Modern Methods in Complex Analysis (AM-137), Volume 137: The Princeton Conference in Honor of Gunning and Kohn. (AM-137) 9781400882571

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Annals of Mathematics Studies

Number 137

Modern Methods in Complex Analysis

Edited by

Thomas Bloom David Catlin John P. D ’Angelo Yum-Tong Siu

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

Copyright © 1996 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 9 8 7 6 5 4 3 2 1

Library of Congress Cataloging-in-Publication Data Modern methods in complex analysis : the Princeton conference in honor of Gunning and Kohn / edited by Thomas Bloom . . . [et al.]. p. cm. — (Annals of mathematics studies : no. 137) Conference held Mar. 16-20, 1992, at Princeton University. Includes bibliographical references. ISBN 0-691-04429-5 (cloth : alk. paper). — ISBN 0-691-04428-7 (pbk. : alk. paper) 1. Mathematical analysis—Congresses. 2. Functions of several complex variables— Congresses. I. Bloom, Thomas. II. Gunning, R. C. (Robert Clifford), 1931- . III. Kohn, Joseph John, 1932- . IV. Series. QA300.M63 1995 515.9—dc20 95-40840 The publisher would like to acknowledge the editors of this volume for providing the camera-ready copy from which this book was printed

CONTENTS PREFACE LIST OF PARTICIPANTS PROGRAM OF TH E CONFERENCE

v ii ix x i ii

THE SCIENTIFIC WORK OF ROBERT C. GUNNING

3

THE SCIENTIFIC WORK OF JOSEPH J. KOHN

16

R aoul B o tt

ON INVARIANTS OF MANIFOLDS

29

M ic h a e l C h r ist

REMARKS ON ANALYTIC HYPO ELLIPTICITY OF db

41

J o h n P . D ’A n g e l o

FIN ITE TY PE CONDITIONS AND SUBELLIPTIC ESTIMATES

63

P h ilip p e E y s s id ie u x a n d N g a im in g M o k

CHARACTERIZATION OF CERTAIN HOLOM ORPHIC GEODESIC CYCLES ON HERM ITIAN LOCALLY SYM M ETRIC MANIFOLDS OF THE NONCOMPACT T Y PE

79

C h a r le s F efferm a n

ON KOHN’S MICROLOCALIZATION OF d PROBLEMS

119

J o h n E r ik F o r n a e s s a n d N essim S ib o n y

COM PLEX DYNAMICS IN HIGHER DIMENSION. II

135

H ans G r a u e r t

SET THEO RETICA L REAL ANALYTIC SPACES

183

R i c h a r d S. H a m i l t o n

AN ISO PERIM ETRIC ESTIMATE FOR THE RICCI FLOW ON THE TW O -SPHERE

191

VI

Contents

R i c h a r d S. H a m i l t o n

ISOPERIM ETRIC ESTIMATES FOR THE CURVE SHRINKING FLOW IN TH E PLANE

201

G. M . H en k in

THE ABEL-RADON TRANSFORM AND SEVERAL COM PLEX VARIABLES

223

A la n M. N a d e l

ON THE ABSENCE OF PERIODIC POINTS FO R THE RICCI CURVATURE OPERATOR ACTING ON THE SPACE OF KAHLER METRICS 277 L o u is N i r e n b e r g THE MAXIMUM PRINCIPLE AND RELATED TOPICS

283

Y u m - T o n g Siu

VERY AMPLENESS CRITERION OF DOUBLE ADJOINTS OF AMPLE LINE BUNDLES

291

F r a n c o is T r e v e s

INTEGRABILITY OF ELLIPTIC OVERDETERM INED SYSTEMS OF NONLINEAR FIRST-ORDER COM PLEX PDE

319

S. M . W e b s t e r

THE HOLOMORPHIC CONTACT GEOM ETRY OF A REAL HYPERSURFACE

327

PREFACE More th an 150 m athem aticians honored Robert C. Gunning and Joseph J. Kohn on the occasion of their sixtieth birthdays by participating in a conference on several complex variables. The conference was held in Fine Hall at Princeton University from March 16 to March 20, 1992. This volume contains 15 papers dedicated to the two honorees. Most of the papers are from th at conference. Several are by m athem aticians unable to attend. The volume also includes articles on the scientific work of both honorees, lists of their Ph.D. students and of the conference participants, and the program for the conference. The organizers acknowledge financial support from the N ational Sci­ ence Foundation and the hospitality of Princeton University for the con­ ference. They also wish to thank Princeton’s M athem atics D epartm ent m anager Scott Kenney for his assistance in organizing the conference.

LIST OF PA R T IC IPA N T S Marco Abate (Seconda Universita di Roma, Italy) Khalid Filali Adib (Purdue) Takao Akahori (Himeji Institute of Technology, Japan) Solomon Alber (U of Nevada, Reno) Herbert Alexander (U of Illinois, Chicago) Eric Am ar (Bordeaux) Salah Baouendi (UC San Diego) David B arrett (U of Michigan) Jose Barros-Neto (Rutgers) Jim Baxter (Washington) Eric Bedford (Indiana U) Steve Bell (Purdue) Shfff Berhanu (Temple) Carlos Berenstein (U of M aryland) Craig J. Benham (Mt. Sinai Medical Center) Bo Berndtsson (Chalmers, Sweden) Edward Bierstone (U of Toronto) Thom as Bloom (U of Toronto) Grigory Bluher (UCLA) Harold Boas (Texas A fe M U ) Raoul B ott (Harvard) W illiam Browder (Princeton) Lutz B ungart (U of W ashington, Seattle) Dan Burns (U of Michigan) David Catlin (Purdue) Der-Chen Chang (U of Maryland) P. C harpentier (Bordeaux) So-chin Chen (SUNY, Albany) Anne-Marie Chollet (U of Lille, France) Young-Bok Chung (Purdue) Salvatore Coen (Universita degli studi di Bologne, Italy) John P. D ’Angelo (U of Illinois) Maklouf Derridj (Orsay, France) Klas Diederich (W uppertal, Germany) Ricardo Diaz (U of Northern Colorado) Gilberto Dini (U degli studi di Firenze, Italy) Pierre Dolbeault (U of Paris VI, France) Avner Dor (Weizmann Inst., Israel) Michael Eastwood (U of Adelaide, Australia) Leon Ehrenpreis (Temple U)

