and an ordered set of r sections σi : S → X (i = 1, . . . , r) of f whose images are disjoint, i.e., Im(σi )∩Im(σj ) = ∅ if i ̸= j, and land in the smooth locus of f . We refer to σi as the i-th marked point of X . A 0-pointed curve is nothing but a curve in the above sense and will be often referred to as an unpointed curve. (ii) Suppose that, for each j ∈ {1, 2}, we are given a k-scheme Sj and an r-pointed curve Xj := (fj : Xj → Sj , {σji }ri=1 ) over Sj . Then, a morphism of r-pointed curves from X1 to X2 is a pair of k-morphisms (ϕ : S1 → S2 , Φ : X1 → X2 )

(126)

such that the following squares form cartesian commutative diagrams: X1 (127)

/ X2

f1

 S1

ASTÉRISQUE 432

Φ

f2



ϕ

/ S2 ,

XO 1

Φ

σ1i

S1

/ X2 O σ2i

ϕ

/ S2

(i = 1, . . . , r).

1.5. THE MODULI STACK OF POINTED STABLE CURVES

39

Definition 1.37. — (i) Let k be an algebraically closed field over k and X a curve over k. A k-rational point x of X is called a nodal point, or an ordinary double point, if the complete local ring b OX,x at x is isomorphic to k[[u, v]]/(uv). (ii) An r-pointed smooth (resp., prestable) curve over S is an r-pointed curve X := (f : X → S, {σi }i ) over S such that, for each geometric point s : Spec(k) → S of S, any point of the fiber Xs := f −1 (s) is a smooth point (resp., either a smooth point or a nodal point). (iii) Let X := (f : X → S, {σi }i ) be an r-pointed curve over S. We shall say that X is of (arithmetic) genus g if, for any geometric point s : Spec(k) → S of S, the equality dimk (H 1 (Xs , OXs )) = g holds. Let X := (X/S, {σi }ri=1 ) be an r-pointed curve over S. Then, for an S-scheme S ′ , the collection of data (128)

S ′ ×S X := ((S ′ ×S X)/S ′ , {(idS ′ , σi ) : S ′ → S ′ ×S X}ri=1 )

forms an r-pointed curve over S ′ . We refer to it as the base-change of X to S ′ . If X is prestable (resp., of genus g), then so is S ′ ×S X . 1.5.2. Pointed stable curves. — Let X := (f : X → S, {σi }ri=1 ) be an r-pointed prestable curve of genus g over S. Since f is a local complete intersection morphism, it follows from the theory of duality that the dualizing sheaf ωX/S on X relative to S exists. We shall set r X log (129) ωX/S := ωX/S ( [σi ]), i=1

where [σi ] (i = 1, . . . , r) denotes the relative effective reduced divisor on X determined log by the image of σi . Note that ωX/S is a line bundle and its fiber over any point of S is of degree 2g − 2 + r. If S = Spec(k) for an algebraically closed field k over k, then the following three conditions are equivalent: log (a) The line bundle ωX/k is ample.

(b) The automorphism group Aut(X ) of X over k (i.e., over the identity morphism of Spec(k)) is finite. (c) The inequality 2g − 2 + r > 0 holds and, for every nonsingular rational component E of X, the number of points in E contained in either the remaining components of X or the image of marked points is at least 3. Definition 1.38. — An r-pointed stable curve (of genus g) over S is an r-pointed prestable curve X (of genus g) over S such that the fiber Xs := s ×S X of X over any geometric point s of S satisfies one of the equivalent three conditions (a)–(c) described above.

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1.5.3. The moduli stack of pointed stable curves. — Suppose that 2g −2+r > 0. Then, r-pointed stable curves of genus g over a k-scheme and morphisms between them form a category (130)

Mg,r → Sch/k

(cf. [50], Definition 1.1) over the category Sch/k , where the projection Mg,r → Sch/k is given by (X/S, {σi }i ) 7→ S. One may verify that Mg,r forms a category fibered in groupoids over Sch/k . Moreover, this category may be represented by a geometrically connected, proper, and smooth Deligne-Mumford stack over k of dimension 3g − 3 + r (cf. [50], Corollary 2.6 and Theorem 2.7; [21], § 5). The stack Mg,r contains a dense open substack
(131) Mg,r ⊆ Mg,r classifying r-pointed smooth curves of genus g. The complement Mg,r \ Mg,r defines a divisor with normal crossings. Denote by (132)

Cg,r := (funiv : Cg,r → Mg,r , {σuniv,i : Mg,r → Cg,r }ri=1 )

the universal family of r-pointed stable curves over Mg,r . Hence, each r-pointed stable curve X of genus g over a k-scheme S determines its classifying morphism ∼ (ϕ, Φ) : X → Cg,r (cf. (126)), inducing an S-isomorphism (idS , Φ) : X → S ×ϕ,Mg,r Cg,r . Example 1.39 (The case of g = 0). — Denote by P1 the projective line over k and by [0], [1], and [∞] the k-rational points of P1 determined by the values 0, 1, and ∞, respectively. After ordering the points [0], [1], [∞], we obtain a unique (up to isomorphism) 3-pointed stable curve (133)

P := (P1 /k, {[0], [1], [∞]})

of genus 0 over k. One may verify that P does not have nontrivial automorphisms and that any 3-pointed stable curve of genus 0 over a k-scheme S is isomorphic to the trivial deformation of P over S. That is to say, we have M0,3 ∼ = M0,3 ∼ = Spec(k). Moreover, assigning the fourth point of each 4-pointed projective line gives a k-iso∼ ∼ morphism M0,4 → P1 \ {[0], [1], [∞]}, which extends to M0,4 → P1 . Next, let r be an integer ≥ 4, and denote by qr : M0,r → M0,r−1 the morphism obtained by forgetting the last marked point and successively contracting any resulting unstable components of each curve classified by M0,r . Then, M0,r+1 can be constructed by blowing-up a certain locus in the fiber product M0,r ×M0,r−1 M0,r of 2 copies of qr . In particular, M0,r (r ≥ 3) can be represented by a k-scheme of dimension r. 1.5.4. Log curves. — Let us introduce the definition of a log curve, which is slightly different from [45], Definition 1.1 (and [2], Definition 4.5) in that our definition does not impose the connectedness condition.

ASTÉRISQUE 432

1.5. THE MODULI STACK OF POINTED STABLE CURVES

41

Definition 1.40. — Let T log be an fs log scheme (cf. Remark 1.41 described below). A log curve over T log is a log smooth integral morphism f log : Y log → T log

(134)

between fs log schemes (cf. [46], § 3) such that the geometric fibers of the underlying morphism f : Y → T are reduced 1-dimensional schemes. (In particular, both TY log /T log and ΩY log /T log are line bundles, and the underlying morphism f : Y → T is flat, according to [46], Corollary 4.5.) Remark 1.41. — A log scheme is called fs, i.e., fine and saturated (cf. [40], § 1.3) if it has étale locally a chart modeled on a fine (= integral and finitely generated) and saturated monoid. The reason for the “fs” assumption in Definition 1.40 is to avoid cusps or worse singularities (cf. [2], Example 4.6). Definition 1.42. — Let T log be an fs log scheme and Y log a log curve over T log . A logarithmic chart, or a log chart for short, on Y log /T log is a pair (U, ∂)

(135)

consisting of an étale Y -scheme U (where we equip U with a log structure pulledback from Y log and denote the resulting log scheme by U log ) and an element ∂ of Γ(U, TU log /T log ) that generates TU log /T log . We shall say that a log chart (U, ∂) is global if U = Y . 1.5.5. The logarithmic structure of a pointed stable curve. — Recall from [45], Theorem 4.5, that Mg,r has a natural log structure given by the divisor at infinity Mg,r \ Mg,r ; we shall denote the resulting fs log stack by log

Mg,r .

(136)

Moreover, by taking the divisor which is the union of the σuniv,i ’s and the pullback of the divisor at infinity of Mg,r , we obtain a log structure on Cg,r ; we denote log

log the resulting log stack by Clog g,r . Both Mg,r and Cg,r are log smooth over k. The structure morphism funiv : Cg,r → Mg,r of Cg,r extends naturally to a morphism log

log funiv : Clog g,r → Mg,r of log stacks, forming a family of log curves. Now, let S be a k-scheme and X := (f : X → S, {σi }ri=1 ) an r-pointed stable curve of genus g over S. Then, X determines its classifying morphism (ϕ, Φ) : X → Cg,r . log

By pulling-back the log structures of Mg,r and Clog g,r , we obtain corresponding log structures on S and X; we denote the resulting schemes by S log and X log respectively. Both S log and X log are fs log schemes, and the morphism f : X → S extends to a log log curve f log : X log → S log over S log (cf. [45], Theorem 2.6). The line bundle ωX/S (cf. (129)) is naturally isomorphic to ΩX log /S log . Remark 1.43. — Recall here the characterization of pointed stable curves in terms of log structures. Let S log := (S, αS : MS → OS ) be an fs log scheme over k. Recall from [45], § 1, Definition 1.2, that a log curve X log := (X, αX : MX → OX ) over S log

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

is called a stable log curve of type (g, r) if, for any log scheme S0log over S log whose underlying scheme S0 is the spectrum of a separably closed field, the following two conditions are fulfilled: — The underlying curve X0 of the fiber X0log := S0log ×S log X log is connected and of genus g, and there exists r pairwise distinct closed points s1 , . . . , sr in the smooth locus of X0 such that the relative characteristic M X log /S log (cf. [45], 0

0

Introduction) of X0log /S0log satisfies (137) M X log /S log ∼ = Ns1 ⊕ · · · ⊕ Nsr ⊕ Zt1 ⊕ · · · ⊕ Ztl . 0

0

Here, {t1 , . . . , tl } is the set of all nodal points on X0 , and for a monoid M and a point x ∈ X0 , we denote by Mx the skyscraper sheaf supported only on x by M . — The equality H 0 (X0 , TX log /S log ) = 0 holds. 0

0

Suppose that 2g − 2 + r > 0. As discussed in [45], the category of r-pointed stable curves of genus g is equivalent to the category of stable log curves of type (g, r). log Indeed, let N denote the stack of stable log curves of type (g, r) over Schlog /k , where log

Schlog /k denotes the category of fs log schemes over k. Also, let us consider Mg,r as a log stack over Schlog over k /k in the natural manner. If we are given an fs log scheme S log

log and an S log -rational point slog → Mg,r , then the log curve S log ×Mlog Clog m : S g,r is a g,r

stable log curve of type (g, r), which determines an S log -rational point slog n of Ng,r . log → 7 s According to [45], § 4, Theorem 4.1, the resulting assignment slog n defined for m various S log ’s determines an isomorphism log ∼

log

Mg,r → Ng,r

(138) of stacks over Schlog /k .

1.6. Local pointed curves and monodromy 1.6.1. Local pointed curves. — Let S be a k-scheme and r an integer ≥ 0. To begin with, let us introduce a local version of a pointed stable curve, as follows. Definition 1.44. —

(i) A local r-pointed curve over S is a collection of data U := (U/S, {σiU : Si → U }ri=1 ),

(139)

where U denotes a curve over S and each σiU (i = 1, . . . , r) denotes a k-morphism from a possibly empty étale S-scheme Si to U satisfying the following condition: there exist an r-pointed curve X := (X/S, {σi }ri=1 ) over S and an étale morphism u : U → X such that, for each i = 1, . . . , r, the equality Si = U ×u,X,σi S holds and σiU coincides with the base-change of σi by u. We shall refer to X as a base for U .

ASTÉRISQUE 432

1.6. LOCAL POINTED CURVES AND MONODROMY

43

(ii) We shall say that a local r-pointed curve U is stable if there exists a base X for U that is stable of constant genus. Remark 1.45. — We regard an r-pointed stable curve X as a local r-pointed stable curve, where a base is taken to be X itself. Let U := (U/S, {σiU : Si → U }ri=1 ) be a local r-pointed curve over S. Suppose that U is obtained by restricting an r-pointed curve X in the manner of Definition 1.44. For each i = 1, . . . , r, we shall denote by [σiU ] (⊆ U ) the scheme-theoretic image of σiU , which is possibly empty. Then, we obtain the relative effective divisor (140)

DU

on U defined as the union of [σiU ]’s. The smooth locus U sm of U \ Supp(DU ) relative to S is a scheme-theoretically dense open subscheme of U . Now, suppose further that X is stable, in particular, both X and S are equipped with the natural log structures discussed before. By pulling-back the log structure of X log , one may equip U with a log structure; if U log denotes the resulting log scheme, we obtain a log curve U log /S log . The log structures of U log and S log depend on the choice of X , but the isomorphism class of ΩU log /S log , hence also that of TU log /S log , is independent of that choice. Moreover, the restriction of the log structure of U log to U sm coincides with the log structure pulled-back from S log . Note that, for each i = 1, . . . , r, there exists a canonical isomorphism (141)

∼

ResU ,i : σiU ∗ (ΩU log /S log ) → OSi ,


called the residue map, which maps the section dlog(x) := x1 · dx ∈ σiU ∗ (ΩU log /S log ) to 1 ∈ OSi for any local function x defining the closed subscheme [σiU ] of U . 1.6.2. Pull-back. — Let U := (U/S, {σiU : Si → U }ri=1 ) be a local r-pointed (stable) curve over S. Then, for an étale U -scheme U ′ , the collection of data (142)

U ′ ×U U := (U ′ /S, {σiU × idU ′ : Si ×U U ′ → U ′ }ri=1 )

forms a local r-pointed (stable) curve over S. We refer to it as the pull-back of U to U ′ . 1.6.3. Monodromy of a connection. — Let U := (U/S, {σi : Si → U }ri=1 ) be a local r-pointed stable curve over S. After choosing a base for U , we obtain a log curve U log /S log extending U/S. In what follows, we suppose that r ≥ 1 and fix an element i ∈ {1, . . . , r} with [σiU ] ̸= ∅. Let π : E → U be a G-bundle on U . As discussed in § 1.2.5, we obtain a log scheme E log defined as E equipped with the log structure pulled-back from U log . Both TU log /S log and TE log /S log are locally free, and the natural morphisms (143)

ιU : TU log /S log → TU/S ,

ιE : TE log /S log → TE /S

are isomorphisms over the scheme-theoretically dense open subscheme U sm of U . It follows that these morphisms are injective over the entire space U . The direct image

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

π∗ (TE log /S log ) ,→ π∗ (TE /S ) of ιE is compatible with the G-actions. Hence, it restricts to an injection
f f log log ,→ T := π∗ (TE /S )G , (144) e ιE : T E /S E /S fitting into the following morphism of short exact sequences: 0

/ gE

(145)

f log log /T E /S id

0

 / gE

dlog E

/ TU log /S log

e ιE

ιU

 f /T E /S

 / TU/S

dE

/0

/ 0,

where the lower horizontal sequence is (66) for the non-logarithmic schemes S, U , and E . Also, by applying, to the lower horizontal sequence, the natural transformation U U (−) → σi∗ (σiU ∗ (−)) arising from the adjunction relation “σiU ∗ (−) ⊣ σi∗ (−)” (i.e., U∗ U “the functor σi (−) is left adjoint to the functor σi∗ (−)”), we obtain a morphism of sequences 0

/ gE

f /T E /S

0

 / σ U (σ U ∗ (gE )) i∗ i

 f )) / σ U (σ U ∗ (T E /S i∗ i

/ TU/S

/0

 / σ U (σ U ∗ (TU/S )) i∗ i

/ 0.

dE

(146)

The two horizontal sequences in this diagram are exact because of the local triviality of the G-bundle E with respect to the étale topology. Composing (145) and (146) gives the following morphism of sequences 0

/ gE

dlog E

f log log /T E /S

(147)

/ TU log /S log

e ι′E

0

ι′U

 f )) / σ U (σ U ∗ (T E /S i∗ i

 / σ U (σ U ∗ (gE )) i∗ i

/0

 / σ U (σ U ∗ (TU/S )) i∗ i

/ 0.

Since the natural surjection TU/S ↠ Coker(ιU ) induces an isomorphism (148)

U U σi∗ (σiU ∗ (TU/S )) ∼ (σiU ∗ (Coker(ιU ))), = σi∗

the right-hand vertical arrow ι′U in (147) turns out to be the zero map. Now, let us take an ℏ-S log -connection ∇ on E . It follows from the above discussion that the composite f )) (149) e ι′ ◦ ∇ : T log log → σ U (σ U ∗ (T E

ASTÉRISQUE 432

U

/S

i∗

i

E /S

1.6. LOCAL POINTED CURVES AND MONODROMY

45

f )). The resulting U U factors through the injection σi∗ (σiU ∗ (gE )) ,→ σi∗ (σiU ∗ (T E /S U U∗ morphism TU log /S log → σi∗ σi (gE ) corresponds, via the adjunction relation U “σiU ∗ (−) ⊣ σi∗ (−),” to a morphism σiU ∗ (TU log /S log ) → σiU ∗ (gE ).

(150)

The image of 1 ∈ Γ(Si , OSi ) via the composite (151)

OSi

Res∨ U ,i

(150)

−−−−→ σiU ∗ (TU log /S log ) −−−→ σiU ∗ (gE )

determines a global section (152)

U∗ µ∇ i ∈ Γ(Si , σi (gE ))



 = Γ(Si , gσiU ∗ (E ) ) .

U Definition 1.46. — We shall refer to µ∇ i as the monodromy (operator) of ∇ at σi .

Example 1.47. — Let us consider the case where E = U ×k G, i.e., E is the trivial G-bundle. Suppose that there exists a local function x ∈ OU defining [σiU ] (⊆ U ). If ∂x denotes the local section of TU log /S log defined as the dual of dx ∈ ΩU log /S log , then the section x · ∂x locally generates TU log /S log . For each S log -connection ∇ on U ×k G, we have (153)

g µ∇ i = ∇ (x · ∂x )| . σi

1.6.4. Change of structure group. — Let G′ and w be as in § 1.2.3, and denote by g′ the Lie algebra of G′ . Also, let (E , ∇) be an ℏ-flat G-bundle on U log /S log . As discussed in § 1.3.7, ∇ induces an ℏ-S log -connection w∗ (∇) on the G′ -bundle E ×G G′ . Then, w (∇) the monodromy µi ∗ of w∗ (∇) at σiU (i = 1, . . . , r) satisfies (154)

w (∇)

µi ∗

= Adw (µ∇ i ),

  where Adw denotes the OU -linear morphism gσiU ∗ (E ) → g′σU ∗ (E ) = g′σU ∗ (E ×G G′ ) ini i duced naturally by w. 1.6.5. Change of parameter. — Let (E , ∇) be as above and c an element of Γ(S, OS ). As discussed in § 1.3.8, ∇ induces a (c · ℏ)-S log -connection c · ∇ on E . Then, the monodromy µc·∇ of c · ∇ at σiU (i = 1, . . . , r) satisfies i µc·∇ = c · µ∇ i i .

(155)

1.6.6. Gauge transformation. — Also, let η be an automorphism of the G-bundle E . η (∇) Then, the monodromy µi ∗ at σiU (i = 1, . . . , r) of the gauge transformation η∗ (∇) by η (cf. Definition 1.20) satisfies (156)

η (∇)

µi ∗

= dη G |

σiU

(µ∇ i )

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CHAPTER 1. LOGARITHMIC CONNECTIONS OVER LOG SCHEMES

(cf. (72) for the definition of dη G ). In particular, if E = U ×k G and η = Lh for a morphism of k-schemes h : U → G, then it follows from Lemma 1.21 that   L (∇) ∇ (157) µi h∗ = AdG (h ◦ σiU )(µ∇ ) = Ad (h) (µ ) . G |σ U i i i

ASTÉRISQUE 432

CHAPTER 2 OPERS ON A FAMILY OF POINTED STABLE CURVES

In this chapter, we study a general theory of (g, ℏ)-opers on pointed stable curves and their moduli for a semisimple Lie algebra g. Compared to previous studies on opers, our formulation would be fairly general because it especially includes the case of nodal curves. The goal of this chapter is to prove the representability of the moduli category classifying (g, ℏ)-opers. This result generalizes results by R. C. Gunning (cf. [31]), A. Beilinson-V. Drinfeld (cf. [13]), and S. Mochizuki (cf. [59]). Also, a key ingredient in studying opers on a pointed curve is an invariant called radius (cf. Definition 2.29). The radius of a given (g, ℏ)-oper at a marked point is defined as an element of the adjoint quotient of g and induced by the monodromy. We will also prove the representability of the moduli category of (g, ℏ)-opers of prescribed radii. Notation and Conventions. — Let k be a perfect field and (G, T) a split semisimple algebraic k-group of adjoint type, where T denotes a maximal torus of G. Let us fix a Borel subgroup B of G defined over k containing T. Write N := [B, B], i.e., the unipotent radical of B. Also, write g, b, n, and t for the Lie algebras of G, B, N, and T, respectively (hence t ⊆ n ⊆ b ⊆ g). 2.1. (g, ℏ)-opers on a log curve 2.1.1. The Cartan decomposition. — Let Γ ⊆ Hom(T, Gm ) (:= the additive group of all characters of T) denote the set of simple roots in B with respect to T. For each β ∈ Hom(T, Gm ), we set  (158) gβ := x ∈ g AdG (t)(x) = β(t) · x for all t ∈ T . Recall that there exists a unique Lie algebra grading M (159) g= gj j∈Z

(i.e., [gj1 , gj2 ] ⊆ gj1 +j2 for j1 , j2 ∈ Z) satisfying the following conditions: — gj = g−j = 0 for all j > rk(g), where rk(g) denotes the rank of g; L L — g0 = t, g1 = α∈Γ gα and g−1 = α∈Γ g−α .

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CHAPTER 2. OPERS ON A FAMILY OF POINTED STABLE CURVES

The associated decreasing filtration {gj }j∈Z on g is defined by M (160) gj := gl , l≥j

which is closed under the adjoint action of B and satisfies the following conditions: — g−j = g and gj = 0 for all j > rk(g); — g0 = b, g1 = n, and [gj1 , gj2 ] ⊆ gj1 +j2 for j1 , j2 ∈ Z. 2.1.2. A filtration on a Lie algebroid. — By means of the filtration {gj }j∈Z defined above, we can construct a filtration on the Lie algebroid associated to a G-bundle equipped with a B-reduction. Let S log be an fs log scheme over k, U log a log curve over S log , and πB : EB → U a B-bundle on U . Denote by πG : (EB ×B G =:) EG → U the G-bundle obtained via change of structure group by the inclusion B ,→ G. Hence, EB specifies a B-reduction ∼ of EG . The natural morphism EB → EG yields an isomorphism gEB → gEG . Moreover, it gives the following injective morphism of short exact sequences: 0

/ bE B

(161)

dlog E

f log /T

e ιg/b

ιg/b

0

/ TU log /S log

B

EB

/S log

 / gE G

f log /T EG

≀ id

 /S log

/0

 / TU log /S log

dlog E

/ 0.

G

Each g (⊆ g) is closed under the adjoint action of B, so induces a subbundle gjEB of gEB (= gEG ). The collection {gjEB }j∈Z forms a decreasing filtration on gEB . Moreover, for each j ∈ Z, we write   fj f log (162) T := Im(e ιg/b ) + gjEB ⊆ T . log log log E /S j

/S

EG

fj

Then, the collection {T

log

EG

G

/S log

}j∈Z forms a decreasing filtration on the vector bun-

fj

f log f0 dle T . The equality TE log /S log = T holds for every j ≥ 0, and the EG /S log E log /S log f natural inclusion gEB (= gEG ) ,→ TE log /S log induces an isomorphism G

(163)

j

∼

fj−1 gEB /gEB → T log j−1

EG

/S log

fj /T log EG

/S log

for every j ≤ 0. Note that each g−α (α ∈ Γ) is closed under the B-action defined Ad

G by the composite B ↠ (B/N =) T −−→ Aut(g−α ). Hence, the decomposition (g−1 =) L −1 0 −α g /g = α∈Γ g gives a canonical decomposition M f−1 f0log (164) T /T = g−α . log log log EB E /S

EG

ASTÉRISQUE 432

/S

G

α∈Γ

49

2.1. (g, ℏ)-OPERS ON A LOG CURVE

2.1.3. The definition of a (g, ℏ)-oper. — We now turn to define a (g, ℏ)-oper on a log curve. Let ℏ be an element of Γ(S, OS ). Definition 2.1. —

(i) Let us consider a pair E ♠ := (EB , ∇)

(165)

consisting of a
B-bundle EB on U and an ℏ-S log -connection ∇ on the G-bundle EG := EB ×B G induced by EB . Then, we shall say that E ♠ is a (g, ℏ)-oper on U log /S log if it satisfies the following two conditions: f−1 — ∇(TU log /S log ) ⊆ T log EG

/S log

;

— for any α ∈ Γ, the composite (166)

TU log /S log

∇ f−1 −→ T log EG

/S log

f−1 ↠T log EG

/S log

f0log /T ↠ g−α EB E /S log G

is an isomorphism, where the third arrow denotes the natural projection with respect to the decomposition (164). For simplicity, a (g, 1)-oper will be referred to as a g-oper. Next, suppose further that U log /S log arises from a local pointed stable curve U together with a choice of its base (e.g., U log = X log for a pointed stable curve X := (X/S, {σi }i )). Then, we will refer to any (g, ℏ)-oper on U log /S log as a (g, ℏ)-oper on U when there is no fear of confusion. (ii) Let E ♠ := (EB , ∇), E ′ ♠ := (EB′ , ∇′ ) be (g, ℏ)-opers on U log /S log . An isomor∼ phism of (g, ℏ)-opers from E ♠ to E ′♠ is defined as an isomorphism EB → EB′ of B-bundles that induces, via change of structure group by B ,→ G, an isomor∼ phism of ℏ-flat G-bundles (EG , ∇) → (EG′ , ∇′ ). Example 2.2 (The case of g = sl2 ). — Suppose that 2 is invertible in k. In the case of g = sl2 = pgl2 , we can describe the definition of an (sl2 , ℏ)-oper a little simpler. As we will define later (cf. (175)), let T (resp., B) denote the maximal torus (resp., the Borel subgroup) of PGL2 consisting of the images, via GL2 ↠ PGL2 , of invertible diagonal matrices (resp., invertible upper triangular matrices). Then, an (sl2 , ℏ)-oper on U log /S log with respect to the pair of choices (T, B) is given as an ℏ-flat PGL2 -bundle (EPGL2 , ∇) on U log /S log equipped with a B-reduction EB on EPGL2 such that the composite (167)

TU log /S log

∇ f −→ T log E

PGL2 /S

log

f log ↠T E

PGL2 /S

log

f log /T E /S log B

log

is an isomorphism. If we choose a log chart (V, ∂) on U /S log and a local trivial∼ ∼ ization V ×k PGL2 → EPGL2 | of EPGL2 inducing V ×k B → EB , then the element V ∇g (∂) ∈ pgln (V ) can be represented by a 2 × 2 matrix of the form ∇g (∂) = ( u∗ ∗∗ ), where the ∗’s indicate arbitrary functions on V and u is a nowhere vanishing function, i.e., an element of Gm (V ). We refer to Example 2.7 for the local description of an (sln , ℏ)-oper for a general n.