X

List o f Participants

Vladim ir Ezhov (Bochum) Gerd Faltings (Princeton) Charles FefFerman (Princeton) Julian Fleron (SUNY, Albany) John Eric Fornaess (Princeton/U of Michigan) Gerald Folland (U of Washington, Seattle) Franc Forstneric (U of Wisconsin) Edward Frankel (University of Nante, France) Estela Gavosto (Princeton) Daryl Geller (SUNY, Stony Brook) Ian G raham (U of Toronto) Hans G rauert (Goettingen, Germany) Sandrine Grellier (France) Peter C. Greiner (U of Toronto) Pengfei Guan (McMaster U) Robert C. Gunning (Princeton) Nicholas Hanges (IAS) Adam Harris (Rice U) Reese Harvey (Rice U) Zheng-Xu He (Princeton) G. Henkin (University of Paris VI, France) Gregor Herbort (W uppertal, Germany) C. Denson Hill (SUNY, Stony Brook) A. Alexandrou Himonas (Notre Dame) Lop-Hing Ho (W ichita State U) Soren Illm an (Princeton) Howard Jacobowitz (IAS) M oonja Jeong (U of Illinois) Shanyu Ji (U of Houston) Qin Jing (Harvard) Leslie Kay (Virginia Polytechnic Institute) N. Kerzman (U of NC, Chapel Hill) Joseph J. Kohn (Princeton) Gen Kom atsu (Osaka U Toyonaka, Japan) Steven Krantz (W ashington U, St. Louis) Oh Nam Kwon (Indiana U) Christine Laurent-Thiebaut (Institut Fourier, France) Jurgen Leiterer (Institut Fourier, France) Laszlo Lempert (Purdue) Moohyun Lee (Purdue) David Lieberman (IDA, Princeton) Haw-Lin Li (Purdue) Qi-keng Lu (IAS/Academ ia Sinica, Beijing, China)

List o f Participants Daowei Ma (U of Chicago) M atei Machedon (Princeton) Andrew M ajda (Princeton/IA S) Richard M andelbaum (U of Rochester) Andrew Markoe (Rider College) Jeffrey McNeal (Princeton) Ngaiming Mok (Columbia) Gerardo Mendoza (Temple) Adrian Nachm an (U of Rochester) ALan Nadel (IAS) Alexander Nagel (U of Wisconsin — Madison) Terrence J. Napier (MIT) Raghavan Narasim han (U of Chicago) W. W hipple Neely (U of W ashington, Seattle) Ed Nelson (Princeton) Louis Nirenberg (NYU-Courant Institute) Alan Noell (Oklahoma State U) Takeo Ohsawa (Nagoya U, Japan) M asami Okada (Courant Institute, NYU) Mikeung Park (Purdue) Mikael Passare (Sweden) Giorgio Patrizio (U degli studi di Roma, Italy) Duong H. Phong (Columbia) Sergey Pinchuk (U of Wisconsin) Karen R. Pinney (Johns Hopkins) John C. Polking (Rice U) Elisa P rato (Princeton) David Prill (U of Rochester) Mihai P u tinar (UC, Riverside) Enrique Ramirez (Mexico) R. Michael Range (SUNY, Albany) Themistockles Rassias (Greece) Jean-Pierre Rosay (U of Wisconsin) Linda Rothschild (UC, San Diego) Halsey Royden (Stanford) Weidong Ruan (Harvard) Norberto Salinas (U of Kansas) Les Saper (Duke) Michael Schneider (Bayreuth) Wolfgang Schwarz (W uppertal, Germany) Angela Selvaggi (U degli studi di Firenze, Italy) Mei-Chi Shaw (Notre Dame) Bernard Shiffman (Johns Hopkins)

Xll Taka Shiota (Harvard) Carlos Simpson (Princeton) Yum-Tong Siu (Harvard) Zbigniew Slodkowski (U of Illinois, Chicago) Mikhail Smirnov (Princeton) A vraham SofFer (Princeton) Nancy K. Stanton (Notre Dame) Elias Stein (Princeton) W ilhelm Stoll (Notre Dame) Edgar Lee Stout (U of Washington, Seattle) Emil J. Straube (Texas A & M U) W illiam Sweeney (Rutgers) Puqi Tang (Purdue) David S. Tartakoff (U of Illinois, Chicago) John Tate (U of Texas, A ustin/Princeton) B.A. Taylor (U of Michigan) Anthony Thom as (Purdue) Andrey Todorov (UC, Santa Cruz) Rodolfo Torres (NYU — Courant Institute) Marvin Tretkoff (Stevens Inst, of Tech.) Frangois Treves (IAS) Alexander Tumanov (Purdue) Bert G. W achsmuth (D artm outh) A jith W aidyaratne (Purdue) Lihe Wang (Princeton) Sidney Webster (U of Chicago) John Wermer (Brown) A rthur W ightm an (Princeton) Andrew Wiles (Princeton) Bun Wong (UC, Riverside) Yeren Xu (U of Washington, Seattle) Stephen Yau (U of Illinois, Chicago)

List o f Participants

PR O G R A M OF TH E C O N FE R E N C E

Monday (March 16, 1992) 9:00

H ans G r a u e r t:

Meromorphic decomposition in the real case

10:30 L ouis N i r e n b e r g : The m axim um principle and principal eigenvalue for second order elliptic operators 1:40

L e o n E h r e n p r e is :

Nonlinear Fourier transform s

2:20 L i n d a R o t h s c h i l d : Analytic discs for real hypersurfaces and generic manifolds in C n 3:00 M e i - C h i S h a w : Local solvability and estim ates for the tan ­ gential Cauchy-Riemann operators 4:00

L e s lie S a p er :

4:40

E r ic B e d fo r d :

L 2 cohomology and the theory of weights Iteration of polynomial autom orphism s in C 2

Tuesday (March 17, 1992) 9:00 J o h n E r i c dimensions 10:30 F r a n c o i s first order ODE 1:40

F orn aess:

T rev es:

Complex analytic dynamics in higher CR structures and complex nonlinear

Stability of embeddings of CR manifolds

L a s z lo L e m p e rt:

2:20 A l e x T u m a n o v : Connections and propogations of analyticity for CR functions 3:00

C a r l o s S im p s o n :

4:00

A la n

N a d e l:

Estim ates for singularly perturbed O D E’s

Geometry of Fano varieties

4:40 M i c h a e l S c h n e i d e r : Com pact manifolds with numerically effective tangent bundles Wednesday (March 18, 1992) 9:00 Y u m - T o n g S iu : Margulis superrigidity as a result of the non­ linear M atsushim a vanishing theorem 10:30 G e n n a d y H e n k i n : Integral representations for analytic va­ rieties with prescribed boundary in complex projective space 1:40

E d B i e r s t o n e : R e s o lu t io n o f s in g u la r it ie s

2:20 R e e s e tions 3:00

H arvey:

T a k e o O h saw a:

Characteristic currents for singular connec­ L 2 cohomology of complex spaces

Program o f the Conference

xiv 4:00 K kernel

la s

A continuity principle for the Bergman

D ie d e r ic h :

4:40 T aka S h io ta : Solitons and algebraic curves

Thursday (March 19, 1992) 9:00

R aou l B o tt:

On invariants of manifolds

10:30 M i k e C h r i s t : Analytic hypoellipticity and a nonlinear eigen­ value problem 1:40 M ifolds

A period m apping for conformal four m an­

ik e E a s t w o o d :

2:20

D a v id C a t lin :

Embeddings of CR structures

3:00

J o h n D ’A n g e l o :

Spherical space forms and CR mappings

4:00 N g a i m i n g M o k : A geometric interpretation of some Eichler autom orphic forms 4:40 G e r a l d equality

F o lla n d :

A simple proof of the sharp Garding in­

Friday (March 20, 1992) 9:00 R a g h a v a n Stein manifolds 10:30

N a r a s im h a n :

C h a r le s F e ffe r m a n :

1:40 M a k l o u f Kohn Laplacian

D e r r id j:

Some remarks on immersions of

Recent developments in d

Estim ates and real analyticity for the

2:20 J e f f M e N e a l : Estim ates for the Bergman kernel on convex domains 3:00 Q i - keng L u : Cauchy-Fantappie formulas for unbounded do­ mains

4:00 T a k a o A k a h o r i : A note on the CR analogue of the TianTodorov theorem 4:40 V l a d i m i r E z h o v : On the problem of equivariant realization of CR autom orphism s

Modern Methods in Complex Analysis

Robert C. Gunning

ROBERT C. G U N N IN G TH E SC IE N T IFIC W O R K OF R O BER T C. G U N N IN G Robert C. Gunning was born on November 27, 1931 in Longmont, Colorado. After graduation from the University of Colorado in 1952 he went on to graduate study at Princeton University where under the direction of S. Bochner he received his doctorate in 1955. Gunning’s work in m athem atics covers a broad range of topics in complex analysis. His early work deals with factors of autom orphy and differential operators compatible with them . For a contractible Stein manifold D with a discrete subgroup T of autom orphism s acting on it, a holomorphic cross section of a holomorphic line bundle over the quotient D / T can be regarded as a holomorphic function which gets m ultiplied by a nowhere zero holomorphic factor, called the factor of automorphy, when an element of T acts on it. Gunning obtained a classification of factors of autom orphy (References 1 and 3). A very im portant contribution of his in this area is his introduction of the Eichler cohomology which has since become a very useful tool in the study of Riem ann surfaces, especially in the com putation of the dimensions of spaces of autom orphic forms and the traces of the Hecke operators on them . G. Bol [B49] observed th at the derivative of order n —1 of a m odular form of degree n —2 is a m od­ ular form of degree —n. Eichler [E57] considered an iterated indefinite integral of order n — 1 of a m odular form of degree —n and defined a bilinear form on m odular forms of degree —n by integrating such an in­ definite integral against another m odular form of degree —n. Eichler used such a bilinear form to generalize A bel’s theorem to obtain a characteri­ zation of a differential on a compact Riem ann surface as the logarithm ic differential of a meromorphic function. Gunning (Reference 10) intro­ duced a new point of view by considering the short exact sequence over a compact Riem ann surface whose middle term is the sheaf of germs of holomorphic autom orphic forms of degree —n with the surjective m ap be­ ing the differentiation of order n + 1. He defined the Eichler cohomology as the cohomology constructed from the global holomorphic sections of the term s of the short exact sequence. From G unning’s point of view the bilinear form which Eichler used can be naturally interpreted as a duality

4

Robert C. Gunning

of cohomology groups. Gunning also considered more general differential operators compatible with factors of autom orphy and obtained classifica­ tion results for such differential operators for the Siegel upper half-plane of rank 2 (References 14 and 15). Another area of G unning’s work is pseudogroups and special coor­ dinate coverings of manifolds. The elements of a pseudogroup are the coordinate transform ation functions between different coordinate charts of a manifold. W ith Bochner (Reference 13) he studied the classification of all pseudogroups of maps ( /i, • • *, f n) of the variables a?i, • • •, x n on dif­ ferentiable manifolds which are defined by a system of linear differential equations of degree t with constant coefficients dsfi dx 31

dx«

0 (^ = 1,2,3,

"3i- -3. e c ) .