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CHAPTER 2. OPERS ON A FAMILY OF POINTED STABLE CURVES

2.1.4. A grading of the adjoint bundle. — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper ∼ on U log /S log . Consider the composite isomorphism TU log /S log → g−α (for each α ∈ Γ) EB L displayed in (166) and the decomposition g = j∈Z gj (cf. (159)). Then, one may verify that each subquotient gjEB /gj+1 of the filtration {gjEB }j∈Z is isomorphic to a EB dirent sum of finite copies of Ω⊗j , i.e., U log /S log (168)

⊗j ∼ gjEB /gj+1 = ΩU log /S log ⊗k gj . EB

⊕rk(g) In particular, we have g0EB /g1EB ∼ . = OU

2.1.5. Pull-back and base-change. — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper on U log /S log , and suppose that we are given an étale U -scheme u : U ′ → U . Then, U ′ is equipped with a log structure pulled-back from U log . The resulting log scheme, which we denote by U ′ log , defines a log curve over S log . The pair of pull-backs (169)

u∗ (E ♠ ) := (u∗ (EB ), u∗ (∇))

(cf. (51), (80)) specifies a (g, ℏ)-oper on U ′ log /S log . Conversely, the formation of a (g, ℏ)-oper has descent with respect to the étale topology on U . Also, let S ′ log be an fs log scheme and slog : S ′ log → S log a morphism of log schemes. Then, the pair of base-changes (170)

s∗ (E ♠ ) := (s∗ (E ), s∗ (∇))

(cf. (52), (82)) forms a (g, s∗ (ℏ))-oper on the log curve (S ′ log ×S log U log )/S ′ log . 2.1.6. Change of parameter. — Let E ♠ := (EB , ∇) be as above and let c ∈ Γ(S, OS× ). Then, the pair (171)

c · E ♠ := (EB , c · ∇)

(cf. (98)) specifies a (g, c · ℏ)-oper on U log /S log .

2.2. Local description and automorphisms of a (g, ℏ)-oper Hereinafter, let S log be an fs log scheme over k, U log a log curve over S log , and ℏ an element of Γ(S, OS ). Also, let us assume in this section that either “ char(k) = 0” or “ char(k) > 2hG ” or “ G = PGLn with 1 < n < char(k)” is fulfilled. 2.2.1. Generators of root spaces. — We shall fix a collection of data (172)

↙ q1 := {xα }α∈Γ ,

where each xα denotes a generator of gα . (If we take two such collections, then they are conjugate by an element of T(k). For that reason, the results obtained in the following discussion are essentially independent of the choice of ↙ q1 .) In particular,

ASTÉRISQUE 432

2.2. LOCAL DESCRIPTION AND AUTOMORPHISMS OF A (g, ℏ)-OPER

51

the pair (B, ↙ q1 ) specifies a pinning of the split semisimple group (G, T). We set X X (173) q1 := xα , ρˇ := ω ˇα, α∈Γ

α∈Γ

where ω ˇ α denotes the fundamental coweight of α, considered as an element of t via differentiation. Then, there is a unique collection {yα }α∈Γ , where yα is a generator of g−α , such that if we write X (174) q−1 := yα ∈ g−1 , α∈Γ

then the set {q−1 , 2ˇ ρ, q1 } forms an sl2 -triple. Example 2.3 (Case of G = PGLn ). — Let us consider the case where G = PGLn (hence g = pgln = sln because of the assumption n < p). We set T (resp., B)

(175)

to be the Borel subgroup (resp., the maximal torus) of PGLn defined as the image, via the natural quotient GLn ↠ PGLn , of the group of invertible upper-triangular matrices (resp., invertible diagonal matrices). If Γ denotes, as before, the set of simple roots in B with respect to T , then we can find a unique collection ↙ q1 := {xα }α∈Γ of so that the resulting sl -triple {q ρ, q1 } can be generators in the root spaces slα 2 −1 , 2ˇ n given by     n−1 0 0 ··· 0 0 0 ··· 0 0      0  1 0 · · · 0 0 n−3 0 ··· 0      0  0 1 · · · 0 0 0 n − 5 ··· 0 (176) q−1 =  ρ= ,  , 2ˇ    . . . .. .. .. .. .. ..  ..   . . . . . . . . . . . . .     0 0 0 · · · −(n − 1) 0 0 ··· 1 0 

0

n−1

0

0

···

0

2(n − 2)

0

···

 0  0  q1 =  .  ..   0

0 .. .

0 .. .

0

0

0

···

0

0

0

0

···

3(n − 2) · · · .. .. . .

0



 0   0   . ..  .    n − 1 0

We suppose that T (resp., B; resp., ↙ q1 ) is always selected as the preferred choice among all maximal tori (resp., all Borel subgroups; resp., all collections of generators in gα ’s) whenever we deal with the algebraic group PGLn . 2.2.2. Local (g, ℏ)-opers. — Let us introduce the local version of a (g, ℏ)-oper, and observe (cf. Proposition 2.8) that, after application of a suitable gauge transformation,

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any such object may be “normalized” with respect to the g−1 -components in the matrix form of that connection. Definition 2.4. — A local (g, ℏ)-oper on U log /S log is an ℏ-S log -connection ∇ on the trivial G-bundle U ×k G such that the pair (U ×k B, ∇) forms a (g, ℏ)-oper on U log /S log . Remark 2.5. — It is clear that any (g, ℏ)-oper is obtained étale locally from a local (g, ℏ)-oper because the underlying B-bundle is locally trivial. Let ∇ be a local (g, ℏ)-oper on U log /S log . Then, by the definition of a (g, ℏ)-oper, we have Im(∇g ) ⊆ OU ⊗k g−1 (cf. (77) for the definition of ∇g ) and, for each α ∈ Γ, the composite
∇g (177) TU log /S log −−→ OU ⊗k g−1 ↠ OU ⊗k (g−1 /g0 ) = OU ⊗k g−1 ↠ OU ⊗k g−α is an isomorphism. Definition 2.6. — Let us keep the above notation. Suppose further that there exists a global log chart (U, ∂) on U log /S log (cf. Definition 1.42). Then, we shall say that ∇ is ↙ q1 -prenormal relative to (U, ∂) if, for any α ∈ Γ, the image of ∂ via (177) coincides with 1 ⊗ yα , or equivalently, the composite (178)

TU log /S log

∇g

−−→ OU ⊗k g−1 ↠ OU ⊗k g−1

sends ∂ to 1 ⊗ q−1 . Example 2.7. — Let us consider the case where G = PGLn (together with T , B, and ↙ q1 defined in Example 2.3). In particular, g = pgln = sln . Suppose that there exists a global log chart (U, ∂) of U log /S log . Then, an ℏ-S log -connection ∇ on U ×k PGLn forms a local (g, ℏ)-oper (resp., a local (g, ℏ)-oper that is ↙ q1 -prenormal relative to (U, ∂)) if and only if the element ∇g (∂) of pgln (U ) (= sln (U )) can be represented by an n × n matrix of the form      ∗ ∗ ∗ ··· ∗ ∗ ∗ ∗ ∗ ··· ∗ ∗      ∗ ··· ∗ ∗  u1 ∗ 1 ∗ ∗ · · · ∗ ∗       0 u2 ∗ · · ·  0 1 ∗ · · · ∗ ∗ ∗ ∗      (179)   resp.,   , 0 u3 · · · ∗ ∗  0 0 0 1 · · · ∗ ∗ .     .. .. . . .. ..   . . .  .. .. .. . .  . . .. ..  . . . . . . . .  0

0

0

···

un−1

∗

0

0

0

···

1

∗

where the ∗’s indicate arbitrary functions on U and u1 , . . . , un−1 in the non-resp’d portion are nowhere vanishing functions, i.e., elements in Gm (U ). Proposition 2.8. — Let ∇ be a local (g, ℏ)-oper on U log /S log . Suppose that there exists a global log chart (U, ∂) on U log /S log . Then, there exists a unique U -rational point h : U → T of T such that Lh∗ (∇) is a ↙ q1 -prenormal local (g, ℏ)-oper relative to (U, ∂).

ASTÉRISQUE 432

2.2. LOCAL DESCRIPTION AND AUTOMORPHISMS OF A (g, ℏ)-OPER

53

× Proof. — For each α ∈ Γ, let uα be the element of Γ(U, OU ) (= Gm (U )) such that the image of ∂ via (177) coincides with uα ⊗ yα . Let h : U → T be an arbitrary U -rational f point of T (⊆ G). The automorphism dLG h (cf. (72)) of TU log ×k G/S log obtained from fj the differential of the left-translation Lh preserves the filtration {T } . U log ×k G/S log j∈Z L −α Hence, it induces an automorphism of α∈Γ gU ×k B via the decomposition M f−1 f0log (180) T /T g−α U ×k G/S log = U ×k B U log ×k G/S log α∈Γ −α ∼ (cf. (164)). This automorphism acts on each component g−α ) as the U ×k B (= OU ⊗k g adjoint action by h; this action coincides with the automorphism −α h−1 α ∈ GL(gU ×k B ) (= Gm (U ))
given by multiplication by (−α) ◦ h = α ◦ (h−1 ) : U → Gm . The assignment h 7→ (h−1 α )α∈Γ determines an isomorphism Y ∼ (181) T(U ) → GL(g−α U ×k B ). α∈Γ

It follows that we can find a U -rational point h of T mapped to (u−1 α )α∈Γ by this
isomorphism. Then, the composite (177) with ∇ replaced by Lh∗ (∇) = dLG h ◦∇ sends ∂ to u−1 α · (uα ⊗ yα ) = 1 ⊗ yα . That is to say, this U -rational point h fulfills the requirement. This completes the existence assertion. Since the uniqueness portion follows immediately from the injectivity of (181), we finish the proof of the assertion.

2.2.3. Automorphisms of a (g, ℏ)-oper. — The following proposition was already proved in [13], § 1.3, proposition (or [7], Proposition 2.1), if U log /S log is an unpointed smooth curve over the field of complex numbers C. Even if k has positive characteristic and U is an arbitrary log curve, we can also prove it by the same argument as in loc. cit. thanks to the condition imposed at the beginning of this section. Proposition 2.9. — (Recall that either “ char(k) = 0,” “ char(k) > 2hG ,” or “ G = PGLn with 1 < n < p” is fulfilled.) Any (g, ℏ)-oper on U log /S log does not have nontrivial automorphisms. Proof. — We only consider the case where either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled because the remaining case follows from an entirely similar discussion (using the result in Remark 1.35). Let E ♠ be a (g, ℏ)-oper on U log /S log . We may assume, after possibly replacing U with its covering in the étale topology, that there exist a global log chart (U, ∂) and a local (g, ℏ)-oper ∇ on U log /S log with E ♠ = (U ×k B, ∇). Moreover, by Proposition 2.8 above, it suffices to consider the case where ∇ is ↙ q1 -prenormal relative to (U, ∂). Let us take an automorphism η of E ♠ , i.e., an automorphism of the trivial G-bundle U ×k G which preserves the B-reduction U ×k B and satisfies η∗ (∇) = ∇. Then, we can find a U -rational point h : U → B of B with η = Lh (cf.

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Example 1.7). Let h : U → T denote the composite h

h:U − → B ↠ (B/N =) T. L −α G G Notice that dLh and dLh act on α∈Γ gU ×k B in the same manner. Hence, Lh∗ (∇) turns out to be ↙ q1 -prenormal relative to (U, ∂) because of the assumption dLG h ◦ ∇ (= Lh∗ (∇) = η∗ (∇)) = ∇. It follows from the uniqueness portion of Proposition 2.8 that h = e, i.e., h factors through the inclusion N ,→ B; we abuse notation by writing h for the resulting
morphism U → N. This morphism induces a U -rational point Log(h) : U → n = g1 (cf. (114)). By Corollary 1.34, the equality Lh∗ (∇) = ∇ implies ∞ X 1 (183) ∇g (∂) = −ℏ · dLog(h)(∂) + · ad(Log(h))s (∇g (∂)). s! s=0 (182)

Now, we shall assume that Log(h) ̸= 0. Then, the positive integer  (184) j0 := max j ∈ Z>0 Log(h) ∈ gj (U ) is well-defined. For each j ∈ Z, let Log(h)j : U → gj and ∇g (∂)j ∈ Γ(U, OU ⊗k gj )
1 g respectively be the j-th components of Log(h) L : U → g ⊆ g and ∇ (∂) ∈ Γ(U, OU ⊗k g) with respect to the decomposition g = j∈Z gj . By the definition of j0 , we see that dLog(h)(∂) ∈ Γ(U, OU ⊗k gj0 ). Since ∇g (∂)−1 = 1 ⊗ q−1 and ∇g (∂)j = 0 for j < −1, the section ad(Log(h)j )s (∇g (∂)j ′ ) lies in Γ(U, OU ⊗k gj0 ) if either “s ≥ 2” or “s = 1 and j + j ′ ≥ j0 ” is satisfied. Hence, (183) implies that, in the component Γ(U, OU ⊗k gj0 −1 ) of Γ(U, OU ⊗k g), we have (185)

(186)

ad(Log(h)j0 )(∇g (∂)−1 ) (= −ad(1 ⊗ q−1 )(Log(h)j0 )) = 0.

But, since j0 > 0, the morphism ad(1 ⊗ q−1 ) : OU ⊗k gj0 → OU ⊗k gj0 −1 is injective. It follows that Log(h)j0 = 0, which contradicts the definition of j0 . Thus, Log(h) must ∼ be zero. Because of the bijectivity of Log : N → n, we have h = e, i.e., η coincides ♠ with the identity morphism of E . This completes the proof of the proposition. 2.2.4. The sheaf of (g, ℏ)-opers. — By taking account of the étale descent property of (g, ℏ)-opers, one may define the stack in groupoids (187)

Op g,ℏ,U log /S log

→ Et/U

over the small étale site Et/U on U whose category of sections over an étale U -scheme U ′ is the groupoid of (g, ℏ)-opers on U ′ log /S log , where the log structure on U ′ is obtained by pulling-back that on U log . It follows from Proposition 2.9 that Op g,ℏ,U log /S log is fibered in equivalence relations (cf. [24], Definition 3.39), i.e., specifies a Set-valued sheaf on Et/U . When there is no fear of confusion, we shall write (188)

ASTÉRISQUE 432

Op g,ℏ

:= Op g,ℏ,U log /S log .

55

2.3. THE CASE OF g = sl2

2.3. The case of g = sl2 We here mention some facts about (sl2 , ℏ)-opers used in the subsequent discussion. These facts will be proved in Chapter 4 later, so it should be noted that we are in advance referring to them without giving their proofs. In this section, we suppose that 2 is invertible in k. 2.3.1. — Let G⊙ be the projective linear group over k of rank 2, i.e., G⊙ := PGL2 , and B⊙ (resp., T⊙ ) the Borel subgroup (resp., the maximal torus) of G⊙ defined as the image, via the natural quotient GL2 ↠ G⊙ , of invertible upper-triangular matrices (resp., invertible diagonal matrices). Let us write N⊙ := [B⊙ , B⊙ ]. Also, write g⊙ , b⊙ , n⊙ , and t⊙ for the Lie algebras of G⊙ , B⊙ , N⊙ , and T⊙ , respectively. In particular, ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ ⊙ we have g⊙ = sl2 , g⊙ 1 = n , g0 = t = b /n , and g−1 = g /b . The set of simple ⊙ ⊙ roots Γ in B with respect to T consists exactly of one element α⊙ . We can choose ⊙ the sl2 -triple {q−1 , 2ˇ ρ⊙ , q1⊙ } given by (189) q1⊙

(= xα⊙ ) :=

0 0

! 1 , ρˇ⊙ (= ω ˇ α⊙ ) := 0

!

1/2

0

0

−1/2

,

⊙ q−1

(= yα⊙ ) :=

0 1

! 0 . 0

In particular, we obtain ↙ q1 ⊙ := {xα⊙ } as in (172) for g⊙ . 2.3.2. Results in Chap. 4: Part I. — Let us write (190)

†

EB⊙ ,ℏ,U log /S log ,

or simply † EB⊙ ,

for the B⊙ -bundle on U constructed in (605) for n = 2. The G⊙ -bundle induced by † EB⊙ via change of structure group by the inclusion B⊙ ,→ G⊙ is denoted by (191)

†

EG⊙ ,ℏ,U log /S log ,

or simply † EG⊙ .

For each log chart (V, ∂) on U log /S log , there exists a specific trivialization (192)

∼

trivB⊙ ,(V,∂) : V ×k B⊙ → † EB⊙ |

V

(cf. (606)) of † EB⊙ restricted to V . The formations of both † EB⊙ and trivB⊙ ,(V,∂) are functorial with respect to pull-back to étale U -schemes, as well as base-change to fs S log -log schemes. Also, let us suppose that U log /S log comes from a local r-pointed stable curve U := (U/S, {σiU : Si → U }) over S with r > 0 together with a choice of its base X . Then, for each i = 1, . . . , r, there exists a canonical trivialization (193)

∼

trivB⊙ ,σiU : Si ×k B⊙ → σiU ∗ († EB⊙ )

of the B⊙ -bundle σiU ∗ († EB⊙ ) on Si (cf. Remark 4.51). 2.3.3. Results in Chap. 4: Part II. — In Definition 4.53, we introduce the notion of a normal (g⊙ , ℏ)-oper, which is a (g, ℏ)-oper of the form († EB⊙ , ∇) for a suitable ℏ-S log -connection ∇ on † EG⊙ . The formation of a normal (g⊙ , ℏ)-oper is closed under

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CHAPTER 2. OPERS ON A FAMILY OF POINTED STABLE CURVES

pull-back to étale U -schemes, as well as base-change to fs S log -log schemes. Also, it follows from Proposition 4.55 that for any (g⊙ , ℏ)-oper E ♠ on U log /S log , there exists uniquely a pair († E ♠ , norE ♠ )

(194)

∼

consisting of a normal (g⊙ , ℏ)-oper on U log /S log and an isomorphism norE ♠ : † E ♠ → E ♠ of (g⊙ , ℏ)-opers. 2.3.4. Results in Chap. 4: Part III. — For j = −1, 0, 1, there exists a canonical isomor∼ ⊗j phism (g⊙ j )† EB⊙ → ΩU log /S log of OU -modules (cf. (617)). In particular, the case of j = 1 gives an isomorphism (195)

⊙

Hom OU (TU log /S log , (n

∼

)† EB⊙ ) → Ω⊗2 . U log /S log

Now, let E ♠ := († EB⊙ , ∇) be a normal (g⊙ , ℏ)-oper on U log /S log and let R ∈ Γ(U, Ω⊗2 ). By identifying R with an OU -linear morphism U log /S log   ⊙ f log TU log /S log → (n )† E ⊙ ⊆ T †E log /S B G⊙

via (195), we obtain the sum f log ∇ + R : TU log /S log → T †E /S log ,

(196)

G⊙

which defines an ℏ-S resulting pair (197)

log

†

-connection on EG⊙ . According to the discussion in § 4.8.3, the ♠ E+R := († EB⊙ , ∇ + R)

♠ forms a normal (g⊙ , ℏ)-oper on U log /S log , and the assignment (E ♠ , R) 7→ E+R is functorial with respect to pull-back to étale U -schemes, as well as base-change to fs S log -log -torsor schemes. Moreover, this assignment determines a structure of Ω⊗2 U log /S log

(198)

Op g⊙ ,ℏ

× Ω⊗2 → Op g⊙ ,ℏ U log /S log

on Op g⊙ ,ℏ , where Ω⊗2 are considered as a sheaf of abelian groups on Et/U in U log /S log an evident manner. In particular, the stalk of Op g⊙ ,ℏ at any geometric point of U is nonempty. 2.4. The G-bundle † EG and the vector bundle † Vg Next, let us go back to the case of a general g. In the rest of this chapter, we assume, unless otherwise stated, that either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled. 2.4.1. Change of structure group arising from an sl2 -triple. — The sl2 -triple {q−1 , 2ˇ ρ, q1 } associated to the fixed collection ↙ q1 determines an injection (199)

ASTÉRISQUE 432

ιg : g⊙ ,→ g,

2.4. THE G-BUNDLE

†

EG

AND THE VECTOR BUNDLE

†

Vg

57

⊙ i.e., ιg satisfies ιg (q1⊙ ) = q1 , ιg (2ˇ ρ⊙ ) = 2ˇ ρ, and ιg (q−1 ) = q−1 . In particular, we have ιg (b⊙ ) ⊆ b.

Proposition 2.10. — There exists a unique injective morphism ιB : B⊙ ,→ B

(200)

of algebraic k-groups which satisfies ιB (T⊙ ) ⊆ T, ιB (N⊙ ) ⊆ N and whose differential at the identity element e ∈ B⊙ coincides with the restriction ιg | ⊙ : b⊙ ,→ b of ιg . b

Proof. — Note that the composite (201)

ιg | ⊙ Log Exp ιN : N⊙ −−→ n⊙ −−−n−→ n −−→ N

becomes a morphism of algebraic k-groups. Also, the following composite forms a morphism of algebraic k-groups: Y ∼ ∼ ιT : T⊙ → GL(n⊙ ) ,→ (202) GL(gα ) → T, α∈Γ

where — the first arrow denotes the isomorphism which, to any h ∈ T⊙ , assigns the automorphism of n⊙ given by multiplication by α⊙ (h) ∈ Gm ; — the second arrow denotes the diagonal embedding under the natural identifications GL(n⊙ ) = GL(gα ) (= Gm ); ∼ Q α — the third arrow denotes the inverse of the isomorphism T → α∈Γ GL(g ) Q which, to any h ∈ T, assigns the collection (hα )α∈Γ ∈ α∈Γ GL(gα ) consisting of the automorphisms hα (α ∈ Γ) given by multiplication by α ◦ h ∈ Gm . In the following, we shall construct a morphism ιB : B⊙ → B extending both the morphisms ιN and ιT . Let us take t ∈ T⊙ and n ∈ N⊙ . Write nt := t−1 · n · t ∈ N⊙ . Also, let us choose 2 × 2 matrices of the forms ( a0 01 ) and ( 10 1b ) representing t and n respec
−1 tively. Then, nt may be represented by the matrix 10 a 1 ·b , and the images Log(n)
−1 and Log(nt ) in n⊙ may be represented by ( 00 0b ) and 00 a 0 ·b respectively. Now, let α ∈ Hom(T, Gm ) be a root with gα ∈ gj (for some j ∈ Z) and let v ∈ gα . By identifying elements of N with automorphisms of g via the inclusion N ,→ (G ∼ =) Aut0 (g), we have a sequence of equalities ! ∞ X 1 −1 s t (ιT (t) · ιN (n ))(v) = ιT (t) (203) · ad(a · b · q1 ) (v) s! s=0 =

∞ X 1 s+j ·a · ad(a−1 · b · q1 )s (v) s! s=0

=

∞ X 1 · ad(b · q1 )s (aj · v) s! s=0

= ιN (n)(aj · v)

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CHAPTER 2. OPERS ON A FAMILY OF POINTED STABLE CURVES

= ιN (n) · ιT (t)(v), where the second equality follows from the fact that ad(a−1 · b · q1 )s (v) ∈ gs+j . This implies that ιT (t) · ιN (nt ) = ιN (n) · ιT (t) for any t ∈ T⊙ , n ∈ N⊙ . Thus, the assignment (n, t) 7→ ιN (n) · ιT (t) determines a well-defined morphism ιB : (N⊙ ⋊ T⊙ =) B⊙ → B of algebraic k-groups, extending both ιT and ιN . It is clear that ιB is a unique morphism satisfying the required conditions. This complete the proof of Proposition 2.10.

2.4.2. A morphism from Op g⊙ ,ℏ to Op g,ℏ . — Let E⊙♠ := (EB⊙ , ∇) be a (g⊙ , ℏ)-oper ⊙

⊙

on U log /S log . We shall write EG⊙ := EB⊙ ×B G⊙ , EB := EB⊙ ×B ,ιB B, and EG := EB ×B G, where ιB denotes the morphism B⊙ ,→ B resulting from Proposition 2.10. Observe that the following diagram is commutative: f log /T

inclusion

b⊙ E ⊙

E ⊙ B

B

(b)

y

inclusion /

bEB (= bEB⊙ )

ιg⊙ /b⊙

(204)

x f log T EB

/S log

(a)

/S log

 g⊙ E ⊙

ιg/b

B

 w gEG ,

(c)

where log induced by ιB ; — (a) is obtained by differentiating the inclusion EBlog ⊙ ,→ EB

— (b) denotes the restriction of (a), which coincides with the restriction ιg | ⊙ : b⊙ ,→ b of ιg twisted by EB⊙ ; b — (c) denotes the injection ιg : g⊙ ,→ g twisted by EB⊙ . On the other hand, the following two commutative square diagrams are cocartesian: b⊙ E ⊙

inclusion

(205)

f log /T

bEB

/S log

E ⊙ B

B

ιg⊙ /b⊙

e ιg⊙ /b⊙



⊙

gE

G⊙

inclusion

f log /T

f log /T

e ιg/b



,

gEG

inclusion

f log /T

Hence, the diagram displayed in (204) induces an OU -linear injection (206)

f log f dιEB⊙ : T E /S log → TE log /S log . G⊙

ASTÉRISQUE 432

/S log

EB

ιg/b



E ⊙ /S log G

inclusion

G

EG

 /S log .

2.4. THE G-BUNDLE

†

EG

AND THE VECTOR BUNDLE

†

59

Vg

By taking account of the local description of Lie bracket operations (cf. (71)), we see that dιEB⊙ becomes a morphism of Lie algebroids. It follows that the composite dι

E ⊙ ∇ f B f ιG∗ (∇) : TU log /S log −→ T log E /S log −−−→ TE log /S log

(207)

G⊙

specifies an ℏ-S

log

G

-connection on EG and the pair ιG∗ (E⊙♠ ) := (EB , ιG∗ (∇))

(208)

forms a (g, ℏ)-oper on U log /S log . The assignment E⊙♠ 7→ ιG∗ (E⊙♠ ) is functorial with respect to pull-back to étale U -schemes, so it determines a morphism of sheaves Op

ιg : Op g⊙ ,ℏ → Op g,ℏ .

(209)

In particular, the stalk of Op g,ℏ at any geometric point of U is nonempty because this is true for g = g⊙ , as mentioned before. 2.4.3. The G-bundle † EG . — By change of structure group via the morphism ιB (resp., the composite of ιB and the inclusion B ,→ G), the B⊙ -bundle † EB⊙ induces a B-bundle (resp., a G-bundle) on U , which we denote by
† † † † (210) EB,ℏ,U log /S log or simply EB resp., EG,ℏ,U log /S log or simply EG . If (V, ∂) is a log chart on U log /S log , then the trivialization trivB⊙ ,(V,∂) induces a trivialization ∼

trivG,(V,∂) : V ×k G → † EG |

(211)

V

†

of the G-bundle EG | on V . The trivial B-reduction V ×k B of V ×k G is compatible V with † EB | via this trivialization. V Also, let us suppose that U log /S log arises from a local r(> 0)-pointed stable curve U := (U/S, {σi }ri=1 ) over S together with a choice of its base X . Then, for each i = 1, . . . , r, we have a canonical trivialization (212)

∼

trivG,σiU : Si ×k G → σiU ∗ († EG )

induced by trivB⊙ ,σiU . Proposition 2.11. — Let (V, ∂) be a log chart on U log /S log and let a ∈ Γ(V, OV× ) (hence, the pair (V, a · ∂) forms a log chart on U log /S log ). Then, the composite automorphism (213)

triv−1 G,(V,∂) ◦ trivG,(V,a·∂)

of the trivial G-bundle V ×k G coincides with the left-translation Lh by h = ιB (( 10 a0 )) ∈ G(V ). In particular, the OV -linear automorphism of OV ⊗k g arising from differentiating triv−1 G,(V,∂) ◦ trivG,(V,a·∂) coincides with the automorphism given by assigning 1 ⊗ v 7→ a−j ⊗ v for any v ∈ gj with j ∈ Z. Proof. — The assertion follows immediately from the definitions of trivG,(V,∂) .

†

EG⊙

and

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2.4.4. The space gad(q1 ) . — Let us consider the space gad(q1 ) of ad(q1 )-invariants, i.e.,  (214) gad(q1 ) := x ∈ g ad(q1 )(x) = 0 . The k-vectorL space gad(q1 ) has dimension equal to rk(g) (cf. [13], § 3.4). The decomposition g = j∈Z gj (cf. (159)) yields a decomposition M ad(q ) (215) gad(q1 ) = gj 1 j∈Z ad(q1 )

on g . Since {q−1 , 2ˇ ρ, q1 } forms an sl2 -triple and either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled, the j-th component gj of g (for each j ∈ Z) can be decomposed into the direct sum ad(q1 )

gj = ad(q−1 )(gj+1 ) ⊕ gj

(216)

(cf. [64], the comment after Chap. 1, § 1.2, Lemme 1.2.1). The subspace gad(q1 ) of g ad(q ) and its components gj 1 for all j are closed under the adjoint action of B⊙ via ιB ad(q )

and satisfies that gj 1 = 0 if j ≤ 0 (cf. [13], § 3.4). If h is a k-vector space equipped with a Gm -action (h, v) 7→ h v (h ∈ Gm , v ∈ h), then we shall write h(1) for the k-vector space h equipped with the Gm -action given by (h, v) 7→ h · h v. Let us identify B⊙ /N⊙ with Gm via the adjoint action ∼ B⊙ /N⊙ → GL(n⊙ ) = Gm . The B⊙ -action on gad(q1 ) factors through the surjection B⊙ ↠ Gm , so it induces a Gm -action on gad(q1 ) . In particular, we obtain a k-vector space gad(q1 ) (1) equipped with a Gm -action.

2.4.5. The vector bundle † Vg . — By the above discussion, we obtain a vector bunad(q ) dle g† E ⊙1 on U of rank equal to rk(g). Write B
ad(q1 ) † † . (217) Vg,U log /S log or simply Vg := ΩU log /S log ⊗ g† E ⊙ B

If Ω× denotes the Gm -bundle corresponding to ΩU log /S log , then † Vg is isomorU log /S log phic to the vector bundle (Ω× )×Gm gad(q1 ) (1). For example, † Vg⊙ is canonically U log /S log isomorphic to Ω⊗2 (cf. § 2.3.4). U log /S log L The decomposition (215) induces a decomposition Vg = j∈Z Vg,j , which gives a L decreasing filtration {Vgj }j∈Z on Vg by putting Vgj := j ′ ≥j Vg,j ′ . According to the discussion in § 2.1.4, we have   ⊗(j+1) ad(q ) Gm ad(q1 ) † (218) Vg,j ∼ ) × g (1) = ΩU log /S log ⊗k gj 1 = (Ω× log log j U /S for every j ∈ Z. Next, the B⊙ -equivariant inclusion gad(q1 ) ,→ g yields an OU -linear injection   (219) ς : † Vg ,→ ΩU log /S log ⊗ g† EB⊙ = ΩU log /S log ⊗ g† EB .

ASTÉRISQUE 432

2.5. ↙q1-NORMALITY AND A TORSOR STRUCTURE ON THE SHEAF OF (g, ℏ)-OPERS

61


Also, since q1 = ιg (q1⊙ ) belongs to gad(q1 ) , the injection ιg restricts to an injec⊙ tion (g⊙ )ad(q1 ) ,→ gad(q1 ) , and hence, induces an OU -linear injection (220)

⊗2 † ιV g : ΩU log /S log ,→ Vg

under the natural identification Ω⊗2 = † Vg⊙ . By using ς (resp., ιV g ), we shall U log /S log regard † Vg (resp., Ω⊗2 ) as an OU -submodule of ΩU log /S log ⊗ g† EB (resp., † Vg ). U log /S log Remark 2.12. — Note that the B⊙ -action on gad(q1 ) factors through the quotient ad(q ) ad(q ) B⊙ ↠ (B⊙ /N⊙ =) T⊙ . Hence, we have g† E ⊙1 = g† E 1×B⊙ T⊙ . On the other hand, as B

B⊙

⊙

mentioned in Remark 4.52, the formation of the T ⊙ -bundle † EB⊙ ×B T⊙ does not depend on the choice of ℏ. It follows that the formation of the vector bundle † Vg does not depend on the choice of ℏ. To be precise, if † Vg′ denotes the vector bundle “ † Vg ” defined by using another ℏ′ instead of ℏ, then (by means of (608) for n = 2) we have ∼ a canonical isomorphism † Vg → † Vg′ .

2.5. ↙ q1 -normality and a torsor structure on the sheaf of (g, ℏ)-opers q1 -normal (g, ℏ)-oper 2.5.1. ↙ q1 -normal (g, ℏ)-opers. — We shall define the notions of a ↙ formulated in the local and global situations respectively. Definition 2.13 (Local case). — Suppose that there exists a global log chart (U, ∂) on U log /S log . Let ∇ be a local (g, ℏ)-oper on U log /S log . Then, we shall say that ∇ is ↙ q1 -normal relative to (U, ∂) if the equality (221)

∇g (∂) = v + 1 ⊗ q−1

holds for some v ∈ Γ(U, OU ⊗k gad(q1 ) ). (In particular, a ↙ q1 -normal local (g, ℏ)-oper relative to (U, ∂) is ↙ q1 -prenormal relative to the same log chart). Definition 2.14 (Global case). — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper on U log /S log . Then, we shall say that E ♠ is ↙ q1 -normal if EB = † EB and, for any log chart (V, ∂) ∗ log log on U /S , the pull-back trivG,(V,∂) (∇), forming a local (g, ℏ)-oper on V log /S log , is ↙ q1 -normal relative to (V, ∂). Remark 2.15 (Different choice of ↙ q1 ). — Let ↙ q1 ′ := {x′α }α∈Γ be another collection of α generators in g ’s. Note that there exists an element h ∈ T with AdG (h)(xα ) = x′α for every α ∈ Γ. Let ι′B denote the morphism resulting from Proposition 2.10 where “↙ q1 ” is replaced by ↙ q1 ′ . Then, the equality Innh ◦ιB = ι′B holds. This implies that if † EB′ denotes the B-bundle obtained by using ι′G in the same manner as † EB , then it is canonically isomorphic to † EB ×B,Innh B. Moreover, for a ↙ q1 -normal (g, ℏ)-oper E ♠ := († EB , ∇), ♠ † ′ the (g, ℏ)-oper Innh∗ (E ) := ( EB , Innh∗ (∇)) turns out to be ↙ q1 ′ -normal. Thus, the notion of ↙ q1 -normality is independent of the choice of ↙ q1 up to gauge transformation by an element of T.

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Remark 2.16 (The case of g = sl2 ). — It is clear that a (g⊙ , ℏ)-oper is ↙ q1 -normal if and only if it is normal in the sense of Definition 4.53. Remark 2.17. — Suppose that there exists a global log chart (U, ∂) on U log /S log . Then, a local (g, ℏ)-oper ∇ is ↙ q1 -normal relative to (U, ∂) (in the sense of Definition 2.13) if and only if the pair (U ×k B, ∇) becomes a ↙ q1 -normal (g, ℏ)-oper (in the sense of Definition 2.14) under the identification U ×k B = † EB given by trivG,(U,∂) . Proposition 2.18. — Let E⊙♠ be a ↙ q1 ⊙ -normal (g⊙ , ℏ)-oper on U log /S log . Then, the ♠ (g, ℏ)-oper ιG∗ (E⊙ ) (cf. (208)) is ↙ q1 -normal. Proof. — The assertion follows from the definitions of ιB and ↙ q1 -normality. 2.5.2. ↙ q1 -normalization. — The following proposition asserts that every isomorphism class of a (g, ℏ)-oper contains exactly one ↙ q1 -normal (g, ℏ)-oper. Proposition 2.19. — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper on U log /S log . Then, there exists uniquely a pair († E ♠ , norE ♠ )

(222)

consisting of a ↙ q1 -normal (g, ℏ)-oper † E ♠ on U log /S log and an isomorphism † ♠ ∼ ♠ norE ♠ : E → E of (g, ℏ)-opers. Moreover, the well-defined assignment E ♠ 7→ († E ♠ , norE ♠ ) commutes with pull-back to étale U -schemes, as well as basechange to fs S log -log schemes. Proof. — Since Op g,ℏ is a sheaf (cf. § 2.2.4), the statement is of local nature. Thus, after possibly pulling-back E ♠ to an étale U -scheme, we may assume that EB = U ×k B and there exists a global log chart (U, ∂) on U log /S log (hence † EG can be identified with U ×k G by means of trivG,(U,∂) ). Moreover, by Proposition 2.8, it suffices to prove the assertion of the case where the local (g, ℏ)-oper ∇ is ↙ q1 -prenormal relative to (U, ∂). For each ℏ-S log -connection ∇0 on U ×k G, we set n o N (∇0 ) := max j ∈ Z ∇g0 (∂) − 1 ⊗ q−1 ∈ Γ(U, OU ⊗k (gad(q1 ) + gj )) , (223) where N (∇0 ) := ∞ if ∇g0 (∂) − 1 ⊗ q−1 ∈ Γ(U, OU ⊗k gad(q1 ) ). Assume now that j0 := N (∇) < ∞. The integer j0 is positive and satisfies the equalities ad(q ) gj0 = ad(q−1 )(gj0 +1 ) ⊕ gj0 1 (cf. (216)) and gj0 +1 ∩ Ker(ad(q−1 )) = 0. Hence, the composite (224)

ad(q−1 )

inclusion

gj0 +1 −−−−−→ gj0 −−−−−→ gj0 → (gad(q1 ) + gj0 )/(gad(q1 ) + gj0 +1 )

turns out to be an isomorphism. Let v ∈ Γ(U, OU ⊗k gj0 +1 ) (= gj0 +1 (U )) be the image of ∇g (∂) − 1 ⊗ q−1 via the composite (225)

OU

⊗k (gad(q1 ) + gj0 ) ↠ OU ⊗k (gad(q1 ) + gj0 )/(gad(q1 ) + gj0 +1 ) ∼

→ OU ⊗k gj0 +1 ,

ASTÉRISQUE 432

2.5. ↙q1-NORMALITY AND A TORSOR STRUCTURE ON THE SHEAF OF (g, ℏ)-OPERS

63

where the second arrow is the isomorphism induced by (the inverse of) (224). Since gj0 +1 is contained in n, the U -rational point h := Exp(v) : U → N (cf. (113) for the definition of Exp) can be defined. By Corollary 1.34, we have ∞ X 1 g (226) Lh∗ (∇) (∂) = −ℏ · dLog(h)(∂) + · ad(Log(h))s (∇g (∂)) s! s=0 = −ℏ · dv(∂) +

∞ X 1 · ad(v)s (∇g (∂)). s! s=0

Both the sections −ℏ · dv(∂) and ad(v)s (∇g (∂)) (s ≥ 2) belong to Γ(U, OU ⊗k gj0 +1 ). Hence, modulo Γ(U, OU ⊗k (gad(q1 ) +gj0 +1 )), we obtain from (226) the following equalities: (227)

Lh∗ (∇)g (∂) ≡ ∇g (∂) + ad(v)(∇g (∂)) ≡ ∇g (∂) − ad(∇g (∂))(v) ≡ ∇g (∂) − ad(1 ⊗ q−1 )(v) ≡ ∇g (∂) − (∇g (∂) − 1 ⊗ q−1 ) ≡ 1 ⊗ q−1 ,

where the third “≡” follows from the condition that ∇ is ↙ q1 -prenormal relative to (U, ∂), and the fourth “≡” follows from the definition of v. This implies that ∇′ := Lh∗ (∇) is a local (g, ℏ)-oper (which is ↙ q1 -renormal relative to (U, ∂)) satisfying the inequality (j0 =) N (∇) < N (∇′ ). Moreover, the left-translation Lh−1 by h−1 ∼ specifies an isomorphism (U ×k B, ∇′ ) → (U ×k B, ∇) of (g, ℏ)-opers. By applying successively the above construction of ∇′ starting from ∇, one obtains a local (g, ℏ)-oper † ∇ on U log /S log with N († ∇) = ∞ (i.e., ↙ q1 -normal) together with an iso∼ morphism (U ×k B, † ∇) → (U ×k B, ∇) of (g, ℏ)-opers. This proves the existence assertion. The uniqueness and the functoriality follow from the above discussion, so we finish the proof of Proposition 2.19. Corollary 2.20. — The underlying B-bundle of any (g, ℏ)-oper on U log /S log may be trivialized Zariski locally on U . Proof. — The assertion follows from Proposition 2.19 and the existence of trivial∼ izations trivG,(V,∂) | : U ×k B → † EB | defined for various log charts (V, ∂) such U ×k B V that V are open subschemes of U . Corollary 2.21. — Suppose that there exists a global log chart (U, ∂) on U log /S log . Let us consider, in the natural manner, OU as a sheaf on Et/U . Given an étale U -scheme V and an element v ∈ Γ(V, OU ⊗k gad(q1 ) ), we shall write ∇v for the ℏ-S log -connection on V ×k G determined by the condition that ∇gv (∂) = 1⊗q−1 +v. Then, the assignment

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v 7→ (V ×k G, ∇v ) defines an isomorphism ∼

Op

triv(U,∂) : OU ⊗k gad(q1 ) → Op g,ℏ

(228) of étale sheaves on U .

Proof. — The assertion is a direct consequence of Proposition 2.19.

2.5.3. A

†

Vg -torsor

ΩU log /S log ⊗ b† EB

structure on Op g,ℏ . — Let us take a global section R of   f log and regard it as an OU -linear morphism TU log /S log → b† EB ⊆ T †E /S log .

Then, for a ↙ q1 -normal (g, ℏ)-oper E ♠ := († EB , ∇) on U log /S log , the pair

G

♠ E+R := († EB , ∇ + R)

(229)

♠ forms a (g, ℏ)-oper. One may verify that E+R is ↙ q1 -normal if and only if R lies ♠ † ↙ in Γ(U, Vg ). Also, any q1 -normal (g, ℏ)-oper E on U log /S log may be obtained ♠ in this way, i.e., coincides with E+R for some R ∈ Γ(U, † Vg ). It follows that the log set of ↙ q1 -normal (g, ℏ)-opers on U /S log , identified with the set Op g,ℏ (U ) because of Proposition 2.19, forms a torsor modeled on Γ(U, † Vg ). This torsor structure is compatible with pull-back to étale U -schemes, as well as base-change to fs S log -log schemes. In particular, by regarding † Vg as a sheaf on Et/U , we obtain a structure of † Vg -torsor

(230)

Op g,ℏ

× † Vg → Op g,ℏ

on the sheaf Op g,ℏ . By construction, the following diagram is commutative: Op g⊙ ,ℏ

(231)

(198)

× Ω⊗2 U log /S log

/ Op ⊙ g ,ℏ

Op

Op

ιg ×ιV g

Op g,ℏ

ιg

 × † Vg

(230)

 / Op . g,ℏ

The following assertion follows from the above discussion together with the fact -torsor with respect to the action given by (198). that Op g⊙ ,ℏ is an Ω⊗2 U log /S log Proposition 2.22. — The sheaf Op g,ℏ may be represented by a relative affine space over U modeled on V(† Vg ). Moreover, the morphism of sheaves (232)

Op g ,ℏ ⊙ Op

Ω⊗2 log

×

U

/S log

†

Vg

→ Op g,ℏ

arising from ιg and (230) together with the commutativity of (231) is an isomorphism.

ASTÉRISQUE 432

2.6. (g, ℏ)-OPERS ON POINTED STABLE CURVES

65

2.6. (g, ℏ)-opers on pointed stable curves 2.6.1. — Let r be an integer ≥ 0, S a k-scheme, and U = (f : U → S, {σiU : Si → U }ri=1 ) a local r-pointed stable curve over S. By choosing a base for U , we obtain log structures on U and S, by which U/S extends to a log curve U log /S log . In this situation, we occasionally use the notation † Vg,U to denote the vector bundle † Vg,U log /S log when  P  ad(q ) there is no fear of confusion. Since dim(gad(q1 ) ) = j∈Z dim(gj 1 ) = rk(g), we have M ∗ (233) σiU († Vg ) = σiU ∗ († Vg,j ) j∈Z (218)

∼ =

M

⊗(j+1)

ad(q1 )

σiU ∗ (ΩU log /S log ) ⊗k gj

j∈Z

∼ =

M

OSi

ad(q1 )

⊗k gj

j∈Z ⊕rk(g) ∼ , = OSi

∼ where the penultimate “ ∼ =” follows from the residue maps ResU ,i : σiU ∗ (ΩU log /S log ) → OSi (cf. (141)).

2.6.2. Cohomology of † Vg . — Let us consider the case of a (non-local) pointed stable curve. Let (g, r) be a pair of nonnegative integers with 2g − 2 + r > 0, S a k-scheme, and X := (f : X → S, {σi }ri=1 ) an r-pointed stable curve of genus g over S. We shall write (234) 2g − 2 + r r 2g − 2 + r r ℵ(g) := · dim(g) + · rk(g), c ℵ(g) := · dim(g) − · rk(g). 2 2 2 2 Both ℵ(g) and c ℵ(g) are verified to be integers. Proposition 2.23. — Recall (cf. (140)) that DX denotes the relative effective divisor on X defined as the union of the images of the σi ’s. Then, the following assertions hold: (i) R1 f∗ († Vg,X ) = R1 f∗ († Vg,X (−DX )) = 0. (ii) f∗ († Vg,X ) (resp., f∗ († Vg,X (−DX ))) is a vector bundle on S of rank ℵ(g) (resp., c ℵ(g)). Moreover, if s : S ′ → S is a morphism of k-schemes, then the natural S ′ -morphism (235)

S ′ ×S V(f∗ († Vg,X )) → V((idS ′ × f )∗ († Vg,X ′ ))
resp., S ′ ×S V(f∗ († Vg,X (−DX ))) → V((idS ′ × f )∗ († Vg,X ′ (−DX ′ ))) , where X ′ := S ′ ×S X , is an isomorphism. ad(q )

Proof. — Because of (218) and the fact that gj 1 = 0 for j ≤ 0, we see that † Vg and † Vg (−DX ) are isomorphic to direct sums of Ω⊗l ’s and Ω⊗l (−DX )’s X log /S log X log /S log

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(for various l ≥ 2) respectively. But, the equalities (236)

R1 f∗ (Ω⊗l ) = R1 f∗ (Ω⊗l (−DX )) = 0 X log /S log X log /S log

hold for any l ≥ 2. It follows that (237)

R1 f∗ († Vg,X ) = R1 f∗ († Vg,X (−DX )) = 0,

thus completing the proof of assertion (i). Moreover, (237) implies (cf. [32], Chap. III, Theorem 12.11 (b)) that both f∗ († Vg,X ) and f∗ († Vg,X (−DX )) are vector bundles on S and the morphisms displayed in (235) are isomorphisms. In what follows, let us compute the ranks of f∗ († Vg,X ) and f∗ († Vg,X (−DX )). ∼ First, we note that the isomorphism Kosg : gad(q1 ) → c defined in § 2.7 (cf. (267)) is Gm -equivariant with respect to the Gm -action on gad(q1 ) (1) and the Gm -action on c that comes from the homotheties on g (i.e., the Gm -action ⋆). Hence, if el (l = 1, . . . , rk(g)) denote the degrees of homogeneous generators ul in Sk (g∨ )G , then Lrk(g) ⊗el † † Vg,X is isomorphic to l=1 ΩX log /S log . The rank of f∗ ( Vg,X ) can be computed by the following sequence of equalities: rk(g)

(238)

rk(f∗ († Vg,X )) =

X

l rk(f∗ (Ω⊗e )) X log /S log

l=1 rk(g)

=

X

((g − 1)(2el − 1) + r · el )

l=1

= (g − 1) · dim(g) +

r · (dim(g) + rk(g)) 2

= ℵ(g), Prk(g) where the third equality follows from the equality l=1 (2el − 1) = dim(g) (cf. [12], 2.2.4, Remark (iv)). On the other hand, by applying the functor R1 f∗ (−) to the short exact sequence r M ⊕rk(g) (239) 0 −→ † Vg,X (−DX ) −→ † Vg,X −→ OSi −→ 0 i=1

(cf. (233)), we obtain (by taking account of (237)) an equality (240)

rk(f∗ († Vg,X )) = rk(f∗ († Vg,X (−DX ))) + r · rk(g).

Thus, (240) and (238) together imply the required computation for the rank of f∗ († Vg,X (−DX )). This completes the proof of assertion (ii). 2.6.3. The moduli stack of (g, ℏ)-opers. — Since the formation of a (g, ℏ)-oper is closed under base-change (cf. § 2.1.5), one may define the Set-valued contravariant functor (241)

Opg,ℏ,X : Sch/S → Set

on Sch/S which, to any S-scheme S ′ , assigns the set of isomorphism classes of (g, ℏ)-opers on S ′ ×S X . In particular, we have Op g,ℏ,X log /S log (X) = Opg,ℏ,X (S).

ASTÉRISQUE 432

2.6. (g, ℏ)-OPERS ON POINTED STABLE CURVES

67

The torsor structure defined in (230) induces a V(f∗ († Vg,X ))-action (242)

actg : Opg,ℏ,X ×S V(f∗ († Vg,X )) → Opg,ℏ,X Op

on Opg,ℏ,X . Also, ιg (cf. (209)) gives rise to a morphism of functors ιOp g : Opg⊙ ,ℏ,X → Opg,ℏ,X .

(243)

This morphism makes the following square diagram commute: Opg⊙ ,ℏ,X ×S V(f∗ (Ω⊗2 )) X log /S log (244)

actg⊙

/ Opg⊙ ,ℏ,X ιOp g

V ιOp g ×ιg

 Opg,ℏ,X ×S V(f∗ († Vg,X )) where ιV g denotes the closed immersion

actg

 / Opg,ℏ,X ,


)) ,→ V(f∗ († Vg⊙ ,X )) = V(f∗ (Ω⊗2 X log /S log

V(f∗ († Vg,X )) induced by ιV g (cf. (220)). The following proposition was already proved in [13] § 3.4, theorem (cf. [7], Lemma 3.6), for the case of smooth curves over C, and in [59], Chap. I, Proposition 2.11, for the case of (arbitrary pointed stable curves but) g = sl2 . Theorem 2.24. — (Recall that either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled.) The following assertions hold: (i) Opg,ℏ,X forms a relative affine space over S modeled on V(f∗ († Vg,X )) with respect to the action actg given by (242). In particular, it may be represented by a relative affine space of relative dimension ℵ(g) (cf. (234)). (ii) The S-morphism (245)

Opg⊙ ,ℏ,X ×

V(f∗ (Ω⊗2log X

/S log

))

V(f∗ († Vg,X )) → Opg,ℏ,X

arising from ιOp and actg (together with the commutativity of (244)) is an g isomorphism. In particular, ιOp is a closed immersion. g Proof. — It follows from Proposition 2.23, (i), that étale locally on S there always exists a (g, ℏ)-oper on X . Hence, the assertions follow from Propositions 2.22. Remark 2.25. — If g is simple, then the morphism ιV g restricts to an isomorphism ∼ † ⊗2 ΩX log /S log → Vg,1 . Hence, (245) becomes an isomorphism (246)

∼

Opg⊙ ,ℏ,X ×S V(f∗ († Vg2 )) → Opg,ℏ,X

(cf. [13], § 3.4, Remark). Remark 2.26. — Let (G′ , T′ ) be another split semisimple algebraic k-group of adjoint type. If k is of positive characteristic, then we suppose that char(k) > 2hG′ . Denote

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by g′ the Lie algebra of G′ , and fix a Borel subgroup B′ of G′ defined over k containing T′ . Also, let E ♠ := (EB , ∇) be a (g, ℏ)-oper on X and E ′ ♠ := (EB′ , ∇′ ) a (g′ , ℏ)-oper on X . Then, B ×k B′ is a Borel subgroup of G ×k G′ and the pair E ♠ ⊠ E ′♠ := (EB ⊠ EB′ , ∇ ⊠ ∇′ )

(247)

(cf. (100), (102)) forms a (g⊕g′ , ℏ)-oper on X . The assignment (E ♠ , E ′♠ ) 7→ E ♠ ⊠E ′♠ commutes with base-change to S-schemes and the resulting S-morphism (248)

Opg,ℏ,X ×S Opg′ ,ℏ,X → Opg⊕g′ ,ℏ,X

is an isomorphism (cf. [13], § 3.4, Remark). 2.6.4. The moduli stack of (g, 0)-opers. — We now specialize to the case of ℏ = 0. Let us refer to a result in Example 4.56 described later, which says that if ℏ = 0, then there exists a canonical decomposition (249)

g⊙ †E

B⊙

⊙ ⊙ = (g⊙ −1 )† EB⊙ ⊕ (g0 )† EB⊙ ⊕ (g1 )† EB⊙ .