An essential property is that, for such a system of linear differential equa­ tions, compositions of solutions should again be solutions. A system is irreducible if no nonsingular linear change of coordinates can make the system independent of some of the variables aq,- • -,x n . They proved th at, for irreducible systems of degree one, the only nontrivial cases are the pseudogroup of differentiable functions, the pseudogroup of holomor­ phic functions, and the pseudogroup of quaternionic analytic functions (defined analogous to the complex-analytic functions). For irreducible systems of degree two, the only possibilities are the pseudogroup of real affine functions and the pseudogroup of complex affine functions. Their result settled the question whether, besides the well-known classes of manifolds such as differentiable manifolds, complex manifolds, etc., there are other manifolds whose coordinate transform ation functions can be defined by such differential equations. For special coordinate coverings, in a series of papers (References 19, 25, 35, and 37) Gunning studied affine and projective complex structures. For those complex structures the manifold adm its a special coordinate covering whose transform ation functions are complex affine transform a­ tions or complex projective linear transform ations. He related the ex­ istence of such structures to the existence of special connections, which is equivalent to the vanishing of certain cohomology classes, and to the vanishing of the curvatures of the connections. Let T be the holomorphic tangent bundle of a compact complex manifold M of complex dimension n. For example, in the case of projective structures he introduced the 1-cocyle {cap} with coefficient in T* 0 T* (8) T for a coordinate covering

Robert C. Gunning

5

with coordinate charts {Ua , za } as follows. ( c °

p

=

2

- j

y

d2z0 f

2 i

d

\

z k

~

where Ca[3k —

s i

< T a /3 k

~

5 % k < T a l3 iJ

k d z “

0

1

d log d e t(dz'p/dzl)

n+ 1

dz*

9

d z < * 0

and 8j is the Kronecker delta. A holomorphic norm al projective connec­ tion is a 0-cochain ba = i,j,k

T%ajkdzi ® dza ® J J P

whose coboundary is the 1 -cocycle {cap} with Tlaj k symmetric in j and k. The transform ation functions for a coordinate covering are all projec­ tive linear transform ations if and only if there is a holomorphic norm al projective connection which is flat in the sense th at the curvature tensor of the connection vanishes. Gunning linked the existence of holomorphic norm al projective connections to some Chern class identities. If M ad­ m its a holomorphic norm al projective connection, then for every weighted homogeneous polynomial Qn- r — Qn- r ( c i , • • *,cn_ r ) E H 2n~ 2r( M , C) the identity crQn- r = (n-f- 1)~r (nJ^ r)cr1Qn- r holds for 1 < r < n. In the special case of a Kaehler M the identity cr = (n + l ) _ r (n^ r) ci holds for 1 < r < n. G unning’s work led to the work of Kobayashi-Ochiai [KO80] th at a compact complex surface adm its a projective structure only when it is the projective plane or a quotient of the ball or when it adm its an affine structure. Using G unning’s techniques his student V itter [V72] classified all possible affine structures on one- and two-dimensional tori, holomorphic families of 1 -tori over a 1-torus, Hopf surfaces, and quotients of abelian surfaces. After a year of post-doctoral work at the University of Chicago, Gun­ ning returned to Princeton as Higgins Lecturer in 1957. He was prom oted to full Professor in 1966. This was also the year th a t he m arried W anda S. Holtzinger. His book with Rossi “Analytic Functions of Several Complex Vari­ ables” was published in 1965 (Reference 17). It was among the first books to give a comprehensive treatm ent of the m ajor topics in the field. To a generation of m athem aticians it was the first book th at guided them into the subject. An open problem in several complex variables is to represent a Stein manifold of complex dimension n with trivial tangent bundle as a Rie­ m ann dom ain spread over C n . In 1967, in a joint paper with N arasim han (Reference 21) Gunning settled the case n = 1.