Let ∇▶ denote the composite (250)

∼

⊙ ∇▶ : TX log /S log → (g⊙ −1 )† EB⊙ ,→ g† E

B⊙

,→ g† EB ,

where the first arrow is the canonical isomorphism mentioned in § 2.3.4, the second arrow denotes the inclusion to the first factor with respect to the decomposition (249), and the last arrow denotes the injection induced by ιg . Then, it specifies a 0-S log -connection on † EG (cf. Example 1.18), and the pair (251)

†

E ▶ := († EB , ∇▶ )

forms a ↙ q1 -normal (g, 0)-oper on U log /S log . This (g, 0)-oper determines a global section S → Opg,0,X , which gives a trivialization (252)

∼

V(f∗ († Vg,X )) → Opg,0,X

of the relative affine space Opg,0,X . This trivialization sends each R ∈ Γ(S, f∗ († Vg,X ))
▶ = Γ(X, † Vg,X ) to † E+R . Now, observe that the Gm -action on gad(q1 ) (1) (cf. § 2.4.4) induces a Gm -action on V(f∗ († Vg,X )); the opposite of this action corresponds to a Z≥0 -grading on the OS -algebra OS ⟨V(f∗ († Vg,X ))⟩. Hence, it defines a Z≥0 -grading on the OS -algebra OS ⟨Opg,0,X ⟩ via the OS -algebra isomorphism (253)

OS ⟨Opg,0,X ⟩

∼

→ OS ⟨V(f∗ († Vg,X ))⟩

induced by (252), 2.6.5. A filtration on OS ⟨Opg,ℏ,X ⟩. — Let us go back to the case of a general ℏ and construct an increasing filtration on the OS -algebra OS ⟨Opg,ℏ,X ⟩. Recall that Opg,ℏ,X admits canonically a structure of relative affine space over S modeled on V(f∗ († Vg,X )). Hence, each point of S has an open neighborhood S ′ on which there exists an isomorphism
∼ (254) S ′ ×S V(f∗ († Vg,X )) = V(f∗′ († Vg,X ′ )) → Opg,ℏ,X ′ (= S ′ ×S Opg,ℏ,X ) ,

ASTÉRISQUE 432

2.7. THE ADJOINT QUOTIENT OF g

69

where X ′ := S ′ ×S X and f ′ denotes the projection S ′ ×S X → S ′ ; this isomorphism corresponds to an isomorphism of OS ′ -algebras (255)

OS ′ ⟨Opg,ℏ,X ′ ⟩

∼

→ OS ′ ⟨V(f∗′ († Vg,X ′ ))⟩.

The Z≥0 -grading on OS ′ ⟨V(f∗′ († Vg,X ′ ))⟩ defines an increasing filtration  j OS ′ ⟨Opg,ℏ,X ′ ⟩ j∈Z (256) ≥0

on OS ′ ⟨Opg,ℏ,X ′ ⟩ so that (255) induces an isomorphism of Z≥0 -graded OS ′ -algebras (257)

∼

gr(OS ′ ⟨Opg,ℏ,X ′ ⟩) → OS ′ ⟨V(f∗′ (Vg,X ′ ))⟩.

Both the formations of (256) and  (257) do not depend on the choice of (254). Hence, the filtrations OS ′ ⟨Opg,ℏ,X ′ ⟩j j for various open subschemes S ′ (⊆ S) may be glued together to obtain an increasing filtration  j (258) OS ⟨Opg,ℏ,X ⟩ j∈Z ≥0

on the OS -algebra OS ⟨Opg,ℏ,X ⟩ which has an isomorphism (259)

∼

gr(OS ⟨Opg,ℏ,X ⟩) → OS ⟨V(f∗ († Vg,X ))⟩

of Z≥0 -graded OS -algebras. Composing (253) and (259) gives the following proposition, by which we can think of OS ⟨Opg,ℏ,X ⟩ as a quantization of OS ⟨Opg,0,X ⟩ in some sense: Proposition 2.27. — There exists a canonical isomorphism of Z≥0 -graded OS -algebras (260)

∼

gr(OS ⟨Opg,ℏ,X ⟩) → OS ⟨Opg,0,X ⟩.

2.7. The adjoint quotient of g In the subsequent discussion, we introduce the definition of radius (cf. Definition 2.29) and study the moduli space of (g, ℏ)-opers of prescribed radii. To begin with, we review some basic facts concerning the quotient of g by the adjoint G-action. Denote by W the Weyl group of (G, T), and consider the GIT quotient g//G (resp., t//W) of g (resp., t) by the adjoint action of G (resp., W), i.e., the spectrum of the ring of polynomial invariants Sk (g∨ )G (resp., Sk (t∨ )W ) on g (resp., t). Let us use the notation (261)

cg , or simply c,

to denote the k-scheme t//W. Also, denote by [0] the k-rational point of c defined as the image of the zero element 0 ∈ t via the quotient t ↠ c. A Chevalley’s theorem asserts (cf. [64], Theorem 1.1.1; [49], Chap. VI, Theorem 8.2) that the k-morphism c → g//G induced by the natural k-algebra morphism Sk (g∨ )G → Sk (t∨ )W is an isomorphism because we have assumed that
either “ char(k) = 0” or “ char(k) > 2hG ” is satisfied. Also, in Sk (g∨ )G ∼ = Sk (t∨ )W , there are rk(g) algebraically independent (relative to k) homogeneous elements, which generates Sk (g∨ )G .

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Denote by χ:g→c

(262)

the morphism of k-schemes defines as the composite of the natural quotient g ↠ g//G ∼ and the inverse of the resulting isomorphism c → g//G. Here, recall the quotient stack [g/G] of g by the adjoint G-action; this stack represents the category fibered in groupoids over Sch/k whose category of sections over a k-scheme T is the groupoid of pairs (E , v) consisting of a G-bundle E on T and an element v ∈ Γ(T, gE ). Then, χ factors through the natural quotient g → [g/G]. We shall denote by [χ] : [g/G] → c

(263)

the resulting morphism. Next, note that c (∼ = Spec(Sk (g∨ )G )) admits a Gm -action (264)

⋆ : Gm ×k c → c; (h, a) 7→ h ⋆ a

coming from the homotheties on g. The k-algebra Sk (g∨ )G is equipped with the grading corresponding to the opposite of this Gm -action. Let us select a choice of homogeneous generators ul ∈ Sk (g∨ )G (l = 1, 2, . . . , rk(g)), which determines an isomor∼ phism of k-algebras k[u1 , . . . , urk(g) ] → Sk (g∨ )G . Under the identification of c with rk(g) the rk(g)-dimensional affine space A using the resulting k-isomorphism
∼ (265) c → Ark(g) := Spec(k[u1 , . . . , urk(g) ]) , the action ⋆ may be expressed as (266)

h ⋆ (a1 , . . . , ark(g) ) = (he1 · a1 , . . . , herk(g) · ark(g) ),

where each el denotes the degree of ul . The action ⋆ extends uniquely to a k-morphism A1 ×k c → c; we also use the notation ⋆ to denote this morphism. Let r be an ×r integer ≥ 0 and S a k-scheme. Denote of r copies of c over k.
by c the product 1 r ×r Then, for each ℏ ∈ Γ(S, OS ) = A (S) and ρ := (ρi )i=1 ∈ c (S), we write ℏ ⋆ ρ := (ℏ ⋆ ρi )ri=1 (∈ c×r (S)). (In the case of r = 0, we set c×r (S) := {∅} and ℏ ⋆ ∅ := ∅.) Finally, let (267)

χ

Kosg : gad(q1 ) ,→ g − →c

be the composite of χ and the inclusion gad(q1 ) ,→ g given by v 7→ v + q−1 for any v ∈ gad(q1 ) . Since either “ char(k) = 0” or “ char(k) > 2hG ” is satisfied, Kosg turns out to be an isomorphism of k-schemes (cf. [64], Lemma 1.2.1). In particular, we have dim(c) = rk(g) = dim(gad(q1 ) ). Moreover, Kosg may be regarded as a Gm -equivariant ∼ isomorphism gad(q1 ) (1) → c.

Example 2.28. — Let us describe the adjoint quotient for PGLn with n < p by using the natural identification pgln = sln . Under this identification, the Lie algebra t of T

ASTÉRISQUE 432

71

2.8. RADII OF (g, ℏ)-OPERS

(cf. Example 2.3) is given as n n o X (268) t := diag(a1 , . . . , an ) ∈ sln a1 , . . . , an ∈ k, aj = 0 , j=1

where diag(a1 , . . . , an ) denotes the n × n diagonal matrix with diagonal entries a1 , . . . , an . For i = 1, . . . , n, denote by λi the element of t∨ defined by λi (diag(a1 , . . . , an )) = ai . Then, we have Sk (t∨ ) = k[λ1 , . . . , λn ]/(λ1 + · · · + λn ). The action of W = Sn (:= the symmetric group of n letters) on t is given by σ · λi := λσ(i) for each σ ∈ Sn . It follows that Sk (t∨ )W coincides with the ring of symmetric polynomials in λ1 , . . . , λn . Now, consider the following elementary symmetric polynomials of λ1 , . . . , λn : n n X X Y st1 := λj , st2 := λj · λj ′ , . . . , stn := (269) λj . j=1

j 0, S a k-scheme, and X := (f : X → S, {σi }ri=1 ) an r-pointed stable curve of genus g over S. Also, write cS := S ×k c. For each i ∈ {1, . . . , r}, we shall set (278)

Radℏ,i : Opg,ℏ,X → cS

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to be the S-morphism given by assigning E ♠ := (EB , ∇) 7→ ρ∇ i . Hence, we obtain an S-morphism r Y Radℏ := Radℏ,i : Opg,ℏ,X → c×r (279) S , i=1

c×r S

where denotes the product of r copies
of cS over S. Given ρ := (ρi )ri=1 ∈ c×r (S) = c×r S (S) , we shall write Opg,ℏ,ρ,X := Rad−1 ℏ (ρ),

(280)

i.e., the closed subscheme of Opg,ℏX defined as the scheme-theoretic inverse image of ρ via Radℏ . That is to say, Opg,ℏ,ρ,X is the S-scheme representing the Set-valued contravariant functor on Sch/S which, to any S-scheme s : S ′ → S, assigns the set of isomorphism classes of (g, ℏ)-opers on S ′ ×S X of radii ρ ◦ s := (ρi ◦ s)ri=1 ∈ c×r (S ′ ). In particular, if r = 0, then we have Opg,ℏ,∅,X = Opg,ℏ,X .

2.9. The moduli space of (g, ℏ)-opers of prescribed radii ˆ i.e., we use the 2.9.1. The action ⋆ on the set of radii. — Let us write A1 = Spec(k[ℏ]), 1 ˆ symbol ℏ to denote the parameter of the affine line A . Also, write A1S := A1 ×k S. For ρ := (ρi )ri=1 ∈ c×r (S), we shall set ρˆ := (ˆ ρi )ri=1 ∈ c×r (A1S ),

(281)

where ρˆi (i = 1, . . . , r) denotes the composite
id 1 ×ρi ⋆ (282) ρˆi : A1S = A1 ×k S −−A−−−→ A1 ×k c − → c. (We set ˆ ∅ := ∅ when
r = 0.) Then, the equality ρˆ ◦ (ℏ, idS ) = ℏ ⋆ ρ holds for any ℏ ∈ Γ(S, OS ) = A1 (S) . ˆ ˆ — Write Xˆ := A1 ×S X , i.e., the base2.9.2. (g, ℏ)-opers with formal parameter ℏ. S change of X to A1S . Let us consider the A1S -scheme   (283) Opg,ℏ, resp., Opg,ℏ, ˆ Xˆ ˆ ρ, ˆ Xˆ . If we regard it as an S-scheme, then it represents the Set-valued contravariant functor on Sch/S assigning, to any S-scheme s : S ′ → S, the set of pairs (ℏ, [E ♠ ]) consisting of ℏ ∈ Γ(S ′ , OS ′ ) = A1 (S ′ ) and the isomorphism class [E ♠ ] of a (g, ℏ)-oper E ♠ on S ′ ×S X (resp., a (g, ℏ)-oper E ♠ on S ′ ×S X of radii ℏ ⋆ ρ ∈ c×r (S ′ )). Denote by   1 1 (284) D : Opg,ℏ, resp., Dρ : Opg,ℏ, ˆ Xˆ → AS ˆ ρ, ˆ Xˆ → AS the S-morphism given by assigning (ℏ, [E ♠ ]) 7→ ℏ. By applying Proposition 2.23, (ii), and Theorem 2.24, (i), to Xˆ , we see that the affine  morphism D forms a relative  † 1 space over A1 modeled on A1 ×S V(f∗ († Vg,X )) ∼ V((id × f ) ( V )) . For each = ˆ ∗ A S

ASTÉRISQUE 432

S

g,X

2.9. THE MODULI SPACE OF (g, ℏ)-OPERS OF PRESCRIBED RADII

75


ℏ ∈ Γ(S, OS ) = A1S (S) , the natural S-morphism   (285) Opg,ℏ,X → S ×ℏ,A1S , D Opg,ℏ, ˆ Xˆ resp., Opg,ℏ,ℏ⋆ρ,X → S ×ℏ,A1S , Dρ Opg,ℏ, ˆ ρ, ˆ Xˆ ˆ±1 ]))-action is an isomorphism. Moreover, Opg,ℏ, ˆ Xˆ has a Gm (= Spec(k[ℏ Gm ×k Opg,ℏ, ˆ Xˆ → Opg,ℏ, ˆ Xˆ

(286)

determined by (c, (ℏ, [E ♠ ])) 7→ (c · ℏ, [c · E ♠ ]) (cf. (171) for the definition of c · E ♠ ), which makes the following square diagram commute: Gm ×k Opg,ℏ, ˆ Xˆ (287)

(286)

/ Op ˆ ˆ g,ℏ,X

idGm ×(D,Radℏ ˆ)

(D,Radℏ ˆ)

 Gm ×k A1S ×S c×r S

(c,ℏ,ϱ)7→(c·ℏ,c⋆ϱ)

 / A1 ×S c×r . S S

The commutativity of this diagram implies that (286) restricts to a Gm -action (288)

Gm ×k Opg,ℏ, ˆ ρ, ˆ ρ, ˆ Xˆ → Opg,ℏ, ˆ Xˆ

× on Opg,ℏ, ˆ ρ, ˆ Xˆ . If ℏ ∈ Γ(S, OS ) (= Gm (S)), then the composite (286)

inclusion

Gm ×k Opg,ℏ,X −−−−−→ Gm ×k D−1 ((Gm )S ) −−−→ D−1 ((Gm )S ),
where (Gm )S := Gm ×k S ⊆ A1S , is an isomorphism. Moreover, the following diagram is commutative: (289)

(289)

Gm ×k Opg,ℏ,X (= (Gm )S ×S Opg,ℏ,X ) (290)

/ D−1 ((Gm )S ) (D,Radℏ ˆ)

idGm ×Radℏ

 (Gm )S ×S c×r S

(c,ϱ)7→(c·ℏ,c⋆ϱ)

 / (Gm )S ×S c×r . S

One may verify that (289) restricts to an isomorphism (291)

∼

Gm ×k Opg,ℏ,ℏ⋆ρ,X → D−1 ρ ((Gm )S ).

In particular, the assignment E ♠ 7→ ℏ · E ♠ determines an isomorphism   ∼ ∼ (292) Opg,1,X → Opg,ℏ,X resp., Opg,1,ρ,X → Opg,ℏ,ℏ⋆ρ,X . 2.9.3. The representability of Opg,ℏ,ℏ⋆ρ,X . — In the following, we prove the representability of Opg,ℏ,ℏ⋆ρ,X by observing the radius map Radℏ specialized at ℏ = 0. Theorem 2.36. — (Recall that either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled.) The morphism (293)

×r 1 (D, Radℏˆ ) : Opg,ℏ, ˆ Xˆ → AS ×S cS

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×r 1 † forms a relative affine space over A1S ×S c×r S modeled on (AS ×S cS )×S V(f∗ ( Vg,X )). In particular, for ℏ
∈ Γ(S, OS ) and ρ ∈ c×r (S), the S-scheme Opg,ℏ,ℏ⋆ρ,X = (D, Radℏˆ )−1 ((ℏ, ρ)) forms a relative affine space modeled on V(f∗ († Vg,X (−DX ))).

Proof. — By Corollary 2.35, the problem is reduced to proving the claim that (D, Radℏˆ ) is smooth and surjective. In order to prove this, we may assume, after possibly changing the base, that S = Spec(k) and k is algebraically closed. First, we shall prove the smoothness of (D, Radℏˆ ). According to Corollary 2.35, the fiber (D, Radℏˆ )−1 ((ℏ, ρ)) (= Opg,ℏ,ℏ⋆ρ,X ) of (D, Radℏˆ ) over each point (ℏ, ρ) ∈ (A1 ×k c×r )(k) forms, if it is nonempty, an affine space modeled on Γ(X, † Vg,X (−DX )). In particular, (D, Radℏˆ )−1 ((ℏ, ρ)) is smooth and has dimension equal to the value (294)

c

1 ×r ℵ(g) = (ℵ(g) + 1) − (1 + r · rk(g)) = dim(Opg,ℏ, ) ˆ Xˆ ) − dim(A ×k c

1 ×r (cf. Proposition 2.23, (ii)). Hence, since both Opg,ℏ, are smooth ˆ Xˆ and A ×k c over k, the smoothness of (D, Radℏˆ ) follows from [3], Chap. V, Proposition (3.5) (or [32], Chap. III, Exercise 10.9). The remaining portion is the surjectivity of (D, Radℏˆ ). To begin with, we shall prove the surjectivity of Res0 : Opg,0,X → c×r . Given a global section s of Ω⊗l X log /klog ∼

⊗l ∗ for l ∈ Z>0 , we shall denote by s(σi ) the image of σi∗ (s) via Res⊗l X ,i : σi (ΩX log /klog ) → k. Recall that there exists a composite isomorphism M M ∼ ∼ ∼ ⊗(j+1) ad(q ) (295) Opg,0,X → Γ(X, † Vg ) → Γ(X, † Vg,j ) → Γ(X, ΩX log /klog ) ⊗k gj 1 , j∈Z>0

j∈Z>0

where the first arrow denotes the inverse to (252) and the third arrow is obtained by (218). Under the identifications given by this composite and Kosg respectively, the morphism Rad0 may be described as a k-linear morphism M M ad(q ) ⊗(j+1) ad(q ) (296) Γ(X, ΩX log /klog ) ⊗k gj 1 → gj 1 j∈Z>0

given by

P

j∈Z>0

P

j s ⊗ v 7→ ⊗(j+1) ΩX log /klog for

j

s(σi ) · v for s ∈

⊗(j+1) Γ(X, ΩX log /klog )

ad(q1 )

and v ∈ gj

. Since the

every j ∈ Z>0 is greater than 2g − 2 + r, (296) turns out to be degree of surjective (cf. [32], Chap. IV, § 4.3, Corollary 3.2, (a)). That is to say, we obtain the surjectivity of Rad0 . We turn to finish the proof of the surjectivity. Suppose, on the contrary, that (D, Radℏˆ ) is not surjective, i.e., there exists a k-rational point (ℏ0 , ρ0 ) of A1 ×k c×r outside the set-theoretic image Im((D, Radℏˆ )) of (D, Rad
ℏˆ ). By the surjectivity of Rad0 proved above, ℏ0 is contained in Gm ⊆ A1 . Denote m by ρG the image of Gm × {ρ0} (⊆ Gm × c×r ) via the lower horizontal ar0 1 m row in (290). The set ρG := (c · ℏ , c ⋆ ρ ) | c ∈ A forms a closed subset 0 0 0 Gm 1 ×r m of A ×k c containing ρ0 . The commutativity of (290) implies that ρG is 0 1 ×r contained in (A ×k c ) \ Im((D, Radℏˆ )). Since (D, Radℏˆ ) is flat as proved already, m Im((D, Radℏˆ )) is open in A1 ×k c×r . It follows that ρG ⊆ (A1 × c×r ) \ Im((D, Radℏˆ )). 0

ASTÉRISQUE 432

77

2.10. THE UNIVERSAL MODULI STACK

m But, (0, 0 ⋆ ρ0 ) belongs to ρG 0 , so this contradicts the surjectivity of Rad0 . Consequently, (D, Radℏˆ ) must be surjective. This completes the proof of this proposition. ⊙

Remark 2.37. — The injection ιg restricts to an injection (g⊙ )ad(q1 ) ,→ gad(q1 ) . Under ⊙ the identifications (g⊙ )ad(q1 ) = cg⊙ , gad(q1 ) = cg given by Kosg⊙ , Kosg respectively, ×r r this injection becomes an injection ιc : cg⊙ ,→ cg . For ρ⊙ := (ρ⊙ i )i=1 ∈ cg⊙ (S), we shall write r ×r ιc∗ (ρ⊙ ) := (ιc ◦ ρ⊙ i )i=1 ∈ cg (S).

(297)

If E⊙♠ is a (g⊙ , ℏ)-oper of radii ρ⊙ , then the induced (g, ℏ)-oper ιG∗ (E⊙♠ ) (cf. (208)) is of radii ιc∗ (ρ⊙ ). Hence, ιOp g (cf. (243)) restricts to a closed immersion ⊙ ⊙ ⊙ ιOp g,ρ⊙ : Opg ,ℏ,ρ ,X → Opg,ℏ,ιc∗ (ρ ),X .

(298)

Also, Opg⊙ ,ℏ,ρ⊙ ,X forms a relative affine space over S modeled on (−DX ))) V(f∗ (Ω⊗2 X log /S log


= V(f∗ († Vg⊙ ,X (−DX )))

and ιOp g,ρ⊙ induces an S-isomorphism (299) V(f∗ (Ω⊗2log

Opg⊙ ,ℏ,ρ⊙ ,X ×

X

/S log

(−DX )))

∼

V(f∗ († Vg,X (−DX ))) → Opg,ℏ,ιc∗ (ρ⊙ ),X .

If, moreover, g is simple, then (299) becomes an S-isomorphism (300)

∼

Opg⊙ ,ℏ,ρ⊙ ,X ×S V(f∗ († Vg2 (−DX ))) → Opg,ℏ,ιc∗ (ρ⊙ ),X

(cf. (246)). Remark 2.38. — Let us keep the notation in Remark 2.26. There exists a canonical identification cg⊕g′ = cg ×k cg′ . Given ρ := (ρi )ri=1 ∈ c×r g (S) and ρ′ := (ρ′i )ri=1 ∈ c×r (S), we obtain g′ (301)

(ρ, ρ′ ) := ((ρi , ρ′i ))ri=1 ∈ c×r g⊕g′ (S).

Then, E ♠ and E ′♠ are of radii ρ and ρ′ respectively if and only if the (g ⊕ g′ , ℏ)-oper E ♠ ⊠ E ′♠ (cf. (247)) is of radii (ρ, ρ′ ). This implies that (248) restricts to an S-isomorphism (302)

∼

Opg,ℏ,ρ,X ×S Opg′ ,ℏ,ρ′ ,X → Opg⊕g′ ,ℏ,(ρ,ρ′ ),X .

2.10. The universal moduli stack In this final section of Chapter 2, we deal with the universal moduli stack of (g, ℏ)-opers, i.e., Opg,ℏ,(ρ,)X in the case of X = Cg,r . Let (g, r) be a pair of nonnegative integers with 2g − 2 + r > 0, ℏ an element of k, and ρ ∈ c×r (k) (where ρ := ∅ if r = 0). Denote by (303)

Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r )

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the category over Sch/k defined as follows: — The objects are the pairs (X , E ♠ ), where X is an r-pointed stable curve of genus g over a k-scheme S and E ♠ is a (g, ℏ)-oper on X (resp., a (g, ℏ)-oper on X of radii ρ); — The morphisms from (X , E ♠ ) to (X ′ , E ′♠ ) are morphisms (ϕ, Φ) : X → X ′ of r-pointed stable curves with E ♠ ∼ = ϕ∗ (E ′♠ ); — The projection Opg,ℏ,g,r → Sch/k (resp., Opg,ℏ,ρ,g,r → Sch/k ) is given by assigning, to each pair (X , E ♠ ) as above, the base k-scheme S of X . In particular, we have Opg,ℏ,∅,g,0 = Opg,ℏ,g,0 . The assignment (X , E ♠ ) 7→ X defines a functor Opg,ℏ,g,r → Mg,r (resp., Opg,ℏ,ρ,g,r → Mg,r ). If we take a k-scheme S and a k-morphism s : S → Mg,r , which classifies an r-pointed stable curve X over S, then there exists a canonical S-isomorphism ∼

S ×s,Mg,r Opg,ℏ,(ρ,)g,r → Opg,ℏ,(ρ,)X .

(304) For simplicity, we write (305)

Opg,g,r := Opg,1,g,r (resp., Opg,ρ,g,r := Opg,ℏ,ρ,g,r ) .

By applying results proved so far, we obtain the following theorem concerning the geometric structure of these moduli categories; this assertion is a generalization of a result for g = g⊙ proved by S. Mochizuki (cf. [59], Chap. I, Theorem 2.3). Theorem 2.39 (cf. Theorem A). — Assume that either “ char(k) = 0” or “ char(k) > 2hG ” is fulfilled. Let ℏ ∈ k and ρ ∈ c×r (k) (where ρ := ∅ if r = 0). Then, Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) may be represented by a relative affine space over Mg,r of relative dimension ℵ(g) (resp., c ℵ(g)) (cf. (234)). In particular, Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) is a nonempty, geometrically connected, and smooth Deligne-Mumford stack over k of dimension 3g − 3 + r + ℵ(g) (resp., 3g − 3 + r + c ℵ(g)). Proof. — The assertion follows from Theorems 2.24, (i), and 2.36 applied to various r-pointed stable curves X of genus g. Finally, note that the Deligne-Mumford stack Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) admits log

a log structure pulled-back from the log structure of Mg,r ; we denote the resulting log stack by   log (306) Oplog resp., Op g,ℏ,g,r g,ℏ,ρ,g,r . log By the above theorem, Oplog g,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) turns out to be log smooth over k.