6

Robert C. Gunning

In 1980-82, Gunning published three papers on Jacobian varieties and th eta functions (References 37, 42, and 43). This work provided the most im portant link in the chain of arguments th at culm inated in the proof of E. Arbarello and C. de Concini of the Novikov conjecture about the Schottky problem. There are two approaches to the Novikov conjecture. One is the approach of Mulase and Shiota in term s of soliton solutions [Sh86]. Another is the approach of E. Arbarello and C. de Concini [AC87] which depends on G unning’s work. Either approach led to a proof of the conjecture. Let M be a compact Riem ann surface of genus g > 1. Let cji, • • -,Lug be a basis over C of the space of all holomorphic 1-forms on M . Let 7 i, • • *, 72g be loops in M which form a basis in the first homology group of M . The g x 2g m atrix ( f uj^) is the period m atrix of M which can be reduced to the form (Ig,Q) with a suitable choice of the basis uji, • • -,tOg, where Q is a sym m etric g x g m atrix with positive definite im aginary part. The abelian variety J — Cn/ ( / y, Q )Z29 is the Jacobian variety of M . Let zq be a point of M . The m ap w which sends z to ( j ; c ^ ) E C 9 modulo {Ig,Q)7?9 embeds M into the Jacobi variety J as a subvariety W \ . The set of all positive divisors d — z\ + • • • -f zr of degree r is m apped to the subvariety Wr of J consisting of all w(d) := Y ^ = i w izv)' bet W f be the subvariety of J consisting of all points of Wr which correspond to divisors d = z\ -f • • • + zr with the property th a t the dimension of the space of meromorphic functions whose divisors > —d is > v. Let 0(w\ f}) be the th eta function Ylneza exp(27ri(^ tnQn + tn w ) ) ) where tn means the transpose of n. Riem ann proved the fundam ental vanishing and singularity theorem th at W g_ x is equal to c — 0 ^, where c is the point in J with 2c representing the canonical divisor of M and 0 ^ is the set of points in J at which all partial derivatives of 0 up to order v vanish. Gunning proved the analog of R iem ann’s vanishing and singularity theorem for second-order theta functions which are defined as follows. The theta function 6 satisfies the functional equation 9(w -f A; Q) = £(A; w)0(w, f2) for A G (/9, £2)Z2fir, where the factor of autom orphy is given by £(A]w) — exp(—2ni tq(w-\- \&q)) whenA = p + ^g. A secondorder th eta function is an entire function f (w ) satisfying f ( w + A) = £(A; w )2f{w ). A basis of the space of all second-order th eta functions is given by fk{w) = 0(2w + Qk; 2Q,)exp(2iritk(w + ^f2Ar)) for k E (Z /2 Z )5f. Let ~02 : C 9 C29 be the m ap whose components are fk for k E (Z /2 Z )fif. The analog of R iem ann’s vanishing and singularity theorem for secondorder th eta functions which Gunning proved asserts th at for any divisor d — z\ + • • • + zn the rank of the n x 29 m atrix /-r*/ ,

x

t-w (d ),

[02 ( w( zi ) + ----- 2

) >• ■•’ 02

,

t-w (d ),\ + -------2—

V

Robert C. Gunning

7

is less than n — v if and only if t £ W%_ 2. The special case of n = 3 says th at 02 ( f (J + w(2i “ *2 - ^ 3))), #2 (§ (t + w(22 - *3 - * i ) ) ) , #2 (|(* + ^(23 — 21 — ^2))) are linearly dependent if and only if t E W\ C *7. The “if” part is the trisecant formula of Fay [F73]. The “only if” part is G unning’s result and it was a breakthrough in the investigation of the Schottky problem. The problem of characterizing, among all sym m etric g x g matrices with positive definite im aginary parts, the ones th a t arise from compact Riem ann surfaces, had been posed by Riem ann and has been a m ajor preoccupation in the subject of Riem ann surfaces ever since. The first significant results beyond R iem ann’s observations th at the matrices are sym m etric and have positive definite im aginary parts were obtained by Schottky in 1888 and the problem has subsequently been called the Schot­ tky, or the Riemann-Schottky, problem. G unning’s converse of Fay’s trisecant formula implies the following. An irreducible principally polar­ ized abelian variety X is the Jacobian variety of some compact Riem ann surface if and only if X contains an irreducible curve V such th a t for any point z on L (r — a —f3 — 7 ) the three points

62

(2 + a), 02 (z + /?),

and #2(2 + 7 ) are collinear. He further reduced the existence of T to the verification of the positivity of the dimension of a subvariety in X . Let the subvariety in X consisting of the set of all z in X such th a t #2 (2 + a), #2 (2 + /?), and #^(2 + 7 ) are collinear. He then showed th at the existence of T is equivalent to the following two conditions: (1) The dimension of Va^ n at the point —a —j3 is positive for some a , /?, 7 in X . (2) There is no complex m ultiplication on X m apping the points (5 — a and 7 —a to 0. W hen these two conditions are satisfied the subva­ riety 2Va^ n will be the curve T with X as its Jacobian. Welters [W83] later gave the following infinitesimal version of G unning’s result when the three points a,/?, 7 approach the same lim it point. The abelian variety X is the Jacobian variety of some compact Riemann surface if and only if there exist constant vector fields D\ ^ 0 and D 2 such th at the dimension of the subvariety defined by { z e X X { z ) A d X { z ) A (D\ + D 2 ) t 2 {z) = 0} at 0 is positive. In th a t case the subvariety is a sm ooth curve whose Jacobian is X . Novikov’s conjecture which was proved by Shiota and Arbarello-De Concini says th a t X is the Jacobian variety of some compact Riem ann surface if and only if there exist vector fields D\ / 0, D 2, D 3, and a complex num ber d4 such th at the following Kodomcev-Petviashvili equation is satisfied. (.D\e)9 - A (D \ 6 ) ( D \ 6 ) + 3 {D 26 ) 2 + 3(D\6)6

8

Robert C. Gunning +3(D10)(D30 ) - (D i D30)6 + dA02 = 0 .

Gunning obtained his generalization of R iem ann’s result to secondorder th eta functions in the framework of generalized th eta functions and vector bundles over Riem ann surfaces and their Jacobian varieties (Ref­ erences 32, 42). While th eta functions are holomorphic sections of line bundles over abelian varieties, generalized th eta functions are holomor­ phic sections of vector bundles over abelian varieties. A vector bundle over the Jacobian variety of a compact Riem ann surface can be pulled back to a vector bundle on the Riem ann surface. The study of holomor­ phic vector bundles over abelian varieties or compact Riem ann surfaces is a very difficult subject. For compact complex manifolds X and Y, if there is a holomorphic family of holmorphic vector bundles V ( x ) over Y param etrized by x E X and if the dimension k of T(Y, V(x )) is indepen­ dent of x, then one can define a holomorphic vector bundle of rank k over X whose fiber at x is T(Y, V{x)). Every point t of the Jacobian variety J of a compact Riem ann surface M of genus g corresponds to a flat line bundle pt on M which is the pullback of a flat line bundle pt over J . For any line bundle £ of degree 1 over M , the dimension of T(M, ptCn+g~ 1) is independent of t when n > g. M attuck [M61] introduced the vector bundle x n over J whose fiber at t is r(M, ptCn+9 1). Such a vector bundle cannot be explicitly described by th eta functions. Gunning (Ref­ erence 42) considered another vector bundle \ over J closely related to second-order th eta functions. Second-order theta functions are holomor­ phic sections of a line bundle L over J . The fiber at t of the vector bundle X th a t Gunning considered is r ( J , ptL). Gunning showed th at x can be explicitly described as a natural extension of x g+1 • From the relation of X9+1 with M and the relation of x with second-order th eta functions, he obtained his analog of R iem ann’s result for second-order th eta functions and a wide class of very useful analytic identities for th eta functions. Besides his original scientific work, Gunning is an exceptionally tal­ ented expositor and teacher. He has directed over 25 Ph.D. students. He has authored books on such diverse topics as m odular forms, com­ plex analytic varieties, Riem ann surfaces, generalized th eta functions, and uniform ization of complex manifolds. He has recently published, in three volumes, an extensive revision of his 1965 classic book with Rossi. Gunning is an enthusiastic and hard working member of the Princeton University community, having served as Chairm an of the D epartm ent of M athem atics from 1976 to 1979 and, since 1989, as Dean of the Faculty. Through his teaching and writing Gunning has had a broad and in­ delible influence on a generation of m athem aticians.

Robert C. Gunning

9

[AC87] E. Arbarello and C. de Conici, Another proof of a conjecture of S. P. Novikov on periods of abelian integrals on Riem ann surfaces, Duke M ath. J. 54 (1987), 163-178. [B49] G. Bol, Invarianten linearer Dilferentialgleichungen, Abh. M ath. Sem. Univ. Hamburg 16 (1949), 1-28. [E57] M. Eichler, Eine Verallgemeinerung der Abelsche Integrale, M ath. Zeitschr. 67 (1957), 267-298. [F73] J. D. Fay, T heta functions on Riem ann surfaces, Lecture Notes in M ath. Vol. 352, Springer-Verlag 1973. [KO80] S. Kobayashi and T. Ochiai, Holomorphic projective structures on compact complex surfaces, M ath. Ann. 249 (1980), 75-94. [M61] A. P. M attuck, Picard bundles, Illinois J. M ath. 5 (1961), 550-564. [Sh86] T. Shiota, Characterization of Jacobian varieties in term s of soliton equations, Invent. M ath. 83 (1986), 333-382. [V72] A. Vitter, Affine structures on compact complex manifolds, Invent. M ath. 17 (1972), 231-244. [W83] G. E. Welters, A characterization of non-hyperelliptic Jacobi vari­ eties, Invent. M ath. 74 (1983), 437-440.

10

Robert C. Gunning

B ibliography of R ob ert C. G unning 1. General factors of automorphy, Proceedings of the National Academy of Sciences, U.S.A., 41 (1955), 496-498. 2. (with S. Bochner) Existence of functionally independent autom or­ phic functions, Proceedings National Academy of Sciences, U.S.A., 41 (1955), 746-752. 3. The structure of factors of automorphy, American Journal of Math­ ematics, 78 (1956), 357-382. 4. Indices of rank and of singularity of Abelian varieties, Proceedings of the National Academy of Sciences, 43 (1957), 167-169. 5. Multipliers on complex homogeneous spaces, Proceedings of the American Mathematical Society, 8 (1957), 394-396. 6. M ultipliers on complex homogeneous spaces, II, Seminars on ana­ lytic functions, Institute for Advanced Study, 1 (1957), 103-110. 7. On V itali’s theorem for complex spaces with singularities, Journal of Mathematics and Mechanics, 8 (1959), 133-142. 8. Factors of autom orphy and other formal cohomology groups for Lie groups, Annals of Mathematics, 69 (1959), 314-326. 9. Homogeneous symplectic m ultipliers, Illinois Journal of Mathemat­ ics, 4 (1960), 575-583. 10. The Eichler cohomology groups and autom orphic forms, Transac­ tions of the American Math. Society, 10 0 (1961), 44-63. 11. On C artan ’s theorems A and B in several complex variables, Annali di Mat., 55 (1961), 1-12. 12. Lectures on modular form s, Annals of M athem atics Studies, 48 (1962), Princeton University Press, 96 pages. 13. (with S. Bochner) Infinite linear pseudogroups of transform ations, Annals of Mathematics 75 (1962), 93-104. 14. Generalized symplectic differential forms and differential operators, Journal of Mathematics and Mechanics, 11 (1962), 703-724. 15. Differential operators preserving relations of automorphy, Transac­ tions of the American Mathematical Society, 108 (1963), 326-352.

Robert C. Gunning

11

16. Connections for a class of pseudogroup structures, Proceedings of the Conference on Complex Analysis, Minneapolis, 1964> SpringerVerlag, Berlin, 1965, 186-194. 17. (with H. Rossi) Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, NJ, 1965, 317 pages, Russian Trans­ lation, Mir, Moscow, 1969, 395 pages. 18. Lectures on Riemann surfaces, Princeton M athem atical Notes 2 , Princeton University Press, 1966, 254 pages. Germ an translation: Bibliographisches Institut, Mannhein, 1972, 276 pages. 19. Special coordinate coverings of Riemann surfaces, Mathematische Annalen, 170 (1967), 67-86. 20. Lectures on vector bundles over Riemann surfaces, Princeton M ath. Notes 6 , Princeton University Press, 1967, 243 pages. 21. (with Raghavan Narasim han) Immersion of open Riem ann surfaces, Mathematische Annalen, 174 (1967), 103-108. 22. Some non-Abelian problems on compact Riem ann surfaces, Pro­ ceedings of the Conference on complex analysis, Rice University, 1967, Rice University Studies, 54 (1968), 39-48. 23. Lectures on complex analytic varieties: The local parametrization theorem, Princeton M athem atical Notes 1 0 , Princeton University Press, 1970, 165 pages. 24. Q uadratic periods of hyperelliptic Abelian integrals, Problems in Analysis, Princeton University Press, 1970, 239-247. 25. Analytic structures on the space of flat vector bundles over a com­ pact Riem ann surface, Several Complex Variables, II, Maryland, 1970, Springer-Verlag, 1971, 47-62. 26. Local moduli for complex analytic vector bundles, Mathematische Annalen, 195 (1971), 51-78. 27. Some m ultivariable problems arising from Riem ann surfaces, Actes, Congres Intern. Math., 2 (1970), 625-626. 28. Complex analytic varieties, Lectures in theoretical physics, New York: Gordon and Breach, 1972, 253-285. 29. Lectures on Riemann surfaces: Jacobi varieties, M athem atical Notes 12, Princeton University Press, 1972, 189 pages.