ASTÉRISQUE 432

CHAPTER 3 OPERS IN POSITIVE CHARACTERISTIC

The aim of this chapter is to study (g, ℏ)-opers in positive characteristic. To begin with, we introduce the logarithmic version of a restricted Lie algebroid and recall the p-curvature associated to a flat principal bundle on a log scheme. After that, we define the notions of a dormant oper and a p-nilpotent oper as opers satisfying certain vanishing conditions on p-curvature. The central characters in our discussion are the moduli spaces classifying these objects. For example, the moduli space of p-nilpotent opers can be obtained as the kernel of the Hitchin-Mochizuki morphism. An important consequence of this chapter is the properness of these moduli spaces, which will be proved by applying the finiteness of the Hitchin-Mochizuki morphism. Notation and Conventions. — Let k be a perfect field of characteristic p > 0 and G a connected smooth affine algebraic k-group with Lie algebra g. 3.1. Frobenius twists and relative Frobenius morphisms 3.1.1. Frobenius morphisms. — Let T be a scheme over k (⊇ Fp := Z/pZ) and f : Y → T a scheme over T . Denote by FT : T → T (resp., FY : Y → Y ) the absolute Frobenius endomorphism of T (resp., Y ). The Frobenius twist of Y over T is, by definition, the base-change Y (1) (:= T ×FT ,T Y )

(307)

of Y via Denote by f (1) : Y (1) → T the structure morphism of Y (1) . The relative Frobenius morphism of Y over T is, by definition, the unique morphism FY /T : Y → Y (1) over T that fits into the following commutative diagram: (308)

Y

FY FY /T

)

Y (1)

f

)  T

f (1)

FT ×idY □ FT

)/

Y f

 / T.

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3.1.2. The Frobenius twist of a pointed stable curve. — Let r be an integer ≥ 0, S a scheme over k, and U := (U/S, {σiU : Si → U }ri=1 ) a local r-pointed stable curve over S. For i = 1, . . . , r, we write U (1)

σi

(309)

:= (idS | , σiU ◦ FS i ) : Si → U (1) (= S ×FS ,S U ) . Si

Then, the collection of data U (1) r }i=1 )

U (1) := (U (1) /S, {σi

(310)

specifies a local r-pointed stable curve over S. If U is taken to be an r-pointed stable curve (of genus g ≥ 0), then the resulting collection U (1) becomes an r-pointed stable curve (of genus g). 3.1.3. Relatively torsion-free sheaves. — Let S be a k-scheme and U a prestable curve over S. If U/S is smooth, then the relative Frobenius morphism FU/S : U → U (1) is finite and faithfully flat of degree p, so the direct image FU/S∗ (OU ) of OU is a vector bundle on U (1) of rank p. This is not true in general because FU/S : U → U (1) is not flat at a non-smooth point of U . But, as is shown in Proposition 3.2 below, FU/S∗ (OU ) satisfies a certain local property even at a non-smooth point, i.e., FU/S∗ (OU ) is relatively torsion-free (cf. [78], Septième partie, Définition 1, or [41], Definition 1.0.1). Any vector bundle on U is relatively torsion-free, and vice versa on the smooth locus of U relative to S. We here recall the definition. Definition 3.1. — Let n be a positive integer and F a coherent OU -module. Then, F is called relatively torsion-free of rank n if it is flat over S and, for each geometric point s ∈ S, the OUs -module F | on the fiber Us := s ×S U is of rank n (cf. [37], Us § 1.2, Definition 1.2.2) and has no associated primes of height one. The following proposition will be used in the discussion of Chapter 6. Proposition 3.2. — For any vector bundle V on U of rank n > 0, the OU (1) -module FU/S∗ (V ) is relatively torsion-free of rank n · p. ⊕n Proof. — Since V is locally isomorphic to OU and FU/S∗ (V ) is flat over S, we may assume, without loss of generality, that S = Spec(k), V = OU , and k is algebraically closed. As explained above, FU/k∗ (OU ) is locally free of rank p over the smooth locus of U (1) . Hence, it suffices to prove that FU/k∗ (OU ) is (relatively) torsion-free of rank p at each nodal point of U . Let us take a nodal point q of U , and write q (1) := FU/k (q) ∈ U (1) (k). Denote by b OU (1) ,q (1) the completions of the stalk of OU (1) (1) b U (1) ,q(1) the maximal ideal at q with respect to its maximal ideal mU (1) ,q(1) and by m of b O (1) (1) . The problem is to prove the torsion-freeness of the m (1) (1) -adic comU

,q

U

,q

pletion b OU,q of the stalk OU,q of OU at q. Recall that b OU,q ∼ = k[[x, y]]/(xy). By using the ∗ b b injection FU/k : OU (1) ,q(1) → OU,q induced by FU/k , we shall identify b OU (1) ,q (1) with the subring k[[xp , y p ]]/(xp y p ) of k[[x, y]]/(xy).

ASTÉRISQUE 432

3.1. FROBENIUS TWISTS AND RELATIVE FROBENIUS MORPHISMS

81

b U (1) ,q(1) ∼ In particular, we have m = xp · k[[xp ]] ⊕ y p · k[[y p ]]). Now, let us define a k[[xp , y p ]]/(xp y p )-linear morphism (311)

(k[[xp , y p ]]/(xp y p )) ⊕ (xp · k[[xp ]] ⊕ y p · k[[y p ]])⊕(p−1) → k[[x, y]]/(xy)

given by (A, (xp · Bi + y p · Ci )p−1 i=1 ) 7→ A +

(312) p

p

p p

p

p−1 X

(xi · Bi + y i · Ci )

i=1 p

for A ∈ k[[x , y ]]/(x y ), and Bi ∈ k[[x ]], Ci ∈ k[[y ]] (i = 1, . . . , p−1). This morphism ⊕(p−1) bU is verified to be bijective. Thus, b OU,q is isomorphic to b OU (1) ,q (1) ⊕ m (1) ,q (1) , which is torsion-free (cf. [41], the discussion at the beginning of § 1.2). This completes the proof of Proposition 3.2.

Also, the following proposition will be used in the proof of PropositionDefinition 3.24. Proposition 3.3. — Let V be a vector bundle on U (1) of rank n > 0. Denote by ∗ wV : V → FU/S∗ (FU/S (V ))

(313)

the OU (1) -linear morphism corresponding, via the adjunction relation ∗ ∗ “FU/S (−) ⊣ FU/S∗ (−),” to the identity morphism of FU/S (V ). Then, wV is injective and its cokernel Coker(wV ) is relatively torsion-free of rank n · (p − 1). Proof. — As the statement is of local nature, we may assume (since V is locally ⊕n isomorphic to OU (1) ) that V = OU (1) =: O . It is verified that wO is injective over the smooth locus U sm of U relative to S. Since the open subscheme U sm of U is scheme-theoretically dense, Next, note that the formation of wO is functorial with respect to base-change to any S-scheme s : S ′ → S in the following sense: if we write U ′ := S ′ ×S U and O ′ := OU ′ (1) , then the square diagram (s × idU )∗ (O ) (314)

(s×idU )∗ (wO )

≀

/ (s × idU )∗ (FU/S∗ (F ∗ (O ))) U/S ≀



O′

wO ′

 / FU ′ /S∗ (F ∗ ′ (O ′ )) U /S

is commutative, where the right-hand vertical arrow arises from the base-change ∼ map (s × idS )∗ (FU/S∗ (OU )) → FU ′ /S∗ (s∗ (OU )). The above discussion applied to O ′ implies the injectivity of wO′ . Hence, wO is universally injective with respect to basechange to S-schemes. By [56], p. 17, Theorem 1, the cokernel Coker(wO ) turns out to be flat over S. Thus, in order to finish the proof, it suffices to consider the

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case where S = Spec(k) and k is algebraically closed. We shall prove the torsionfreeness of Coker(wO ) at a nodal point (resp., a smooth point) q (1) of U (1) . Let us keep the notation in the proof of Proposition 3.2. According to the discussion there, the mU (1) ,q(1) -adic completion of wO may be identified with the inclusion b OU (1) ,q (1) ,→ ⊕(p−1) ⊕p b b b b O (1) (1) ⊕ m (resp., O (1) (1) ,→ O ) into the first factor. Hence, the U

,q

U (1) ,q (1)

U

,q

U (1) ,q (1)

⊕(p−1) b⊕(p−1) bU completion of Coker(wO ) at q is isomorphic to m (1) ,q (1) (resp., OU (1) ,q (1) ), which is torsion-free of rank p − 1. This completes the proof of Proposition 3.3. (1)

3.2. Lie algebroids in positive characteristic In this section, we recall the definition of a restricted Lie algebra. The p-curvature of a flat G-bundle will be defined by means of the structure of restricted Lie algebra on the associated Lie algebroid. 3.2.1. Restricted Lie algebras. — First, let us recall the notion of a restricted Lie algebra. Definition 3.4. — Let R be a (unital, associative, and commutative) k-algebra. A restricted Lie algebra (or a p-Lie algebra) over R is a Lie algebra h over R together with a map (−)[p] : h → h satisfying the following conditions: — (ad(v))p = ad(v [p] ) for all v ∈ h; — (a · v)[p] = ap · v [p] for all a ∈ R, v ∈ h; Pp−1 — (v + u)[p] = v [p] + u[p] + i=1 si (v,u) for all v, u ∈ h, where si (v, u) denotes the i coefficient of ti−1 in the expression ad(vt + u)p−1 (v) ∈ h[t] (:= h ⊗R R[t]). We will refer to the operation (−)[p] as the p-power operation on h. Example 3.5. — Let h be a unital associative (but possibly noncommutative) R-algebra. Then, the bracket operation [v, u] := vu−uv and the p-power operation v [p] := v p make h into a restricted Lie algebra over R. For example, the space of R-linear endomorphisms EndR (V ) of R-module V gives a typical example of a restricted Lie algebra over R. 3.2.2. The p-power operation on the tangent bundle. — Let f : Y → T be a scheme over a k-scheme T . In the natural fashion, we can define the notion of a restricted OT -Lie algebra on Y (i.e., a sheaf of restricted OT -Lie algebra in the sense of [52], § 4.1). Just as in the case of EndR (V ) mentioned above, the sheaf End f −1 (OT ) (V ) defined for each OY -module V (cf. § 1.1.1) has a natural structure of restricted OT -Lie algebra on Y with p-power operation v [p] := v p (= the p-th iteration of v) for any local section v ∈ End f −1 (OT ) (V ). Recall that the tangent bundle TY /T may be identified with the subsheaf of End f −1 (OT ) (OY ) consisting of locally defined derivations on OY

ASTÉRISQUE 432

3.2. LIE ALGEBROIDS IN POSITIVE CHARACTERISTIC

83

relative to T . The p-power operation on End f −1 (OT ) (OY ) restricts to a structure of restricted OT -Lie algebra (315)

(−)[p] : TY /T → TY /T

on TY /T . Indeed, let us take a local section ∂ of TY /T , which corresponds to a locally defined derivation. The p-th iterate ∂ [p] := ∂ p of the derivation ∂ is verified to form a derivation, i.e., specifies a local section of TY /T . This implies that TY /T is closed under the p-power operation on End f −1 (OT ) (OY ). More generally, this discussion can be generalized to log schemes. Let f : Y log → T log be a morphism of fine log schemes over k. If ∂ := (D, δ) (cf. (55)) is the logarithmic derivation corresponding to a local section of TY log /T log , then the pair   (316) ∂ [p] = (D, δ)[p] := (Dp , FY∗ ◦ δ + Dp−1 ◦ δ) specifies a logarithmic derivation. Moreover, by [65], Proposition 1.2.1 (or, [67], Chap. V, § 4, Theorem 4.1.4), the equality   (317) (a · ∂)[p] = ap · ∂ [p] + (a · ∂)p−1 (a) · ∂ = ap · ∂ [p] − a · ∂ p−1 (ap−1 ) · ∂ holds for any local section a ∈ OY . The sheaf TY log /T log together with the resulting operation (318)

(−)[p] : TY log /T log → TY log /T log

forms a restricted OT -Lie algebra on Y . If the log structures on Y log and T log are trivial, the p-power structure on TY log /T log coincides with (315).

3.2.3. The restricted Lie algebra of an algebraic group. — The Lie algebra of any smooth affine algebraic group admits a structure of restricted Lie algebra. Indeed, the p-power operation ∂ 7→ ∂ [p] on TG/k defines a structure of restricted Lie algebra (319)

(−)[p] : g → g

on the Lie algebra g via the identification of g with the space of left-invariant vector fields on G. Occasionally, this p-power operation will be regarded as an endomorphism of the k-scheme g. According to [77], Chap. I, § 8, the second corollary of Theorem I.8.1, a structure of restricted Lie algebra on g is uniquely determined if the Killing form on g is nondegenerate. Let G′ be another connected smooth affine algebraic k-group with Lie algebra g′ and w : G → G′ a morphism of algebraic k-groups. Then, the differential dw : g → g′ preserves the p-power operation (cf. [84], 4.4.9). For example, when G′ = GL(g) and w = AdG , this fact just means the first condition appearing in Definition 3.4. Also, when G′ = G and w = Innh for h ∈ G, it says that the automorphism  AdG (h) = dInnh | of g preserves the p-power operation. This implies that, for a e G-bundle E on Y , the p-power operation on g induces a structure of restricted OY -Lie

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algebra (−)[p] : gE → gE

(320)

on the adjoint bundle gE . In the case where E = Y ×k G, the p-power operation on OY ⊗k g (= gE ) is given by a ⊗ v 7→ ap ⊗ v [p] for any a ∈ OY and v ∈ g. Moreover, the assignment (E , R) 7→ (E , R)[p] := (E , R[p] ) (for a G-bundle E on a k-scheme and a global section R of gE ) defines a k-endomorphism (−)[p] : [g/G] → [g/G]

(321)

of the quotient stack [g/G], which is compatible with the p-power operation on g via the natural quotient g → [g/G]. 3.2.4. Restricted Lie algebroids. — In the following, we introduce the notion of a restricted Lie algebroid defined on a log scheme. The non-logarithmic version can be found in [52], § 4.2, Definition 4.2. Definition 3.6. —

(i) A restricted Lie algebroid on Y log /T log is a triple (V , α, (−)[p] )

(322)

consisting of a Lie algebroid (V , α) on Y log /T log (cf. Definition 1.11) and a map of sheaves (−)[p] : V → V satisfying the following conditions: — (V , (−)[p] ) forms a restricted OT -Lie algebra on Y ; — The anchor map α preserves the p-power operation; — The equality (a · ∂)[p] = ap · ∂ [p] + α(a · ∂)p−1 (a) · ∂

(323)

holds for any local sections a ∈ OY , ∂ ∈ V . When there is no fear of confusion, we will only use the symbol V to indicate the restricted OT -Lie algebroid (V , α, (−)[p] ). (ii) Let V , V ′ be restricted Lie algebroids on Y log /T log . A morphism of restricted Lie algebroids from V to V ′ is defined as a morphism of Lie algebroids V → V ′ (cf. Definition 1.13) preserving the p-power operation. 3.2.5. Restricted Lie algebroids associated to G-bundles. — Let π : E → Y be a G-bundle on Y . The p-power operation on TE log /T log restricts to a p-power operaf log log = (π (T log log ))G
. By construction, this tion (−)[p] on the Lie algebroid T E

/T

∗

E

/T

operation  is consistent,via restriction, with the p-power operation on the adjoint f log log constructed in § 3.2.3, and the anchor map dlog preserves bundle gE ⊆ T E /T E the p-power operation. Moreover, by considering the local situation (i.e., the case f log log . That is to say, the triple where E = Y ×k G), we can verify (323) for T E /T (324)

f log log , dlog , (−)[p] ) (T E /T E

forms a restricted Lie algebroid on Y log /T log associated to E .

ASTÉRISQUE 432

85

3.3. p-CURVATURE ON AN ℏ-FLAT G-BUNDLE

3.3. p-curvature on an ℏ-flat G-bundle Next, we shall recall the notion of p-curvature and discuss basic properties concerning p-curvature. Roughly speaking, the p-curvature of a connection measures the obstruction to the compatibility of the p-power operation ℏp−1 · (−)[p] on TY log /T log f log log defined above. and that on T E

/T

3.3.1. p-curvature. — Let E be as above. Also, let ℏ be an element of Γ(T, OT ) and ∇ a flat ℏ-T log -connection on E . For each local section ∂ ∈ TY log /T log , we write p

(325)

ψ ∇ (∂) := ∇(∂)[p] − ℏp−1 · ∇(∂ [p] ).

Then, the following assertion holds: Lemma 3.7. — Let ∂, ∂1 , ∂2 be local sections of TY log /T log , and let a be a local section of OY . (Suppose that these sections are defined on a common open subset of Y .) Then, the following equalities hold: p

(326)

ψ ∇ (a · ∂) = ap · p ψ ∇ (∂),

p

ψ ∇ (∂1 + ∂2 ) = p ψ ∇ (∂1 ) + p ψ ∇ (∂2 ).

Proof. — The first equality follows from the following sequence of equalities: p

(327)

ψ ∇ (a · ∂) = ∇(a · ∂)[p] − ℏp−1 · ∇((a · ∂)[p] ) = (a · ∇(∂))[p] − ℏp−1 · ∇(ap · ∂ [p] + (a · ∂)p−1 (a) · ∂) = ap · ∇(∂)[p] + dlog (a · ∇(∂))p−1 (a) · ∇(∂) E − ℏp−1 · ∇(ap · ∂ [p] ) − ℏp−1 · ∇((a · ∂)p−1 (a) · ∂) (( (·( (( · ℏ(· ( ∂)p−1 (a) ∇(∂) = ap · ∇(∂)[p] + (a( ( ((( ((a) p−1 p−1 p ·a ·( ∂)( · ∂) −ℏ · ∇(a · ∂ [p] ) − ∇((ℏ ( ( ( ( p [p] p−1 [p] = a · ((∇(∂)) − ℏ · ∇(∂ )) = ap · p ψ ∇ (∂),

where the second and third equalities follow from (317) and (323) respectively. To verify the second equality, we use the flatness of ∇. Indeed, since ∇ preserves the Lie bracket operation, the equality si (∇(∂1 ), ∇(∂2 )) = hp−1 · ∇(si (∂1 , ∂2 )) holds for every i (cf. Definition 3.4 for the definition of si ). Hence, we have (328)

p

ψ ∇ (∂1 + ∂2 ) = ∇(∂1 + ∂2 )[p] − ℏp−1 · ∇((∂1 + ∂2 )[p] ) = (∇(∂1 ) + ∇(∂2 ))

[p]

−ℏ

p−1

·∇

[p] ∂1

+

[p] ∂2

+

p−1 X si (∂1 , ∂2 )

!

i i=1 p−1   X 2 )) si (∇(∂1 ),  ∇(∂ = ∇(∂1 )[p] + ∇(∂2 )[p] +  i   i=1  p−1 p−1  X 1 , ℏ · ∇(s ∂2 )) i (∂ [p] [p]  − ℏp−1 · ∇(∂1 ) − ℏp−1 · ∇(∂2 ) −  i   i=1

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=



   [p] [p] ∇(∂1 )[p] − ℏp−1 · ∇(∂1 ) + ∇(∂2 )[p] − ℏp−1 · ∇(∂2 )

= p ψ ∇ (∂1 ) + p ψ ∇ (∂2 ), thus completing the proof of the second equality. The above lemma implies that the assignment FY−1 (∂) 7→ ∂ ∈ TY log /T log ) defines an OY -linear morphism (329)

p

p

ψ ∇ (∂) (for any

f log log . ψ ∇ : FY∗ (TY log /T log ) → T E /T

Definition 3.8. — We will refer to p ψ ∇ as the p-curvature of ∇. Also, ∇ is called p-flat (or p-integrable) if p ψ ∇ = 0. Remark 3.9. — Since dlog satisfies dlog ◦ ∇ = ℏ · idTY log /T log and preserves the E E   p-power operation, the image of p ψ ∇ is contained in gE = Ker(dlog ) . Thus, the E morphism p ψ ∇ may be regarded as an OY -linear morphism FY∗ (TY log /T log ) → gE , or equivalently, a global section of FY∗ (ΩY log /T log ) ⊗ gE . Proposition 3.10. — Let us keep the above notation. Also, let η be an automorphism of the G-bundle E and denote by AdG (η) the automorphism of gE induced by η via change of structure group by AdG . Then, the equality (330)

p

ψ η∗ (∇) = AdG (η) ◦ p ψ ∇

(cf. (83) for the definition of η∗ (∇)) of morphisms FY∗ (TY log /T log ) → gE holds. In particular, the p-flatness of an ℏ-T log -connection is closed under gauge transformations. Proof. — The assertion follows immediately from the definition of p-curvature to∼ f f log log → gether with the fact that the automorphism dη G : T TE log /T log (cf. (72)) E /T preserves the p-power operation and restricts to AdG (η). 3.3.2. Change of structure group. — Let G′ be another connected smooth affine algebraic k-group with Lie algebra g′ and w : G → G′ a morphism of algebraic k-groups. Also, let (E , ∇) be an ℏ-flat G-bundle on Y log /T log . Since the differential dw : g → g′ of w preserves the p-power operation, the morphism of Lie algebroids f log log → T f G ′ log log (cf. (73)) preserves the p-power operation, i.e., (dw)E : T E /T (E × G ) /T defines a morphism of restricted Lie algebroids. Hence, the p-curvature p ψ w∗ (∇) of the ℏ-T log -connection w∗ (∇) (cf. (96)) on E ×G G′ satisfies (331)

p

ψ w∗ (∇) = (dw)E ◦ p ψ ∇ .

In particular, the p-flatness of ∇ implies the p-flatness of w∗ (∇), and vice versa if dw is injective.

ASTÉRISQUE 432

87

3.3. p-CURVATURE ON AN ℏ-FLAT G-BUNDLE

3.3.3. Change of parameter. — Let (E , ∇) be as above and c an element of Γ(T, OT ). Then, the p-curvature p ψ c·∇ of the flat (c · ℏ)-T log -connection c · ∇ (cf. (98)) is given by p

(332)

ψ c·∇ = cp · p ψ ∇ .

In particular, the p-flatness of ∇ implies the p-flatness of c · ∇, and vice versa if c ∈ Γ(T, OT× ). 3.3.4. Products. — Let G′ be as in § 3.3.2, and let (E , ∇) (resp., (E ′ , ∇′ )) be an ℏ-flat G-bundle (resp., an ℏ-flat G′ -bundle) on Y log /T log . As discussed in § 1.3.9, the product (E ⊠ E ′ , ∇ ⊠ ∇′ ), forming an ℏ-flat G ×k G′ -bundle, can be constructed. Note that the f ′ log log (cf. (101)) preserves the p-power f log log ⊕ T f ′ log log ,→ T inclusion T E /T E /T (E ⊠E ) /T f log log ⊕ T f ′ log log is operation, where the p-power operation on the direct sum T E

/T

E

/T

taken to be the operation obtained naturally from both factors. Hence, the p-curvature p ∇⊠∇′ ψ of ∇ ⊠ ∇′ satisfies, via this inclusion, the following equality: p

(333)

′

′

ψ ∇⊠∇ = p ψ ∇ ⊕ p ψ ∇ .

In particular, ∇ ⊠ ∇′ is p-flat if and only if both ∇ and ∇′ are p-flat. 3.3.5. Canonical connections. — We shall construct a canonical connection on the Frobenius pull-back of a G-bundle. Let π : G → Y (1) be a G-bundle on Y (1) . Denote by πF : FY∗/T (G ) → Y

(334)

the G-bundle on Y defined as the base-change of G via FY /T : Y → Y (1) . Also, denote by
(335) FG /Y /T : FY∗/T (G ) := Y ×Y (1) ,π G → G the projection to the second factor (i.e., FG /Y /T := FY /T × idG ), which preserves the G-action. Differentiating this morphism yields an OFY∗/T (G ) -linear isomorphism (336)

TF ∗

Y /T

∼

(G )log /Y log

→ FG∗/Y /T (TG log /Y (1)log ).

Consider a morphism of short exact sequences of OFY∗/T (G ) -modules (337) 0

/ TF ∗ (G )log /Y log Y /T ≀ (336)

0

 / F∗ ( T log /Y (1)log ) G G /Y /T

/ TF ∗ (G )log /T log Y /T dFG /Y /T

 / F∗ ( T G log /T log ) G /Y /T

/ π ∗ (TY log /T log ) F

/0

dFY /T

 / π ∗ (F ∗ (TY (1)log /T log )) F Y /T

/ 0,

where — the upper horizontal sequence is obtained by differentiating πF ;

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— the lower horizontal sequence is obtained by differentiating π and successively pulling-back via FG /Y /T under the natural isomorphism ∼

πF∗ (FY∗/T (TY (1)log /T log )) → FG∗/Y /T (π ∗ (TY (1)log /T log ));

(338)

— the middle and right-hand vertical arrows are obtained by differentiating the morphisms FG /Y /T and FY /T respectively. By applying the functor πF ∗ (−)G to this diagram, we obtain a morphism of short exact sequence of (339) 0

/ gF ∗ (G ) Y /T

dlog F∗

f∗ /T F

Y /T

Y /T

 / πF ∗ (F ∗ (TG log /T log ))G G /Y /T

 / F ∗ (gG ) Y /T

/ TY log /T log

/0

πF ∗ (dFY /T )G

πF ∗ (dFG /Y /T )G

≀

0

(G )

(G )log /T log

 / F ∗ (TY (1)log /T log ) Y /T

/ 0.

Since d(FY∗/T (a)) = 0 for any a ∈ OY (1) , the right-hand vertical arrow πF ∗ (dFY /T )G vanishes identically on Y . Hence, the middle vertical arrow πF ∗ (dFG /Y /T )G factors through the injection FY∗/T (gG ) ,→ πF ∗ (FG∗/Y /T (TG log /T log ))G . The resulting f∗ ∗ morphism T (g ) determines, via the left-hand vertical arrow log log → F ∼

FY /T (G )

/T

Y /T

G

gFY∗/T (G ) → FY∗/T (gG ), a split surjection of the upper horizontal sequence in (339). Thus, the corresponding split injection f∗ (340) ∇can : T log log → T log log Y

G

log

/T

FY /T (G )

/T

FY∗/T (G ).

of that sequence specifies a T -connection on We will refer to it as the log ∗ canonical T -connection on FY /T (G ) (cf. [47], Theorem (5.1), for the non-logarithmic version). Moreover, given ℏ ∈ Γ(T, OT ), we obtain an ℏ-T log -connection (341)

can ∇can G ,ℏ := ℏ · ∇G

and refer to it as the canonical ℏ-T log -connection. Proposition 3.11. — Let us keep the above notation. Then, the canonical ℏ-T log -connection ∇can is flat and p-flat. G ,ℏ Proof. — By (99) and (332), it suffices to consider the case where ℏ = 1. Moreover, since the problem is of local nature, we can assume that G = Y (1) ×k G, i.e., the trivial G-bundle. Then, since F(Y (1) ×k G)/Y /T = FY /T × idG , the T log -connection ∇can G   triv ∗ (1) coincides with the trivial connection ∇ (cf. (78)) on Y ×k G = FY /T (Y ×k G) . The flatness and p-flatness of ∇triv can be verified immediately by considering the various definitions involved, so this completes the proof of the proposition.

ASTÉRISQUE 432

3.4. DORMANT/p-NILPOTENT (g, ℏ)-OPERS AND THEIR MODULI SPACES

89

3.4. Dormant/p-nilpotent (g, ℏ)-opers and their moduli spaces In this section, we introduce dormant (g, ℏ)-opers and p-nilpotent (g, ℏ)-opers, which are certain (g, ℏ)-opers characterized in terms of p-curvature. Moreover, the moduli spaces classifying these opers will be defined. Hereinafter, suppose that (G, T) is a split semisimple algebraic k-group of adjoint type (where T denotes a maximal torus T of G) with p > 2hG . Let us fix a Borel subgroup B of G defined over k containing T. 3.4.1. The adjoint quotient of the p-power operation. — Since G is a Chevalley group, all the groups G, T, and B can be defined over the prime field Fp . It follows that the induced k-scheme c (cf. (261)) arises from an Fp -scheme cFp , which means that c = cFp ×Fp k. In particular, the Frobenius twist c(1) is isomorphic to c itself, and hence, the relative Frobenius morphism Fc/k may be regarded as a k-endomorphism of c. We denote by c(Fp ) the (finite) set of Fp -rational points of cFp , i.e., the set of elements in c(k) invariant under Fc/k . The element [0] ∈ c(k) lies in c(Fp ). For each integer r ≥ 0, we write [0]×r := ([0], [0], . . . , [0]) ∈ c(Fp )×r ,

(342)

where [0]×0 := ∅. The following proposition will be used to prove Proposition 3.13 described below. Proposition 3.12. — The square diagrams g (343)

χ

/c

[g/G] Fc/k

(−)[p]

 g

χ

 /c

[χ]

/c

(−)[p]

Fc/k

 [g/G]

[χ]

 /c

are commutative (cf. (321) for the definition of the endomorphism (−)[p] of [g/G]). Proof. — The p-power operation (−)[p] on g is compatible with the adjoint G-action, so it induces an endomorphism F ′ of c via the quotient χ : g → c. On the other hand, the p-power operation (−)[p] on g restricts to that on the Lie
algebra t of T, which coincides with the Frobenius morphism Ft/k : t → t = t(1) (cf. [84], 4.4.10. Example (2)). Recall that c can be obtained as the quotient of t by the W-action, and the quotient t ↠ c is compatible with both the Frobenius morphisms and the p-power operations. Hence, F ′ must coincide with Fc/k . This completes the commutativity of the left-hand square in (343), which also implies the commutativity of the right-hand square. 3.4.2. Radius of a p-flat connection. — Let r be a positive integer, S a k-scheme, and U := (U/S, {σiU : Si → U }ri=1 ) a local r-pointed stable curve over S. After choosing a base for U , we obtain a log curve U log /S log extending U/S. Also, let ℏ be an element

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of Γ(S, OS ) and (E , ∇) an ℏ-flat G-bundle on U log /S log . For each i ∈ {1, . . . , r}, the restriction σiU ∗ (p ψ ∇ ) of p ψ ∇ via σiU may be regarded as an element of Γ(Si , gσiU ∗ (E ) ) under the identification gσiU ∗ (E ) = σiU ∗ (FU∗ (ΩU log /S log ) ⊗ gE ) given by the isomorphism
∼ (344) = FS∗i (σiU ∗ (ΩU log /S log )) → OSi FS∗i (ResU ,i ) : σiU ∗ (FU∗ (ΩU log /S log )) (cf. (141)). By the definition of p-curvature, the following equality holds: [p] σiU ∗ (p ψ ∇ ) = (µ∇ − ℏp−1 · µ∇ i ) i .

(345)

Proposition 3.13. — Let (E , ∇) be as above, and suppose further that ∇ is p-flat. Moreover, we shall fix an element i ∈ {1, . . . , r}. Then, the equality   p−1 p−1 = ρℏi ·∇ (346) Fc/k ◦ ρ∇ ⋆ ρ∇ i =ℏ i of elements of c(Si ) holds (cf. (264) for the definition of ⋆). In particular, the following assertions hold: (i) If, moreover, ℏ ∈ Γ(S, OS× ), then ℏ−1 ⋆ ρ∇ i ∈ c(Fp ); (ii) if, moreover, ℏ = 0 and Si is reduced, then ρ∇ i = [0] ∈ c(Fp ). [p] Proof. — Since ∇ is p-flat, (345) is equivalent to the equality (µ∇ = ℏp−1 · µ∇ i ) i . By Proposition 3.12, we have a sequence of equalities U∗ ∇ Fc/k ◦ ρ∇ i = Fc/k ◦ [χ]((σi (E ), µi ))

(347)

[p] = [χ]((σiU ∗ (E ), µ∇ i ) ) [p] = [χ]((σiU ∗ (E ), (µ∇ i ) ))

= [χ]((σiU ∗ (E ), ℏp−1 · µ∇ i )) = ℏp−1 ⋆ ρ∇ i , thus completing the proof of (346). ∼ Next, we consider assertions (i) and (ii). Let us fix an isomorphism c → Ark(g) := Spec(k[u1 , . . . , urk(g) ]) as in (265). The element ρ∇ i can be written, via this isomorphism, as   rk(g) (348) ρ∇ ∈ Ark(g) (Si ) i = (a1 , . . . , ark(g) ) =: (aj )j=1 for some aj ∈ Γ(Si , OSi ) (j = 1, . . . , rk(g)). Under this equality, we can describe ℏ−1 ⋆ ρ∇ i as −e1 ℏ−1 ⋆ ρ∇ · a1 , . . . , ℏ−erk(g) · ark(g) ). i = (ℏ

(349)

rk(g)

rk(g)

Moreover, (346) reads (apj )j=1 = (ℏ(p−1)·ej · aj )j=1 . If ℏ ∈ Γ(S, OS× ) (i.e., the assumption in (i) is satisfied), then, for every j = 1, . . . , rk(g), we have (350)

apj = ℏ(p−1)·ej · aj ⇐⇒ (ℏ−ej · aj )p = ℏ−ej · aj ⇐⇒ ℏ−ej · aj ∈ Fp .

ASTÉRISQUE 432

3.4. DORMANT/p-NILPOTENT (g, ℏ)-OPERS AND THEIR MODULI SPACES

91

This implies that ℏ−1 ⋆ ρ∇ i ∈ c(Fp ). On the other hand, if ℏ = 0 and Si is reduced (i.e., the assumption in (ii) is satisfied), then (351)

apj = ℏ(p−1)·ej · aj (= 0) ⇐⇒ aj = 0.

That is to say, the equality ρ∇ i = 0 holds. This completes the proof of Proposition 3.13. 3.4.3. Dormant/p-nilpotent (g, ℏ)-opers. — Let S log be an fs log scheme over k and U log a log curve over S log . For a line bundle L on U , we shall write L × for the Gm -bundle associated to L . Now, let (E , ∇) be an ℏ-flat G-bundle on U log /S log . Then, the pair (E , p ψ ∇ ) specifies a global section of the quotient stack [(FU∗ (ΩU log /S log )× ×Gm g)/G] over U ; the composite of this section with the morphism (352)

∗ [(FX (ΩU log /S log )× ×Gm g)/G] → FU∗ (ΩU log /S log )× ×Gm c

induced by χ : g → c specifies a U -rational point (353)

p

× Gm ∗ c)(U ) ψ/∇ /G ∈ (FU (ΩU log /S log ) ×

of FU∗ (ΩU log /S log )× ×Gm c. By Proposition 3.10, p ψ/∇ /G depends only on the isomorphism class of (E , ∇). Here, denote by (354)

[0]U ∈ (FU∗ (ΩU log /S log )× ×Gm c)(U )

the U -rational point determined by [0] ∈ c(k). Remark 3.14 (Change of structure group). — Let G′ , g′ , and w be as in § 3.3.2. Suppose further that G′ is a split semisimple algebraic k-group of adjoint type with p > 2hG′ . The adjoint G-action on g is compatible with the adjoint G′ -action on g′ via w and its differential dw : g → g′ . Hence, dw induces a k-morphism cg → cg′ by taking the respective quotients of g and g′ . This morphism preserves the Gm -action, so it yields a U -morphism (355)

(dw)U : FU∗ (ΩU log /S log )× ×Gm cg → FU∗ (ΩU log /S log )× ×Gm cg′ .

One may verify that (331) induces the following equality: (356)

p

w (∇)

ψ//G∗

= (dw)U ◦ p ψ/∇ /G .

w (∇)

∗ p In particular, p ψ/∇ /G = [0]U implies ψ//G

= [0]U .

Definition 3.15. — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper on U log /S log . We shall say that E ♠ is dormant (resp., p-nilpotent) if p ψ ∇ = 0 (resp., p ψ/∇ /G = [0]U ). Zzz...

p-nilp

3.4.4. The moduli spaces Opg,ℏ,X and Opg,ℏ,X . — Let (g, r) be a pair of nonnegative integers with 2g − 2 + r > 0 and X := (f : X → S, {σi }ri=1 ) an r-pointed stable curve of genus g over S. Denote by   Zzz... p-nilp (357) Opg,ℏ,X resp., Opg,ℏ,X

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(cf. [72] for the notation “Zzz...”!) the subfunctor of Opg,ℏ,X classifying dormant (g, ℏ)-opers (resp., p-nilpotent (g, ℏ)-opers). For each ρ ∈ c×r (S), we obtain the subfunctor   Zzz... p-nilp (358) Opg,ℏ,ρ,X resp., Opg,ℏ,ρ,X of Opg,ℏ,ρ,X classifying dormant (g, ℏ)-opers (resp., p-nilpotent (g, ℏ)-opers) of radii ρ. These functors may be represented by S-schemes and we have a commutative diagram of S-schemes Zzz...

/ Opp-nilp g,ℏ,ρ,X

/ Opg,ℏ,ρ,X



 / Opp-nilp g,ℏ,X

 / Opg,ℏ,X ,

Opg,ℏ,ρ,X (359) Zzz...

Opg,ℏ,X

where the arrows are all closed immersions and both sides of squares are cartesian. Proposition 3.16. — Suppose that r > 0, and let ρ be an element of c×r (S). Then, the following assertions hold: Zzz...

(i) Suppose further that ℏ ∈ Γ(S, OS× ). Then, Opg,ℏ,ρ,X = ∅ unless ℏ−1 ⋆ρ ∈ c(Fp )×r . In particular, we have a Zzz... Zzz... (360) Opg,ℏ,X = Opg,ℏ,ℏ⋆ϱ,X . ϱ∈c(Fp )×r Zzz...

(ii) Suppose further that ℏ = 0 and S is reduced. Then, Opg,ℏ,ρ,X = ∅ unless ρ = [0]×r ∈ c×r (k) (⊆ c×r (S)). In particular, we have Zzz...

Zzz...

Opg,0,X = Opg,0,[0]×r ,X .

(361)

Proof. — Assertion (i) (resp., (ii)) follows directly from Proposition 3.13, (i) (resp., (ii)). Proposition 3.17. — Let us fix a collection ↙ q1 as in (172). (i) Let E⊙♠ := (EB⊙ , ∇) be a (g⊙ , ℏ)-oper on X . Denote by ιG∗ (E⊙♠ ) := (EB , ιG∗ (∇)) the (g, ℏ)-oper associated to E⊙♠ and ↙ q1 (cf. (208)). Then, the equality p

(362)

ψ ιG∗ (∇) = dιEB⊙ ◦ p ψ ∇

holds (cf. (206) for the definition of dιEB⊙ ). In particular, E⊙♠ is dormant if and only if ιG∗ (E⊙♠ ) is dormant. (ii) The closed immersion ιOp q1 (cf. (243)) g : Opg⊙ ,ℏ,X ,→ Opg,ℏ,X associated to ↙ restricts to a closed immersion (363)

ASTÉRISQUE 432

p Op ιg

Zzz...

Zzz...

: Opg⊙ ,ℏ,X → Opg,ℏ,X .

3.5. THE HITCHIN-MOCHIZUKI MORPHISM

93

Moreover, this morphism makes the following commutative diagram cartesian: p Op ιg

Zzz...

Opg⊙ ,ℏ,X (364)

/ OpZzz... g,ℏ,X inclusion

inclusion

 Opg⊙ ,ℏ,X

 / Opg,ℏ,X .

ιOp g

Proof. — The former assertion of (i) follows from the fact that dιEB⊙ preserves the p-power operation. The latter assertion of (i) follows from the injectivity of dιEB⊙ . Finally, assertion (ii) follows directly from assertion (i). Remark 3.18. — Let us keep the notation in Remarks 2.26 and 2.38 (under the assumption p > 2hG ). Then, it follows from (333) that (248) restricts to an S-isomorphism (365)

Zzz...

Zzz...

∼

Zzz...

Opg,ℏ,X ×S Opg′ ,ℏ,X → Opg⊕g′ ,ℏ,X .

If r > 0, then for ρ ∈ cg (Fp )×r and ρ′ ∈ cg′ (Fp )×r , (302) restricts to an S-isomorphism (366)

Zzz...

Zzz...

∼

Zzz...

Opg,ℏ,ρ,X ×S Opg′ ,ℏ,ρ′ ,X → Opg⊕g′ ,ℏ,(ρ,ρ′ ),X .

Also, the same assertions hold for the moduli space of p-nilpotent (g ⊕ g′ , ℏ)-opers (of radii (ρ, ρ′ )). 3.5. The Hitchin-Mochizuki morphism 3.5.1. — Let us fix a collection ↙ q1 := {xα }α of generators of gα ’s (for simple positive roots α in B) as in (172). This collection associates elements q1 (cf. (173)) and q−1 ∼ (cf. (174)), and hence we obtain an isomorphism Kosg : gad(q1 ) → c (cf. (267)). We shall denote by (367)

Ψℏ : cS → cS

the S-endomorphism of cS := S ×k c given by assigning (368)

[p] λ 7→ Ψℏ (λ) := χ ◦ ((q−1 + Kos−1 − ℏp−1 · (q−1 + Kos−1 g (λ)) g (λ)))

for any λ ∈ cS (S ′ ), where S ′ is an S-scheme. If r > 0, then for each ρ := (ρi )ri=1 ∈ c×r (S) (= cS (S)×r ), we shall write (369)

Ψℏ (ρ) := (Ψℏ (ρi ))ri=1 ∈ c×r (S).

Remark 3.19. — Let E ♠ := (EB , ∇) be a (g, ℏ)-oper on X . Then, for every i = 1, . . . , r, we have (370)

χ(σi∗ (p ψ ∇ )) = Ψℏ (ρ∇ i ).

To prove this equality, we may assume that E ♠ is ↙ q1 -normal (cf. Propositions 2.19 and 3.10). Under this assumption, the monodromy µ∇ i of ∇ satisfies

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−1 ∇ ∇ σi∗ (µ∇ i ) = q−1 + Kosg (ρi ). Hence, (345) deduces (370). In particular, Ψℏ (ρi ) = [0] if E ♠ is p-nilpotent.

Example 3.20. — Suppose that ℏ = 0. Then, it follows from Proposition 3.12 that the equality Ψ0 (λ) = Fc/k ◦ λ

(371)

holds for λ ∈ cS (S ′ ), where S ′ is any S-scheme.  Thatis to say, Ψ0 coincides with the (1) relative Frobenius morphism FcS /S : cS → cS = cS of cS over S. 3.5.2. A grading on the coordinate ring of cS . — The OS -algebra OS ⟨cS ⟩ correspondL ing to the affine S-scheme cS has a Z-grading OS ⟨cS ⟩ = j∈Z OS ⟨cS ⟩j induced by the Gm -action on c opposite to the action ⋆. This grading is positive, meaning that OS ⟨cS ⟩j = 0 if j < 0. By letting M j (372) OS ⟨cS ⟩ := OS ⟨cS ⟩j ′ , j ′ ≤j

we obtain an increasing filtration {OS ⟨cS ⟩j }j∈Z≥0 on OS ⟨cS ⟩. Proposition 3.21. — to Ψℏ satisfies

(i) The OS -algebra endomorphism Ψ∗ℏ of OS ⟨cS ⟩ corresponding Ψ∗ℏ (OS ⟨cS ⟩j ) ⊆ OS ⟨cS ⟩p·j

(373) for each integer j ≥ 0.

(ii) For each integer j ≥ 0, we shall denote by grj (Ψ∗ℏ ) the OS -linear morphism

j j−1 (374) OS ⟨cS ⟩j = OS ⟨cS ⟩ /OS ⟨cS ⟩ → OS ⟨cS ⟩p·j = OS ⟨cS ⟩p·j /OS ⟨cS ⟩p·j−1 induced by Ψ∗ℏ | (375)

. Then, the OS -algebra endomorphism   M M gr(Ψ∗ℏ ) := grj (Ψ∗ℏ ) : OS ⟨cS ⟩ →  OS ⟨cS ⟩p·j ⊆ OS ⟨cS ⟩ OS ⟨cS ⟩j

j∈Z≥0

j∈Z≥0

of OS ⟨cS ⟩ coincides with the OS -algebra endomorphism Fc∗S /S : OS ⟨cS ⟩ → OS ⟨cS ⟩ induced by FcS /S . (iii) Ψℏ is finite and faithfully flat of degree prk(g) . Proof. — To prove the assertions, it suffices to consider the case where S = A1 := ˆ and ℏ = ℏ. ˆ Indeed, a general case can be deduced from this case by baseSpec(k[ℏ]) change via the morphism ℏ : S → A1 . Also, we may assume, without loss of generality, that k is algebraically closed. First, we shall consider assertion (i). Let α : G ×k gad(q1 ) → g be the morphism given by (h, v) 7→ AdG (h)(q−1 + v). Also, let Ψg,ℏˆ be the A1 -endomorphism of A1 ×k g

ASTÉRISQUE 432

95

3.5. THE HITCHIN-MOCHIZUKI MORPHISM

ˆp−1 · v. Then, for each (h, v) ∈ G ×k gad(q1 ) , we have given by v 7→ v [p] − ℏ (376) χ ◦ Ψg,ℏˆ ◦ α((h, v)) = χ ◦ Ψg,ℏˆ (AdG (h)(q−1 + v)) ˆp−1 · AdG (h)(q−1 + v)) = χ((AdG (h)(q−1 + v))[p] − ℏ ˆp−1 · (q−1 + v))) = χ(AdG (h)((q−1 + v)[p] − ℏ ˆp−1 · (q−1 + v)) = χ((q−1 + v)[p] − ℏ [p] ˆp−1 · (q−1 + Kos−1 (χ(v)))) = χ((q−1 + Kos−1 −ℏ g (χ(v))) g

= Ψℏˆ ◦ χ(v) = Ψℏˆ ◦ χ(AdG (h)(v)) = Ψℏˆ ◦ χ ◦ α((h, v)). This implies χ ◦ Ψg,ℏˆ ◦ α = Ψℏˆ ◦ χ ◦ α. The elements q−1 + v for various v ∈ gad(q1 ) are regular (cf. [64], the comment preceding Lemma 1.2.3), so the morphism α is dominant. Hence, we have χ ◦ Ψg,ℏˆ = Ψℏˆ ◦ χ. If ˆ ⊗k Sk (g∨ ) → k[ℏ] ˆ ⊗k Sk (g∨ ) Ψ∗g,ℏˆ : k[ℏ]

(377)

ˆ denotes the k[ℏ]-algebra endomorphism corresponding to Ψg,ℏˆ , then the equality χ ◦ Ψg,ℏˆ = Ψℏˆ ◦ χ induces a commutative square diagram   inclusion / ˆ ⊗k Sk (g∨ ) ˆ ⊗k Sk (g∨ )G k[ℏ] Γ(A1 , OA1 ⟨cA1 ⟩) = k[ℏ] Ψ∗ ˆ g,ℏ

Ψ∗ ˆ ℏ

(378)

   ˆ ⊗k Sk (g∨ )G Γ(A , OA1 ⟨cA1 ⟩) = k[ℏ] 1

inclusion

 ˆ ⊗k Sk (g∨ ). / k[ℏ]

By the definition of Ψg,ℏˆ , we obtain the inclusion relation (379)

ˆ ⊗k S≤j (g∨ )) ⊆ k[ℏ] ˆ ⊗k S≤p·j (g∨ ) Ψ∗g,ℏˆ (k[ℏ] k k ′

∨ ′ ∨ for each j ∈ Z≥0 , where S≤j k (g ) (j ∈ Z≥0 ) denotes the k-subspace of Sk (g ) con′ sisting of elements of degree ≤ j . Since the inclusion   ˆ ⊗k Sk (g∨ )G ,→ k[ℏ] ˆ ⊗k Sk (g∨ ) (380) Γ(A1 , OA1 ⟨cA1 ⟩) = k[ℏ]

preserves the grading, (379) and the commutativity of (378) together imply that (381)

Ψ∗ℏˆ (Γ(A1 , OA1 ⟨cA1 ⟩j )) ⊆ Γ(A1 , OA1 ⟨cA1 ⟩p·j ).

This completes the proof of assertion (i). ˆ Next, we shall consider assertion (ii). Denote by Ψ∗g,0 the k[ℏ]-algebra endomor∨ 1 ˆ ⊗k Sk (g ) corresponding to the endomorphism A ×k g → A1 ×k g given phism of k[ℏ]

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by the p-power operation (−)[p] on g. Also, let gr(Ψ∗g,ℏˆ ) and gr(Ψ∗g,0 ) denote the en  ˆ ⊗k Sk (g∨ ) = gr(k[ℏ] ˆ ⊗k Sk (g∨ )) induced, via taking gradings, domorphisms of k[ℏ] from Ψ∗g,ℏˆ and Ψ∗g,0 respectively. By definition, the equality gr(Ψ∗g,ℏˆ ) = gr(Ψ∗g,0 ) holds. Hence, the commutativity of (378) implies the commutativity of the following square diagram: Γ(A1 , OA1 ⟨cA1 ⟩)

inclusion

ˆ ⊗k Sk (g∨ ) / k[ℏ] gr(Ψ∗ g,0 )

gr(Ψ∗ ˆ) ℏ

(382)

 Γ(A1 , OA1 ⟨cA1 ⟩)

inclusion

 ˆ ⊗k Sk (g∨ ). / k[ℏ]

On the other hand, by Proposition 3.12,   the inclusion (380) is compatible with the ∗ ∗ endomorphisms Fc 1 /A1 = gr(Fc 1 /A1 ) , gr(Ψ∗g,0 ). Thus, we have gr(Ψ∗ℏˆ ) = Fc∗ 1 /A1 , A A A as desired. Finally, we shall consider assertion (iii). By assertion (ii), Γ(A1 , OA1 ⟨cA1 ⟩) turns out to be a finite gr(Ψ∗ℏˆ )(Γ(A1 , OA1 ⟨cA1 ⟩))-module. It follows from a routine argument that Γ(A1 , OA1 ⟨cA1 ⟩) is a finite Ψ∗ℏˆ (Γ(A1 , OA1 ⟨cA1 ⟩))-module, i.e., Ψℏˆ is finite. Since the domain and codomain of Ψℏˆ are irreducible, smooth, and of the same dimension, one may verify that Ψℏˆ is also faithfully flat. The computation of its degree may be ˆ = 0” accomplished by computing the degree of its fiber Ψℏˆ |ˆ over the point “ ℏ ℏ=0 in A1 . But, Ψℏˆ |ˆ coincides with Fc/k (cf. Example 3.20), so its degree is given
ℏ=0 by pdim(c) = prk(g) . This completes the proof of assertion (iii). 3.5.3. The spindles

L N

(383)

X

and

L N∇

. — Denote by L N L N∇ X (resp., X)

X

L N (cf. [72] for the notation “ ” that represents a spindle) the Set-valued contravariant functor on Sch/S which, to any S-scheme s : S ′ → S, assigns the set of mor∗ phisms S ′ ×S X → (FX (ΩX log /S log ))× ×Gm c (resp., S ′ ×S X → Ω× ×Gm c) X log /S log × over X. Here, for each line bundle LL , we denote by L the Gm -torsor correspondN L N∇ ing to L . One may verify that both and X X may be represented by a finite type S-scheme (cf. (396) in the proof of Proposition 3.23). There exists a canonical L N∇ L N∇ identification X (1) = ( X )(1) between S-schemes. Next, let us suppose r > 0. Given an S-scheme s : S ′ → S and an element w L N L N∇ that ′ ′ of X (S ) (resp., X (S )), we obtain, for each i ∈ {1, . . . , r}, the composite S-morphism (384)

w|

σi

∼

σ ∗ (w)

∼

∼

: S ′ → S ′ ×S S −−i−−→ σi∗ (Ω)× ×Gm c → OS× ×Gm c → cS (= S ×k c) ,

∗ where Ω := FX (ΩX log /S log ) (resp., Ω := ΩX log /S log ) and the third arrow arises ∼ from the isomorphism σi∗ (Ω) → OS induced by ResX ,i (cf. (141)). The assignment

ASTÉRISQUE 432

97

3.5. THE HITCHIN-MOCHIZUKI MORPHISM

w 7→ (w| )ri=1 defines an S-morphism σi   N N∇ ⊕ :L ⊕ ,∇ : L ×r ×r (385) Rad⊗ resp., Rad⊗ , X → cS X → cS where c×r S denotes the product of r copies of cS over S. 3.5.4. Sleeping Beauty. — Denote by

κ

(386)

ℏ

: Opg,ℏ,X →

L N

X

κ

(cf. [72] for the notation “ ” that represents an old fairy) the S-morphism which, to each (g, ℏ)-oper E ♠ := (EB , ∇), assigns p ψ/∇ /G . Let us set L N (387) ☛X : S → X (cf. [72] for the notation “☛” that represents a princess’s finger) to be the morphism ∗ determined by [0]X : X → (FX (ΩX log /S log ))× ×Gm c (cf. (354)). This morphism makes the following square diagram cartesian:  Zzz...  p-nilp inclusion / Opg,ℏ,X ⊆ Opg,ℏ,X Opg,ℏ,X

κ

(388)  S

☛X

/

L N

X

ℏ

.

Lemma 3.22. — Suppose that r > 0, and let E ♠ := (EB , ∇) be a (g, ℏ)-oper on X . Then, for each i = 1, . . . , r, we have the following equality of elements in c(S): p

(389)

ψ/∇ /G |

σi

= Ψℏ (ρ∇ i ).