12

Robert C. Gunning

30. Some special complex vector bundles over Jacobi varieties, Inven­ tories Mathematicae, 22 (1973), 187-210. 31. Lectures on complex analytic varieties: Finite analytic mappings, M athem atical Notes 1 4 , Princeton University Press, 1974,163 pages. 32. Rieman surfaces and generalized theta functions, Springer-Verlag, Berlin and New York, 1976, 165 pages. 33. Complex numbers and complex variables, McGraw-Hill Yearbook of Science and Technology (1977), 174-176. 34. On the divisor order of vector bundles of rank two on a Riem ann surface, Bulletin Inst. Math. Academia Sinica, 6 (1978), 295-303. 35. On the uniformization of complex manifolds: the role of connec­ tions, M ath. Notes 22, Princeton University Press, 1978, 141 pages. 36. M athem atics, A Princeton Companion, A. Leitch, ed., Princeton University Press, 1978, 316-319. 37. Affine and projective structures on Riemann surfaces, Riemann sur­ faces and related topics: Proceedings of the 1978 Stony Brook Con­ feren c eA n n a ls of Math. Studies, 97 (1980), 225-244. 38. On the period classes of Prym differentials, II, J. fu r die reine und angew. Math., 319 (1980), 153-171. 39. On projective covariant differentiation, E.B. Christoff el, The influ­ ence of his work on mathematics and the physical sciences, Basel: Birkhauser Verlag, 1981, 584-591. 40. Complex numbers and complex variables, McGraw-Hill Encyclope­ dia of Science and Technology (1982), 466-471. 41. Review of “Families of m eromorphic functions on compact Riem ann surfaces” by M. Namba, Bulletin of the American Math. Society, 4 (1981), 353-357. 42. On generalized theta functions, American Journal of Mathematics, 104 (1982), 183-208. 43. Some curves in abelian varieties, Inventiones Mathematicae, 66 (1982), 377-389. 44. An identity for Abelian integrals, Global analysis - Analysis on Manifolds, Teubner Texte zur Math., 571, Teubner, Leipzig (1983), 126-130.

Robert C. Gunning

13

45. Riem ann surfaces and their associated W irtinger varieties, B u ll Amer. Math. Soc., 1 1 (1984), 287-316. 46. Some identities for Abelian integrals, Amer. (1986), 39-74.

Jour.

Math., 108

47. On th eta functions for Jacobi varieties, Algebraic Geometry, Bowdoin 1985, aProc. Symposia in Pure Math., 46, part 1” , Amer. M ath. Soc. (1986), 89-98. 48. Analytic identities for th eta functions, Theta Functions, Bowdoin 1987, uProc. Symposia in Pure Math., 49, part 1 ” , Amer. M ath. Soc. (1989), 503-516. 49. Holomorphic Functions of Several Variables, W adsworth and Brooks, Cole Pacific Grove, California, 1990 Vol. I (Function Theory), 203 pp.; Vol. II (Local Theory), 218 pp.; Vol. I ll (Hom ologicalTheory), 194 pp. 50. The Collected Papers of Salomon Bochner (editor), Amer. M ath. Soc. (1991): Vol. I (762 pp.), Vol. II (790 pp.), Vol. Ill (732 pp.), Vol. IV (446 pp.).

Robert C. Gunning

14

List of P h .D . S tudents of R ob ert C. G unning Andrew Campbell Craig Benham Thom as Bloom Grigory Bluher Michael Eastwood Robert Ephraim Mike G ilm artin Xavier Gomez-Mont Richard Ham ilton Eric Jablow Sheldon Katz Richard Koch Henry Laufer Richard M andelbaum J. Peter Matelski Vernon Alan Norton Cris Poor David Prill John Ries M artha K atzin Simon Yum-Tong Siu John Snively Charles Stenard John Stutz A lbert V itter, III Bun Wong David Yuen