(cf. (367) for the definition of Ψℏ ). Proof. — The formations of both sides of (389) depend only on the isomorphism class of E ♠ and the problem is of local nature, Hence, after possibly replacing S with its covering in the Zariski topology, one may assume that there exists a log chart (U, ∂) on X log /S log with Im(σi ) ⊆ U and E ♠ is ↙ q1 -normal (cf. Proposition 2.19). If ∼

τi : g(S) (= Γ(S, OS ⊗k g)) → Γ(S, gσi∗ († EG ) )

(390)

∼

denotes the isomorphism induced naturally by trivG,(U,∂) : U ×k G → † EG | (cf. (211) U

−1 ∇ and (275)), then the equality µ∇ i = τi (q−1 + Kosg (ρi )) holds. Hence, we have

(391) p ∇ ψ//G |

σi

= [χ]((σi∗ († EG ), σi∗ (p ψ ∇ ))) [p] = [χ]((σi∗ († EG ), (µ∇ − ℏp−1 · µ∇ i ) i )) ∇ [p] ∇ = [χ]((σi∗ († EG ), τi (q−1 + Kos−1 − ℏp−1 · τi (q−1 + Kos−1 g (ρi )) g (ρi )))) ∇ [p] ∇ = [χ]((σi∗ († EG ), τi ((q−1 + Kos−1 − ℏp−1 · (q−1 + Kos−1 g (ρi )) g (ρi )))))

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CHAPTER 3. OPERS IN POSITIVE CHARACTERISTIC

∇ [p] ∇ = [χ]((S ×k G, (q−1 + Kos−1 − ℏp−1 · (q−1 + Kos−1 g (ρi )) g (ρi )))) ∇ [p] ∇ = χ ◦ ((q−1 + Kos−1 − ℏp−1 · (q−1 + Kos−1 g (ρi )) g (ρi )))

= Ψℏ (ρ∇ i ), where the second equality follows from (345). This completes the proof of the assertion.

Opg,ℏ,X (392)

κ

The above lemma implies the commutativity of the following diagram: ℏ

N /L

X ⊗ Rad⊕

Radℏ

 c×r S

Ψ×r ℏ

 / c×r , S

κ

κ

×0 ×r where Ψ×r ℏ denotes the r-th power of Ψℏ . (We set cS := S and Ψℏ := idS when ×r ×r ×0 r = 0.) Given an element ρ ∈ c (S) = cS (S), where c (S) := {∅} and ρ := ∅ if r = 0, we shall write   L N N∇ ⊗ ⊕ )−1 (ρ) resp., L ⊗ ⊕ ,∇ )−1 (ρ) , (393) := (Rad := (Rad ρ,X ρ,X L N L N∇ i.e., the closed subscheme of X (resp., X ) defined to be the scheme-theretic ⊗ ⊕ ⊗ ⊕ ,∇ inverse image of ρ via Rad (resp., Rad ). By the commutativity of (392), the morphism ℏ restricts to an S-morphism L N (394) ℏ,ρ : Opg,ℏ,ρ,X → Ψℏ (ρ),X .

3.5.5. The Hitchin-Mochizuki morphism. — Given an element ρ of c×r (S), we shall denote by   N∇ L N L N∇ L N ⊗ ⊕ :L ⊗ ⊕ (395) FX/S resp., FX/S,ρ : ρ,X (1) → ρ,X X (1) → X the S-morphism given by pull-back via FX/S . ⊗ ⊕ (resp., F ⊗ ⊕ Proposition 3.23. — The morphism FX/S X/S,ρ ) is a closed immersion. ⊗ ⊕ ⊗ ⊕ , it suffices to prove Proof. — Since FX/S,ρ may be obtained as the restriction of FX/S the non-resp’d assertion. ∼ Let us fix an isomorphism c → Spec(k[u1 , . . . , urk (g)]) as in (265). If L is a line bundle on either X or X (1) , then the isomorphism fixed right now allows us to identify Lrk(g) ⊗el L × ×Gm c with the relative affine space associated to , i.e., we have L × ×Gm l=1 L L rk(g) c∼ = V( l=1 L ⊗el ). Hence, there exist isomorphisms of S-schemes rk(g)

(396)

L N∇

X

∼

(1)

→

(1) V(f∗ (

M l=1

ASTÉRISQUE 432

rk(g) ⊗el ΩX (1)log /S log )),

L N

X

∼

→ V(f∗ (

M l=1

⊗el ∗ FX (ΩX log /S log ))).

99

3.5. THE HITCHIN-MOCHIZUKI MORPHISM

For l = 1, . . . , rk(g), we shall write 

⊗el ∗ l (397) ωl : Ω⊗e (ΩX → FX/S∗ (FX/S (1)log /S log )) X (1)log /S log

 ⊗el ∗ ∼ (ΩX )) = FX/S∗ (FX log /S log

for the morphism of OX (1) -modules corresponding, via the adjunction relation ⊗el ∗ ∗ “FX/S (−) ⊣ FX/S∗ (−),” to the identity morphism of FX/S (ΩX (1)log /S log ). The ∗ l FX/S∗ (FX (Ω⊗e )) is relatively torsion-free of rank p (cf. PropoX log /S log sition 3.2) and ωl is easily verified to be injective over the scheme-theoretically dense open subscheme X (1)sm of X (1) \ Supp(DX (1) ) (cf. § 1.6). It follows that the morphism ωl , hence also its direct image f (1) (ωl ), is injective. Since el ≥ 2, the following equalities hold:

OX (1) -module

(1)

R1 f∗ (ΩeXl (1)log /S log ) = 0,

(398)

(1)

⊗el ∗ R1 f∗ (FX/S∗ (FX (ΩX log /S log )))



 ⊗p·el = R1 f∗ (ΩX = 0. log /S log )

(1)

Hence, both the domain and codomain of f∗ (ωl ) are locally free (cf. [32], Chap. III, (1) Theorem 12.11 (b)), and the formation of f∗ (ωl ) is functorial with respect to S. By considering the above argument with S replaced by various S-schemes, we see (1) that f∗ (ωl ) is universally injective with respect to base-change to S-schemes. In (1) particular, Coker(f∗ (ωl )) turns out to be flat over S (cf. [56], p. 17, Theorem 1), and the dual (1)

(1)

(1)

⊗el ⊗el ∗ ∨ ∨ f∗ (ωl )∨ : f∗ (FX/S∗ (FX (ΩX → f∗ (ΩX log /S log ))) (1)log /S log )

(399)

⊗ ⊕ is obtained by (taking the of f∗ (ωl ) is surjective for all l. The morphism FX/S spectrum of) the surjection (1)

rk(g)

SOS (

(400)

M

rk(g) (1)

⊗el ∗ ∨ f∗ (FX/S∗ (FX (ΩX log /S log ))) ) ↠ SOS (

l=1

M

(1)

⊗el ∨ f∗ (ΩX (1)log /S log ) ),

l=1

Lrk(g)

(1) f∗ (ωl )∨ ,

arising from l=1 proof of the assertion.

so it defines a closed immersion. This completes the

κ

κ

κ

κ

Proposition-Definition 3.24. — Let ρ be an element of c×r (S). Then, the morphism ℏ ⊗ ⊕ (resp., F ⊗ ⊕ (resp., ℏ,ρ ) factors through the closed immersion FX/S X/S,Ψℏ (ρ) ). That is to say, there exists a unique morphism   L N∇ L N∇ ∇ resp., ∇ (401) ℏ : Opg,ℏ,X → ℏ,ρ : Opg,ℏ,ρ,X → X (1) Ψℏ (ρ),X (1) satisfying

∇ ℏ

∇ ℏ,ρ )

(resp.,

⊗ ⊕ resp., FX/S,Ψ ◦ ℏ (ρ)

∇ ℏ,ρ

=

κ

ℏ

κ



=

κ

κ

κ

We will refer to

∇ ℏ

κ

⊗ ⊕ ◦ FX/S

(402)

 ℏ,ρ

.

as the Hitchin-Mochizuki morphism.

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κ

Proof. — It suffices to consider the non-resp’d assertion because the resp’d assertion is entirely similar. In what follows, let us keep the notation in the proof of Proposition 3.23. Let E ♠ be a (g, ℏ)-oper on X , which determines an S-rational point sE ♠ of Opg,ℏ,X ; its image ℏ (sE ♠ ) corresponds, via the second isomorphism in (396), to an element rk(g)

(403)

RE ♠ ∈ Γ(X (1) ,

M

⊗el ∗ FX/S∗ (FX (ΩX log /S log ))).

l=1

κ

By an argument similar to the argument discussed in the proof of [19], Proposition 3.1 (or [53], Proposition 3.2), we see that the restriction RE ♠ | sm of RE ♠ to X the scheme-theoretically dense open subscheme X sm of X \ Supp(DX ) lies in the Lrk(g) L image of l ωl is relatively torsion-free (cf. l=1 wl |X sm . Since the cokernel of Lrk(g) Proposition 3.3), RE ♠ itself lies in the image of l=1 wl . By considering this fact with E ♠ replaced with the base-change to every S-scheme, we conclude that ℏ factors ⊗ ⊕ . This completes the proof of the assertion. through FX/S

Opg,ℏ,X (404)

κ

According to Proposition-Definition 3.24, the commutative diagram (392) induces a commutative square diagram ∇ ℏ

X

⊕ ,∇ Rad⊗

Radℏ

 c×r S

N∇ /L (1)

Ψ×r ℏ

 / c×r . S

In particular, since both Ψ×r ℏ and Radℏ are surjective (cf. Proposition 3.21, (iii), and ⊕ ,∇ is surjective. Theorem 2.36), we see that Rad⊗

3.6. Varying the parameter ℏ

κ

ˆ ˆ We shall discuss dormant and p-nilpotent (g, ℏ)-opers with universal parameter ℏ. After that, we describe the degeneration of the Hitchin-Mochizuki morphism ℏ at ℏ = 0. Zzz... ˆ ˆ 3.6.1. The Gm -action on Opg,ℏ, ˆ Xˆ . — Let us keep the notation in § 2.9 (e.g., ℏ, X ,

and D, etc.). Also, let us take an element ρ ∈ c×r (S). It follows from (332) that the Gm -action on Opg,ℏ, ˆ Xˆ (resp., Opg,ℏ, ˆ ρ, ˆ Xˆ ) defined in (286) (cf. (288)) restricts to a Gm -action   Zzz... Zzz... Zzz... Zzz... (405) Gm × Opg,ℏ, resp., Gm × Opg,ℏ, → Opg,ℏ, ˆ Xˆ → Opg,ℏ, ˆ Xˆ ˆ ρ, ˆ ρ, ˆ Xˆ ˆ Xˆ

ASTÉRISQUE 432

101

3.6. VARYING THE PARAMETER ℏ

Zzz...

Zzz...

on Opg,ℏ, ). If ℏ is an element of Γ(S, OS× ), then this morphism ˆ Xˆ (resp., Opg,ℏ, ˆ ρ, ˆ Xˆ restricts to an isomorphism ∼

Zzz...

Zzz...

Gm ×k Opg,ℏ,X → D−1 ((Gm )S )) ∩ Opg,ℏ, ˆ Xˆ

(406)


Zzz... Zzz... ∼ resp., Gm ×k Opg,ℏ,ℏ⋆ρ,X → D−1 . ˆ ρ, ρ ((Gm )S )) ∩ Opg,ℏ, ˆ Xˆ In particular, the assignment E ♠ 7→ ℏ · E ♠ determines an isomorphism   Zzz... Zzz... Zzz... Zzz... ∼ ∼ (407) Opg,1,X → Opg,ℏ,X resp., Opg,1,ρ,X → Opg,ℏ,ℏ⋆ρ,X over S. Equations (406) and (407) are compatible with (291) and (292) respectively. p-nilp

3.6.2. The Gm -action on Opg,ℏ, ˆ Xˆ . — On the other hand, the Gm -action on c defined p by (c, ϱ) 7→ c ⋆ ϱ (for any c ∈ Gm , ϱ ∈ c) induces a Gm -action L N L N (408) Gm ×k Xˆ → Xˆ . It follows from (332) that the following square diagram is commutative: (291)

(409)

κ

idGm ×

/ Op ˆ ˆ g,ℏ,X

κ

Gm ×k Opg,ℏ, ˆ Xˆ ˆ ℏ

 L N Gm ×k Xˆ

(408)

ˆ ℏ

N /L Xˆ .

Since the inverse image of Im(☛Xˆ ) via (408) coincides with Gm ×k Im(☛Xˆ ), the above diagram induces a Gm -action   p-nilp p-nilp p-nilp p-nilp (410) Gm × Opg,ℏ, → Op resp., G × Op → Op m ˆ Xˆ ˆ ρ, ˆ ρ, ˆ Xˆ g,ℏ, g,ℏ, ˆ Xˆ g,ℏ, ˆ Xˆ p-nilp

p-nilp

Zzz...

Zzz...

on Opg,ℏ, ) compatible with that on Opg,ℏ, ). ˆ Xˆ (resp., Opg,ℏ, ˆ ρ, ˆ Xˆ (resp., Opg,ℏ, ˆ ρ, ˆ Xˆ ˆ Xˆ Moreover, if ℏ ∈ Γ(S, OS× ), then (291) restricts to (411)

p-nilp

∼

p-nilp

Gm ×k Opg,ℏ,ℏ⋆ρ,X → D−1 ((Gm )S )) ∩ Opg,ℏ, ˆ Xˆ
p-nilp p-nilp ∼ resp., Gm ×k Opg,ℏ,ℏ⋆ρ,X → D−1 , ˆ ρ, ρ ((Gm )S )) ∩ Opg,ℏ, ˆ Xˆ

which is compatible with (406). In particular, the assignment E ♠ 7→ ℏ·E ♠ determines an S-isomorphism (412)

p-nilp

∼

p-nilp

Opg,1,ρ,X → Opg,ℏ,ℏ⋆ρ,X .

3.6.3. Degeneration at ℏ = 0. — We now specialize to the case of ℏ = 0. Since the ∼ isomorphism Kosg : gad(p1 ) → c (cf. (267)) is Gm -equivariant with respect to the Gm -action on gad(p1 ) (1) and the Gm -action ⋆ on c, it induces an isomorphism N∇ ∼ L (413) V(f∗ († Vg,X )) → X

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CHAPTER 3. OPERS IN POSITIVE CHARACTERISTIC

L N∇ of S-schemes (cf. the proof of Proposition 2.23). In particular, X is a relative affine spaces over S of relative dimension ℵ(g) (cf. Proposition 2.23, (ii)). For an element ρ := (ρi )ri=1 ∈ c×r (S) (where c×r (S) := {∅} and ρ := ∅ if r = 0), denote by V(f∗ († Vg,X ))ρ

(414)

the closed subscheme of V(f∗ († Vg,X )) corresponding to Proposition 3.25. —

ρ,X

(⊆

L N∇

X

) via (413).

(i) The square diagram (252)

V(f∗ († Vg,X )) (415)

L N∇

/ Opg,0,X

∼

κ

(413) ≀

 L N∇ N∇ (1)
/L X (1) = ( X)

L N ∇

X

F⊕ ⊗∇

X /S

κ

is commutative. In particular,

∇ 0

∇ 0

is finite and faithfully flat of degree pℵ(g) .

(ii) Let ρ be an element of ρ ∈ c×r (S). Then, (252) restricts to an S-isomorphism ∼

V(f∗ († Vg,X ))ρ → Opg,0,ρ,X ,

(416)

and (415) restricts to a commutative diagram (416)

V(f∗ († Vg,X ))ρ

/ Opg,0,ρ,X

κ

∼

≀

(417)

L N∇

ρ,X

F⊕ ⊗∇

∇ 0,ρ

 L N∇ (1)  = ( , (1) Ψℏ (ρ),X ρ,X )

N∇ /L

ρ,X /S

κ

where the left-hand vertical arrow denotes the restriction of (413) to V(f∗ († Vg,X ))ρ . c ℵ(g) In particular, ∇ . 0,ρ is finite and faithfully flat of degree p Proof. — Assertion (i) follows from the definitions of the various morphisms in (415) and Proposition 3.12. Assertion (ii) follows from assertion (i) together with the result of Example 3.20. The following assertion is a direct consequence of the above proposition. Corollary 3.26. — Suppose that we are given ρ ∈ c×r (S) with Ψℏ (ρ) = [0]×r . Then, p-nilp p-nilp Opg,0,X (resp., Opg,0,ρ,X ) is finite and faithfully flat over S of degree pℵ(g) (resp., c

p-nilp

p-nilp

p ℵ(g) ). Moreover, Opg,0,X (resp., Opg,0,ρ,X ) is, Zariski locally on S, isomorphic to the S-scheme S ×k Spec(k[ε1 , . . . , εℵ(g) ]/(εp1 , . . . , εpℵ(g) ))

(418) 

ASTÉRISQUE 432

resp., S ×k Spec(k[ε1 , . . . , εc ℵ(g) ]/(εp1 , . . . , εpc ℵ(g) ))



103

3.7. THE FINITENESS OF THE HITCHIN-MOCHIZUKI MORPHISM

(cf. the following remark for a more detailed description of this S-scheme). Remark 3.27. — For a vector bundle V on S, we write Jp (V ) for the relative affine scheme over S determined by the OS -algebra SOS (V ∨ )/⟨SpOS (V ∨ )⟩, i.e., the quotient of the symmetric OS -algebra SOS (V ∨ ) by the ideal sheaf generated by the homogenous local sections of degree p. Then, by taking account of (218), (388), and Proposition 3.25, p-nilp p-nilp we can describe Opg,0,X (resp., Opg,0,ρ,X ) as rk(g) p-nilp

Opg,0,X

(419)

∼ = Jp (f∗ († Vg )) ∼ =

Y

⊗(j+1)

ad(q1 )

Jp (f∗ (ΩX log /S log ))×dim(gi

)

j=1





rk(g)

p-nilp resp., Opg,0,ρ,X ∼ = Jp (f∗ († Vg (−DX ))) ∼ =

Y

⊗(j+1)

ad(q ) ×dim(gi 1 )

Jp (f∗ (ΩX log /S log (−DX )))

,

j=1

where “

Q ” means product over S and (−)×d means d-th power product over S. Zzz...

Corollary 3.28. — Opg,0,X is finite over S. Moreover, if S is reduced, then we have Zzz... (Opg,0,X )red ∼ = S,

(420)

where the left-hand side denotes the reduced scheme associated to Opg,0,X . Proof. — The assertion follows from Corollary 3.26.

3.7. The finiteness of the Hitchin-Mochizuki morphism

κ

The goal of this section is to obtain a detailed understanding of the morphism

∇ ℏ(,ρ)

Zzz...

for an arbitrary ℏ. As a corollary, the finiteness of Opg,ℏ,X will be proved. L N∇ 3.7.1. The filtration on OS ⟨ X ⟩. — As constructed in (258), there exists a canonical filtration {OS ⟨Opg,ℏ,X ⟩j }j∈Z≥0 on the OS -algebra OS ⟨Opg,ℏ,X ⟩. On the other L N∇hand, the Gm -action on c opposite to “⋆” (cf. (264)) induces a Z≥0 -grading OS ⟨ X ⟩ = L L N∇ L N∇ j∈Z≥0 OS ⟨ X ⟩j on OS ⟨ X ⟩. By letting M L N∇ j L N∇ OS ⟨ X ⟩ := OS ⟨ X ⟩j ′ , (421) j ′ ≤j

κ

L N∇ we obtain an increasing filtration {OS ⟨ X ⟩j }j∈Z≥0 . Let us denote by L N∇ ∇∗ (422) ℏ : OS ⟨ X (1) ⟩ → OS ⟨Opg,ℏ,X ⟩.

κ

the OS -algebra morphism induced by

∇ ℏ

. Then, we obtain the following assertion:

κ

Lemma 3.29. — For any j ∈ Z≥0 , the inclusion relation L N∇ ∇∗ j p·j (423) ℏ (OS ⟨ X (1) ⟩ ) ⊆ OS ⟨Opg,ℏ,X ⟩ is fulfilled.

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CHAPTER 3. OPERS IN POSITIVE CHARACTERISTIC

Proof. — To prove this lemma, one may assume, after possibly changing the base S to its Zariski open covering, that there exists a (g, ℏ)-oper on X . If we fix a ↙ q1 -nor♠ † mal (g, ℏ)-oper E := ( EB , ∇) on X (cf. Proposition 2.19), then the assignment ♠ R 7→ E+R := († EB , ∇ + R) (cf. (229)) for each R ∈ f∗ († Vg,X ) determines an S-iso∼ morphism τX : V(f∗ († Vg,X )) → Opg,ℏ,X . The OS -algebra isomorphism ∼

∗ τX : OS ⟨Opg,ℏ,X ⟩ → OS ⟨V(f∗ (Vg,X ))⟩

(424)

corresponding to τX preserves the filtration because of the definition of {OS ⟨Opg,ℏ,X ⟩j }j . ∗ Next, let us consider the Z≥0 -grading on OS ⟨V(f∗ (FX (ΩX log /S log ) ⊗ g† EG ))⟩ induced by the Gm -action on g opposite to the action coming from the homotheties on g; this grading defines, in the natural manner, an increasing filtration  ∗ j (425) OS ⟨V(f∗ (FX (ΩX log /S log ) ⊗ g† EG ))⟩ . j∈Z ≥0

†

∗ V(f∗ (FX (ΩX log /S log ) ⊗ g† EG ))

Denote by ψX : V(f∗ ( Vg,X )) → the S-morphism given by R 7→ p ψ ∇+R . By the definition of p-curvature, the OS -algebra morphism ∗ ∗ ψX : OS ⟨V(f∗ (FX (ΩX log /S log ) ⊗ g† EG ))⟩ → OS ⟨V(f∗ († Vg,X ))⟩

(426)

corresponding to ψX satisfies ∗ ∗ ψX (OS ⟨V(f∗ (FX (ΩX log /S log ) ⊗ g† EG ))⟩j ) ⊆ OS ⟨V(f∗ († Vg,X ))⟩p·j

(427)

for every j ∈ Z≥0 . L N ∗ (ΩX log /S log ) ⊗ g† EG )) → X denote the S-morphism Moreover, let χX : V(f∗ (FX induced by χ : g → c. The morphism χ is Gm -equivariant, so the OS -algebra morphism L N ∗ χ∗X : OS ⟨ X ⟩ → OS ⟨V(f∗ (FX (428) (ΩX log /S log ) ⊗ g† EG ))⟩

κ

κ

corresponding to χX preserves the filtration. ∗ −1 ∗ The L composite (τX ) ◦ ψX ◦ χ∗X coincides with the OS -algebra morphism N ∗ ℏ . Hence, the desired inclusion relaℏ : OS ⟨ X ⟩ → OS ⟨Opg,ℏ,X ⟩ induced by tion follows from (402) together with the various filtration-preservation properties of OS -algebra morphisms obtained above.

κ

3.7.2. The finiteness of the Hitchin-Mochizuki morphism. — For each j ∈ Z≥0 , denote by grj ( ∇∗ ℏ ) the morphism L N∇ L N∇ j L N∇ j−1
OS ⟨ X ⟩j = OS ⟨ X ⟩ /OS ⟨ X ⟩ (429) → grp·j (OS ⟨Opg,ℏ,X ⟩)

κ

∇∗ ℏ )

  M L N ∇  grp·j (OS ⟨Opg,ℏ,X ⟩) ⊆ gr(OS ⟨Opg,ℏ,X ⟩). gr( ∇∗ ℏ ) : OS ⟨ X⟩→

κ

(430)

the direct sum gr(

κ

∇∗ L∇ j (cf. Lemma 3.29). Then, ℏ |O ⟨N S X⟩ ∇∗ ) determines an OS -algebra morphism ℏ

κ

induced by L j j∈Z≥0 gr (

j∈Z≥0

ASTÉRISQUE 432

:=

3.7. THE FINITENESS OF THE HITCHIN-MOCHIZUKI MORPHISM

105

κ

κ

Proposition 3.30. — Let us keep the notation in § 2.9. We shall consider the OA1S -algebra morphism L N∇ ∇∗ 1 (431) ˆ Xˆ ⟩ ˆ : OA1S ⟨ Xˆ (1) ⟩ → OAS ⟨Opg,ℏ, ℏ   L N ∇ 1 ⟨ resp., ∇∗ : O ⟩ (1) ⟩ → OA1 ⟨Opg,0,X ˆ ˆ A ˆ X 0 S S

κ

defined to be the morphism ∇∗ (cf. (422)) where the triple “(S, X , ℏ)” is replaced ℏ ˆ (resp., (A1 , Xˆ , 0)). Then, the following diagram is commutative: by (A1S , Xˆ , ℏ) S (260)

gr(OA1S ⟨Opg,ℏ, ˆ Xˆ ⟩) d

/ OA1 ⟨Op g,0,Xˆ ⟩ S ;

∼

(432) ∇∗ ˆ ) ℏ

κ

κ

gr(

OA1 ⟨ S

∇∗ ˆ 0

L N∇

Xˆ ⟩.

κ

κ

In particular, the morphism gr( ∇∗ ℏ ) does not depend on the choice of ℏ ∈ Γ(S, OS ) and ∇∗ coincides with 0 under the identification gr(OS ⟨Opg,ℏ,X ⟩) = OS ⟨Opg,0,X ⟩ given by (260). ∗ ∗ ∗ Proof. — Let ψX ) denote the morphism ψX (cf. (426)) with X and ˆ ,ℏ ˆ (resp., ψXˆ ,ˆ 0 ˆ ˆ ℏ replaced by X and ℏ (resp., 0) respectively. Then, by considering the definition ∗ of p-curvature, we see that the morphism gr(ψX ˆ ,ℏ ˆ ), i.e., the graded morphism asso∗ ∗ ciated to ψXˆ ,ℏˆ , coincides with ψXˆ ,0 . This implies (from the discussion in the proof of Lemma 3.29) the desired commutativity of (432). The latter assertion follows from ˆ = ℏ” in A1 . this result by taking the fiber over the point “ ℏ S

κ

κ

By using the above proposition, we obtain the following assertion, generalizing [59], Chap. I, Theorem 2.3, and the first unpublished version of [19], § 1, Theorem 1.1. L N∇ L N∇
1 Corollary 3.31. — (i) The morphism ∇ ˆ Xˆ → AS ×S ˆ : Opg,ℏ, X = Xˆ is fiℏ ℵ(g) nite and faithfully
flat of degree p . In particular, for each p-nilp ℏ ∈ Γ(S, OS ) = A1 (S) , the structure morphism Opg,ℏ,X → S of the S-scheme L N∇ p-nilp Opg,ℏ,X (resp., the Hitchin-Mochizuki morphism ∇ ℏ : Opg,ℏ,X → X ) is ℵ(g) finite and faithfully flat of degree p .

κ

×r (ii) Let ρ be an elementL Nof∇ c (S) (where ρ := ∅ if r = 0). Then, the morphism c ∇ ℵ(g) . In ˆ Xˆ → ˆ : Opg,ℏ,ρ, Ψℏ (ρ),Xˆ (1) is finite and faithfully flat of degree p ℏ,ρ
1 ×r particular, if ℏ is an element of Γ(S, OS ) = A (S) with Ψℏ (ρ) = [0] , then p-nilp c the projection Opg,ℏ,[0]×r ,X → S is finite and faithfully flat of degree p ℵ(g) .

Proof. — We only consider assertion (ii) because the proof of assertion (i) follows from a similar (and relatively simple) L N∇ argument. First, let us prove the former assertion of (ii). Both Opg,ℏ,ρ, ˆ Xˆ and Ψ ˆ (ρ),Xˆ are flat over S, so we may assume, after ℏ

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CHAPTER 3. OPERS IN POSITIVE CHARACTERISTIC

κ

κ

taking the respective fibers, that S = Spec(k). Because of Proposition 3.25, (i), and the commutativity of (432) proved in Proposition 3.30, gr(OA1 ⟨Opg,ℏ, ˆ Xˆ ⟩) is a finite L N∇ L N∇ gr(OA1 ⟨ Xˆ ⟩)-module. Hence, OA1 ⟨Opg,ℏ, ˆ Xˆ ⟩ is a finite OA1 ⟨ Xˆ ⟩-module, or equiv∇ alently, the morphism ∇ may be obtained from the finite is finite. Note that ˆ ˆ ρ,ℏ ℏ

κ

κ

κ

morphism ∇ ˆ by restricting the domain and codomain to their closed schemes. This ℏ ∇ implies the finiteness of ∇ ˆ . Moreover, since the domain and codomain of ˆ are ρ,ℏ ρ,ℏ

κ

κ

κ

κ

irreducible, smooth, and of the same dimension, we see that ∇ ˆ is also faithfully flat. ρ,ℏ Now we compute its degree. To this end, it suffices to consider the fiber over 0 ∈ A1 , i.e., compute the degree of ∇ 0,ρ . But, the computation in this case has already finished c ∇ ℵ(g) in Proposition 3.25, (ii), so we have deg( ∇ . This completes ˆ ) = deg( ρ,0 ) = p ρ,ℏ the proof of the former assertion. The latter assertion follows directly from the former assertion. Zzz...

Corollary 3.32. — For any ℏ ∈ Γ(S, OS ), Opg,ℏ,X is finite over S. p-nilp

Proof. — The assertion follows from the finiteness of Opg,ℏ,X proved in CorolZzz...

p-nilp

lary 3.31, (i), and the fact that Opg,ℏ,X is a closed subscheme of Opg,ℏ,X . 3.8. The universal moduli stacks of dormant/p-nilpotent (g, ℏ)-opers This final section deals with the moduli stacks of dormant/p-nilpotent (g, ℏ)-opers on the universal family of pointed stable curves. We describe several facts concerning the geometric structures of these stacks deduced from results obtained so far. Let (G, T) be, as before, a split semisimple algebraic k-group of adjoint type. We suppose that p > 2hG . Also, let (g, r) be a pair of nonnegative integers with 2g − 2 + r > 0, ℏ an element of k, and ρ an element of c×r (k) (where c×r (k) := {∅} and ρ := ∅ if r = 0). 3.8.1. Dormant case. — First, let us consider the moduli stack of dormant (g, ℏ)-opers. For each ℏ ∈ k, we shall denote by   Zzz... Zzz... (433) Opg,ℏ,g,r resp., Opg,ℏ,ρ,g,r the closed substack of Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) (cf. (303)) classifying dormant Zzz... Zzz... Zzz... (g, ℏ)-opers. For simplicity, we write Opg,g,r := Opg,1,g,r (resp., Opg,ρ,g,r := Zzz... Zzz... Zzz... Opg,1,ρ,g,r ). The Deligne-Mumford stack Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) is equipped log

with the log structure pulled-back from Mg,r ; we denote the resulting log stack by   Zzz... Zzz... log log (434) . Opg,ℏ,g,r resp., Opg,ℏ,ρ,g,r Theorem 3.33 (cf. Theorem C). — (i) If ℏ ∈ k × , then we have a Zzz... Zzz... (435) Opg,ℏ,ℏ⋆ϱ,g,r = Opg,ℏ,g,r . ϱ∈c×r (Fp )

ASTÉRISQUE 432

3.8. THE UNIVERSAL MODULI STACKS OF DORMANT/p-NILPOTENT (g, ℏ)-OPERS

107

Zzz...

In particular, Opg,ℏ,ℏ⋆ρ,g,r (for each ρ ∈ c×r (Fp )) may be represented by a (possibly empty) proper Deligne-Mumford stack over k which is finite over Mg,r . (ii) If ℏ = 0, then we have Zzz...

Zzz...

Opg,0,[0]×r ,g,r = Opg,0,g,r .

(436)

Moreover, the natural projection Zzz...

(Opg,0,g,r )red → Mg,r

(437)

Zzz...

is an isomorphism of k-stacks, where (Opg,0,g,r )red denotes the reduced stack Zzz... Zzz... associated to Opg,0,g,r . In particular, Opg,0,g,r is nonempty and geometrically irreducible of dimension 3g − 3 + r. Proof. — Assertions (i) and (ii) follow from Corollary 3.16, (i) and (ii), respectively.

Here, we shall refer to the following Mochizuki’s theorem; this result concerns the moduli stack of dormant g⊙ (= sl2 )-opers and was proved in the work of p-adic Teichmüller theory (cf. Remark 2.33 for the correspondence between g⊙ -opers and torally indigenous bundles). Zzz...

Theorem 3.34. — The stack Opg⊙ ,g,r is a nonempty, smooth, and proper DeligneZzz...

Mumford stack over k of dimension 3g − 3 + r. Also, the projection Opg⊙ ,g,r → Mg,r Zzz...

(resp., Opg⊙ ,ρ,g,r → Mg,r ) is finite, faithfully flat, and generically étale. Finally, Zzz...

Zzz...

Opg⊙ ,g,0 (resp., Opg⊙ ,ρ,g,r for r > 0) is geometrically connected if it is nonempty. Proof. — See [60], Chap. II, § 2.3, Theorem 2.8. Remark 3.35. — [60], Chap. II, § 2.3, Theorem 2.8, does not mention the nonemptiness Zzz... of Opg⊙ ,g,r , i.e., the moduli stack of dormant torally indigenous bundles; the statement ρ,dor

Zzz...

is only concerned with its substack Opg⊙ ,ρ,g,r (= N g,r in the notation of loc. cit.), and whether it is empty or not depends completely on the data of radii. But, the Zzz... nonemptiness of Opg⊙ ,g,r can be immediately verified. Indeed, to verify this property, let us consider a totally degenerate curve, i.e., a pointed stable curve obtained by gluing together 3-pointed projective lines at their marked points (cf. Definition 7.15 for the precise definition). We can find the molecule structure on such a curve such that its restriction to (the normalization of) each rational component forms the atom each of whose radius is 1. Then, by [60], Introduction, Theorem 1.3, (2), the indigenous bundle corresponding to this molecule is dormant. It follows that the fibers of the Zzz... projection Opg⊙ ,g,r → Mg,r over the points classifying totally degenerate curves are Zzz...

always nonempty, in particular, we have Opg⊙ ,g,r ̸= ∅.

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Zzz... ∼ OpZzz... Remark 3.36. — Suppose that ℏ ∈ k × . Then, since Opg⊙ ,ℏ,g,r = g⊙ ,g,r and Zzz... Zzz... ∼ Op ⊙ Opg⊙ ,ℏ,ℏ⋆ρ,g,r = g ,ρ,g,r (cf. (407)), the same assertion as Theorem 3.34 holds if Zzz...

Zzz...

Zzz...

Zzz...

Opg⊙ ,g,r (resp., Opg⊙ ,ρ,g,r ) is replaced with Opg⊙ ,ℏ,g,r (resp., Opg⊙ ,ℏ,ℏ⋆ρ,g,r ). Remark 3.37. — It is an important problem to determine for what elements Zzz... ρ ∈ c×r (Fp ) the associated moduli stack Opg,ℏ,ℏ⋆ρ,g,r with ℏ ∈ k × is nonempty. We can give a complete answer to this problem for g = g⊙ (cf. [60], Introduction, § 1.2, Theorem 1.3, or the discussion of § 7.8 in the present manuscript). But, little is known about the general case at the time of writing the present manuscript. Zzz... One of the facts that can be easily verified is that Opg,ℏ,ℏ⋆ρ,g,r = ∅ unless ρ ∈ (c(Fp ) \ {[0]})×r . Indeed, suppose, on the contrary, that there exist an r-pointed stable curve X of genus g and a dormant (g, ℏ)-oper E ♠ := (EB , ∇) on X of radii ρ := (ρi )i with ρi0 = [0] for some i0 ∈ {1, . . . , r}. After possibly applying Proposition 2.19 and changing the base space of X , let us assume that k is algebraically closed, X is defined over k, and E ♠ is ↙ q1 -normal. The monodromy µ∇ i0 of ∇ at the i0 -th marked point σi0 of X may be expressed as µ∇ = 1 ⊗ q + R for −1 i0 some R ∈ gad(q1 ) . But, R must be zero because Kosg is an isomorphism and ρi0 = [0]. It follows that [p] σi∗0 (p ψ ∇ ) = (µ∇ − ℏp−1 · µ∇ i0 ) i0

(438)

[p]

= 1 ⊗ q−1 − ℏp−1 · (1 ⊗ q−1 ) = −ℏp−1 ⊗ q−1 ̸= 0. This contradicts the dormancy condition on E ♠ . Consequently, we have proved the desired property. Zzz...

For the moduli stack Opg,ℏ,g,r with g general, the following assertion can be deduced from results proved so far together with the above theorem due to S. Mochizuki: Theorem 3.38 (cf. Theorem C). —

Zzz...

(i) Opg,ℏ,g,r may be represented by a proper Zzz...

Deligne-Mumford stacks over k, and the natural projections Opg,ℏ,g,r → Mg,r is finite. (ii) The assignment (X , E ♠ ) 7→ (X , ιG∗ (E ♠ )) (cf. (208)) determines a closed immersion Zzz...

Zzz...

Opg⊙ ,ℏ,g,r ,→ Opg,ℏ,g,r

(439)

Zzz...

over Mg,r . In particular, Opg,ℏ,g,r is nonempty, of dimension 3g − 3 + r, and has an irreducible component that dominates Mg,r . Proof. — The assertion follows from Proposition 3.17, (ii), Corollary 3.32, and Theorem 3.34.

ASTÉRISQUE 432

3.8. THE UNIVERSAL MODULI STACKS OF DORMANT/p-NILPOTENT (g, ℏ)-OPERS

109

κ

κ

3.8.2. p-nilpotent case. — Next, let us describe an assertion concerning the geometric structure of the universal moduli stack classifying p-nilpotent (g, ℏ)-opers. Suppose that Ψℏ (ρ) = [0]×r (where Ψℏ (ρ) := ∅ if ρ = ∅). As in the case of a pointed stable L N∇ L N∇ curve over a scheme, we can construct (1) ), i.e., the relative (1) (resp., Ψℏ (ρ),Cg,r Cg,r L N∇ L N∇ affine space (resp., ) for the case where X is taken to be the X (1) Ψℏ (ρ),X (1) universal family of pointed curves Cg,r . Then, the Hitchin-Mochizuki morphism (cf. (401)) for Cg,r defines a morphism  L N∇  L N∇ ∇ ∇ (440) resp., : Op → (1) (1) g,ℏ,ρ,g,r ℏ,univ : Opg,ℏ,g,r → ℏ,ρ,univ C Ψ (ρ),C g,r

g,r

ℏ

over Mg,r . Also, denote by  p-nilp resp., Opg,ℏ,ρ,g,r L N∇ L N∇ the inverse image of the zero section of C (1) (resp., Ψ 

(1) ℏ (ρ),Cg,r

g,r

∇ ℏ,ρ,univ ),

) via

κ

p-nilp

Opg,ℏ,g,r

(441)

∇ ℏ,univ

(resp.,

(i) The morphism

finite and faithfully flat of degree p

ℵ(g)

c

(resp., p

∇ ℏ,univ ℵ(g)

(resp.,

κ

Theorem 3.39 (cf. Theorem B). —

κ

κ

i.e., the closed substack of Opg,ℏ,g,r (resp., Opg,ℏ,ρ,g,r ) classifying p-nilpotent (g, ℏ)-opers. The following assertion is a generalization of a result for g = g⊙ proved by S. Mochizuki (cf. [59], Chap. I, Theorem 2.3), as well as of a result for general g’s by T.-H. Chen-X. Zhu (cf. the first unpublished version of [19], § 1, Theorem 1.1). ∇ ℏ,ρ,univ )

is

p-nilp

). In particular, Opg,ℏ,g,r

p-nilp

(resp., Opg,ℏ,ρ,g,r ) may be represented by a nonempty and proper DeligneMumford stack over k of dimension 3g − 3 + r, and the natural projecp-nilp p-nilp tion Opg,ℏ,g,r → Mg,r (resp., Opg,ℏ,ρ,g,r → Mg,r ) is finite and faithfully flat of c degree pℵ(g) (resp., p ℵ(g) ). (ii) If, moreover, ℏ = 0, then the natural projection (442)

p-nilp

(Opg,0,g,r )red → Mg,r p-nilp

is an isomorphism of k-stacks, where (Opg,0,g,r )red denotes the reduced stack p-nilp p-nilp associated to Opg,0,g,r . In particular, Opg,0,g,r is geometrically irreducible. Proof. — The assertions follow from Corollaries 3.26 and 3.31, (i) and (ii).

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CHAPTER 4 FLAT VECTOR BUNDLES AND DIFFERENTIAL OPERATORS

In this chapter, we consider the description of (sln , ℏ)-opers in terms of differential operators and vector bundles. It is well-known that sln -opers on a smooth curve correspond bijectively to certain differential operators between line bundles on that curve. To extend the situation (e.g., to the case of a family of possibly singular stable curves), we introduce a generalized theta characteristic, i.e., an (n, ℏ)-theta characteristic. After choosing an (n, ℏ)-theta characteristic ϑ, we define the notion of an (n, ℏ, ϑ)-projective connection and a (GLn , ℏ, ϑ)-oper. For example, an (n, ℏ, ϑ)-projective connection is defined as a certain n-th order differential operator such that the principal symbol is 1 and the subprincipal symbol is characterized by ϑ. The main result of this chapter establish a generalization of the correspondence mentioned above, which connects sln -opers, (GLn , ℏ, ϑ)-opers, and (n, ℏ, ϑ)-projective connections. Moreover, we discuss the equivalence between the dormancy condition on opers in positive characteristic and the fullness of root functions of the corresponding projective connections. Notation and Conventions. — Let us fix a perfect field k and a positive integer n. 4.1. Logarithmic ℏ-connections on vector bundles In Definition 1.17, we already defined a logarithmic connection on a principal bundle. In what follows, we formulate and study the version defined on a vector bundle. As proved in § 4.3 later, these definitions of a connection are equivalent under the identification between rank n vector bundles and GLn -bundles. 4.1.1. Logarithmic ℏ-connections Let T log := (T, αT : MT → OT ), Y log := (Y, αY : MY → OY ) be fine log schemes over k and f log : Y log → T log a k-morphism of log schemes. Also, let ℏ be an element of Γ(T, OT ) and V a rank n vector bundle on Y (or more generally, an OY -module). To simplify the notation, we write T := TY log /T log and Ω := ΩY log /T log . Definition 4.1. — An ℏ-T log -connection on V is an f −1 (OT )-linear morphism (443)

∇:V →Ω⊗V

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satisfying ∇(a·v) = ℏ·da⊗v +a·∇(v) for any local sections a ∈ OY , v ∈ V . Each local section in the kernel Ker(∇) is called horizontal. We will refer to any 1-T log -connection as a T log -connection. Example 4.2 (The case of ℏ = 0). — A 0-T log -connection on V is nothing but an OY -linear morphism V → Ω ⊗ V , or equivalently, an element of Γ(Y, Ω ⊗ End OY (V )). In particular, there always exists a 0-T log -connection on any vector bundle. A 0-T log -connection are often called a Higgs field. 4.1.2. Curvature. — Now, let ∇ : V → Ω ⊗ V be an ℏ-T log -connection on V . Given a local section ∂ of T (= Ω∨ ), we denote by ∇∂ the f −1 (OT )-linear endomorphism of V defined as (∂ ⊗ idV ) ◦ ∇. Also, for local sections ∂1 , ∂2 ∈ T , we write ψ ∇ (∂1 ∧ ∂2 ) := [∇∂1 , ∇∂2 ] − ∇[∂1 ,∂2 ] .

(444)

It is verified that ψ ∇ (∂1 ∧∂2 ) = −ψ ∇ (∂2 ∧∂1 ) and ψ ∇ ((a·∂1 )∧∂2 ) = ψ ∇ (∂1 ∧(a·∂2 )) = a · ψ ∇ (∂1 ∧ ∂2 ) for any local sections ∂1 , ∂2 ∈ T and a ∈ OY . Hence, the assignment ∂1 ∧ ∂2 7→ ψ ∇ (∂1 ∧ ∂2 ) determines an OY -linear morphism ψ∇ :

(445)

2 ^

T

→ End f −1 (OT ) (V ).

Definition 4.3. — We shall refer to ψ ∇ as the curvature of ∇. Also, we shall say that ∇ is flat (or integrable) if ψ ∇ = 0. (In particular, if Y log /T log is a log curve, then any ℏ-S log -connection on V is automatically flat because T is a line bundle.) Remark 4.4. — It is immediately verified that the image of ψ ∇ lies in the subsheaf End OY (V ) of End f −1 (OT ) (V ). Hence, ψ ∇ may be regarded as an OY -linear morV  V2 2 phism T → End OY (V ), or equivalently, a global section of Ω ⊗ End OY (V ). Definition 4.5. — as a pair

(i) An ℏ-flat (vector) bundle on Y log /T log (of rank n) is defined V := (V , ∇)

(446)

consisting of a vector bundle V on Y (of rank n) and a flat ℏ-T log -connection ∇ on V . If, moreover, V is of rank 1, then such a pair V is called an ℏ-flat line bundle on Y log /T log . For simplicity, a 1-flat bundle is called a flat (vector) bundle. (ii) Let V := (V , ∇) and V ′ := (V ′ , ∇′ ) be ℏ-flat bundles on Y log /T log . A morphism of ℏ-flat bundles from V to V ′ is an OY -linear morphism η : V → V ′ preserving the ℏ-T log -connection, i.e., satisfying ∇′ ◦ η = (idΩ ⊗ η) ◦ ∇. Example 4.6 (Trivial bundle). — As a trivial example of an ℏ-flat bundle, we have the pair (447)

ASTÉRISQUE 432

(OY , ℏ · d)

4.1. LOGARITHMIC ℏ-CONNECTIONS ON VECTOR BUNDLES

113

consisting of the structure sheaf OY and ℏ · d, i.e., the universal logarithmic derivation OY → Ω multiplied by ℏ; we will refer to it as the trivial ℏ-flat (line) bundle. Note that each ℏ-T log -connection ∇ on OY can be described as ∇ = ℏ · d + ∇GL1 for a unique 1-form ∇GL1 ∈ Γ(Y, Ω) (= HomOY (OY , Ω ⊗ OY )). Then, ∇ is flat if and only if ∇GL1 is closed, i.e., d∇GL1 = 0. In general, for any ℏ-T log -connection ∇ on OY⊕n , there exists a unique element
∇GLn ∈ HomOY (OY⊕n , Ω ⊗ (OY⊕n )) = Γ(Y, Ω ⊗ End OY (OY⊕n )) (448) satisfying ∇ = ℏ · d⊕n + ∇GLn . 4.1.3. Pull-back and base-change. — Let V be a vector bundle on Y and ∇ an ℏ-T log -connection on V . Also, let Y ′ log be a fine log scheme equipped with a log étale morphism ulog : Y ′ log → Y log . Then, we can pull-back ∇ to obtain an ℏ-T log -connection on u∗ (V ). Indeed, because of the log étaleness of ulog , the induced morphism u∗ (Ω) → Ω′ := ΩY ′ log /T log is an isomorphism. Under the identification u∗ (Ω) = Ω′ given by this isomorphism, we define the pull-back u∗ (∇) of ∇ to be the f −1 (OT )-linear morphism (449)

u∗ (∇) : u∗ (V ) = OY ′ ⊗u−1 (OY ) u−1 (V ) → Ω′ ⊗ u∗ (V ) = OY ′ ⊗u−1 (OY ) u−1 (Ω ⊗ V ) given by a ⊗ v 7→ da ⊗ v + a ⊗ ∇(v) for any local sections a ∈ OY ′ , v ∈ V . This ∗ morphism forms an ℏ-T log -connection on u∗ (V ) and satisfies ψ u (∇) = u∗ (ψ ∇ ). In particular, the flatness of ∇ implies the flatness of u∗ (∇). Also, the base-change of ∇ to a fine T log -log scheme can be constructed. To verify this, suppose that we are given a fine log scheme T ′ log over k equipped with a k-morphism tlog : T ′ log → T log . Write Y ′ log := T ′ log ×T log Y log (where the fiber product is taken in the category of fine log schemes over k) and use the notation t∗ (−) to denote base-change to Y ′ . Then, Ω′ := ΩY ′ log /T ′ log may be identified with t∗ (Ω). By means of this identification, we can define the base-change (450)

t∗ (∇) : t∗ (V ) → Ω′ ⊗ t∗ (V ) ∗

of ∇ in the same manner as (449). It is immediate that ψ t the flatness of ∇ implies the flatness of t∗ (∇).

(∇)

= t∗ (ψ ∇ ), and hence,

4.1.4. Determinant. — We here recall several constructions of ℏ-flat bundles by using a given ℏ-flat bundle. Let us fix a vector bundle V on Y of rank n and an ℏ-T log -connection ∇ : V → Ω⊗V on V . The determinant of ∇ is defined as the ℏ-T log -connection det(∇) : det(V ) → Ω ⊗ det(V ) Vn on the determinant bundle det(V ) = V of V given by n X (452) det(∇)(v1 ∧ v2 ∧ · · · ∧ vn ) = v1 ∧ · · · ∧ ∇(vj ) ∧ · · · ∧ vn (451)

j=1

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for v1 , . . . , vn ∈ V . The curvature ψ det(∇) of det(∇) satisfies ψ det(∇) = tr(ψ ∇ )

(453)

under the natural identification tr(End OY (V )) = OY . In particular, the flatness of ∇ implies the flatness of det(∇). 4.1.5. Dual. — Let V and ∇ be as above. The dual of ∇ is defined as the ℏ-T log -connection ∇∨ : V ∨ → Ω ⊗ V ∨

(454)

on the dual bundle V ∨ of V determined by ∇∨ (h)(v) = d(h(v)) − (idΩ ⊗ h) ◦ ∇(v)

(455)

for local sections h ∈ V ∨ , v ∈ V . It is immediate that (V ∨∨ , ∇∨∨ ) ∼ = (V , ∇). Under ∨ the natural identification End OY (V ∨ ) = End OY (V ), the curvature ψ ∇ of ∇∨ satisfies ∨

ψ ∇ = −t ψ ∇ ,

(456)

where t ψ ∇ denotes the transpose of ψ ∇ . In particular, the flatness of ∇ is equivalent to that of ∇∨ . 4.1.6. Tensor product. — Next, let V ′ be another vector bundle on Y and ∇′ an ℏ-T log -connection on V ′ . The tensor product of ∇ and ∇′ is the ℏ-T log -connection ∇ ⊗ ∇′ : V ⊗ V ′ → Ω ⊗ (V ⊗ V ′ )

(457)

on the tensor product V ⊗ V ′ determined by ∇ ⊗ ∇′ (v ⊗ v ′ ) = ∇(v) ⊗ v ′ + v ⊗ ∇′ (v ′ )

(458)

for local sections v ∈ V , v ′ ∈ V ′ . In particular, for each positive integer m, ∇ induces an ℏ-T log -connection ∇⊗m : V ⊗m → Ω ⊗ V ⊗m

(459)

′

⊗m

on the m-fold tensor product V ⊗m of V . The curvature ψ ∇⊗∇ (resp., ψ ∇ (resp., ∇⊗m ) can be expressed as ′

ψ ∇⊗∇ = ψ ∇ ⊗ idV′ + idV ⊗ ψ ∇

(460) 

resp., ψ ∇

⊗m

=

m X

) of ∇⊗∇′

′

 ⊗(j−1)

idV

⊗(m−j) 

⊗ ψ ∇ ⊗ idV

.

j=1 ⊗m

In particular, if V is a line bundle, then we have ψ ∇ = m · ψ ∇ . Moreover, (460) ′ ⊗m ′ implies that ∇ ⊗ ∇ (resp., ∇ ) is flat if both ∇ and ∇ are flat (resp., ∇ is flat). ∼

4.1.7. Gauge transformation. — Let V , V ′ be vector bundles on Y and η : V → V ′ an OY -linear isomorphism. Then, for an ℏ-T log -connection ∇ on V , the composite (461)

ASTÉRISQUE 432

η −1

∇

idΩ ⊗η

η∗ (∇) : V ′ −−→ V −→ Ω ⊗ V −−−−→ Ω ⊗ V ′

4.2. LOGARITHMIC DIFFERENTIAL OPERATORS WITH PARAMETER ℏ

115

specifies an ℏ-T log -connection on V ′ . Definition 4.7. — Suppose that V ′ = V , i.e., η is an automorphism of V . Then, we shall refer to η∗ (∇) as the gauge transformation of ∇ by η. The following assertion may be regarded as an extension of the discussion in Remark 1.35 to the case of gauge transformation by an automorphism that is not necessarily unipotent. Proposition 4.8. — Let η be an automorphism of OY⊕n and ∇ an ℏ-T log -connection on OY⊕n . Then, the gauge transformation η∗ (∇) of ∇ by η satisfies the following equality:
(462) η∗ (∇)GLn = −ℏ · η −1 dη + η ◦ ∇GLn ◦ η −1 = ℏ · ηd(η −1 ) + η ◦ ∇GLn ◦ η −1 (cf. (448) for the definition of (−)GLn ), where dη denotes the image of η via the morphism End OY (OY⊕n ) → Ω⊗ End OY (OY⊕n ) given by (aij )ij 7→ (daij )ij under the natural 2 identification End OY (OY⊕n ) = OY⊕n . Proof. — The assertion follows from the definition of η∗ (∇). 4.2. Logarithmic differential operators with parameter ℏ 4.2.1. The ring of logarithmic differential operators. — Let Y log /T log and ℏ be as above. Let us consider the Lie algebroid (T , ℏ · [−, −], ℏ · idT ), i.e., the tangent Lie algebroid rescaled by ℏ. Denote by