Joseph J. Kohn

JO SEPH J. KO HN T H E SC IE N T IFIC W O R K OF JO S E P H J. K O H N The work of J. J. Kohn on the Cauchy-Riemann equations and related operators has fostered an intense interaction between partial differential equations and the theory of functions of several complex variables. It has led to widely applicable analytic techniques and to delicate geometric questions of current interest. This synopsis aims merely to indicate the depth and breadth of K ohn’s life and work. Joseph J. Kohn was born in Prague, Czechoslovakia on May 18, 1932. Seven years later his family moved to a small town in Ecuador and three years after th at they moved to Quito, Ecuador. His family came to the United States in 1945 and lived in New York City. Kohn attended high school in New York, received his BS degree from MIT in 1953, and went to Princeton for graduate study. At Princeton he became D. C. Spencer’s student and began work on the 3-Neumann problem. Spencer approached complex analysis in several variables by trying to do Hodge theory on domains in complex Euclidean space or in complex manifolds. For simplicity here we consider a sm oothly bounded pseudoconvex domain Q in complex Euclidean space C n . Suppose th a t a is a differential (p , q) form with square integrable coefficients th a t satisfies da = 0. The 5-Neum ann problem is to construct the solution u to the Cauchy-Riemann equation du — a th at is orthogonal to the kernel of d , and to prove regularity results for u in term s of a. This particular solution, now generally known as the Kohn solution to the 3-Neumann problem, can be expressed as d N a where N is the inverse to the com­ plex Laplacian d d + dd . The 5-Neumann problem is a boundary value problem, because the condition th at a differential form be in the dom ain of d is a boundary condition resulting from integration by parts. We dis­ cuss the technical difficulties of the S-Neumann problem after including some additional biographical information. Although he had not yet solved the 9-Neumann problem, Kohn re­ ceived his Ph.D. in 1956. He remained in Princeton for two more years, as an instructor at the University and as a visiting mem ber at the Institute

Joseph J. Kohn

17

for Advanced Study. In 1958 he went to Brandeis University. In 1962 Kohn solved the A" called the signature operator of M . Its index is therefore also a numerical invariant of M . And much more generally, given any representation of SO (n), say on P , we have a corresponding bundle V over M associated to the frame bundle of g , and one can “tw ist” ds by V to obtain a new elliptic system ds V : r(A + ® V) — ►r(A- V). All these constructions, of course, vary sm oothly with the Riemann structure, hence index(d5 ® F ) yields a numerical invariant of M for every representation V , of S O (m), m = dim M . . Over the years we have learned th at the invariants created by this pro­ cedure are no greater - or less - than the Pontryagin and Chern numbers discussed earlier, but I want to emphasize th at these questions became relevant only after we understood ellipticity in its most general form. To explain why, let me presume - for this audience - to “remind you” how ellipticity is defined. Given a local expression for a linear operator pjOi D = Y a «{ x ) ^ ~ ^ w dxa where a runs over multi-indexes and the A a (x) are n x n m atrix functions, the “symbol of D n is defined as the expression (D ;()= \a \= m

where the £'s are real variables and the sum is taken over the highest order term s only. Then a x{ D ) is called elliptic if and only if for any system (£) with £ 0, (Tx (D;£) is a nonsingular matrix.

34

Raoul B o tt Hence when D is elliptic, the assignment of x rx p ;* )

K| = l

defines a m ap of (n — l)-sphere gn-1 _ ^ Q L ^

t

and the homotopy class of this element - le t’s call it [D] - is a fundam ental topological invariant of the elliptic system on every component of M . I like to call it the “virulence” of D. Now the only elliptic operator we all knew in the 40’s and 50’s was really the Laplacian of the Hodge theory - and its “virulence” is 0! On the other hand, the signature operator ds has a nontrivial virulence in even dimensions, and it is for th at reason th at the index problems of ds 0 V ju st described, yield such a rich harvest of invariants as we vary V. In this connection it has always struck me as quite marvelous - nearly a proof of the existence of God - th at in its Euclidean incarnation the famous Dirac operator, describing the behaviour of the electron, has vir­ ulence 1 - th at is, its symbol \$\ generates 7r*(GL(TV, C )), N 0, in all dimensions. But to continue the story of our manifold invariants, let me now take the point of view of the topologist proper, who does not care a fig about analysis and geometry. His m ain concern is to sort out just how essen­ tial the differentiability - so im plicit in our constructions - was for the invariants we have created, and from the structure theory of bundles it then soon became apparent th a t as far as the Pontryagin numbers were concerned, differentiability is used only to the extent th at we have to have a tangent bundle T M to M . Indeed, the classification theory of bundles then guarantees th at T M is pulled back from the universal bundle by a m ap f r : M — > B G L ( n ), into the classifying space of the full linear group, well determ ined up to homotopy, and one can recover the Pontryagin classes purely from this map! In fact, H *(B G L(n)) is generated precisely by the “universal” Pontryagin class pi E H 4l(B G L (n )) which have the property th at f r P i = Pi-

In short, our geometric constructions represented f^ P i. Finally, to show th a t our “Index invariants” really always am ounted to some Pontryagin

On Invariants o f Manifolds

35

and Euler numbers was of course rather more difficult, but is an im m e­ diate consequence of the Atiyah-Singer Index theorem. This then summarizes for you what we have known for a long time, and I would now like to briefly contrast it with some of the new invariants which were thrust upon us to a large extent by the physicists - and in particular by the young m an I see characteristically sitting halfway towards the back in this hall; th a t is, - Ed W itten. If we call what I have described so far as an interaction of analysis and topology, what I am going to describe now is better described as the effect of “super-analysis” on topology. And here “super” is really m eant, not in its technical sense, but in accordance with its usage in “supernatural” ! So let me describe the “new integrals” of E d ’s which we are deciphering in m any ways at this moment. One starts with an oriented 3-manifold and a compact Lie group G, with Lie algebra g. We traditionally write A for the space of 1 -forms on M , with values in g, and for simplicity think of A E A simply as a m atrix valued 1-form on M . Consider now the function S m : *4 -> R defined by

s“ {A)=^ L TriAdA+l Aa)It should be clear th a t under this integral we have defined an ordinary 3form and hence S m {A) is a well-defined number. Actually the expression under the integral was well known in the m athem atical world, and is called the Chern-Simons fo rm , and they constructed it explicitly to have the following functorial properties. Let Q be the space of sm ooth maps of M into G, and consider the following affine action of Q on A: g £ G

sends

A

to

g ~ l Ag + g ~ 1dg.

This is then the famous action of the “group of gauge transform ations” which was found to be so very im portant in the physics of this century. Well, the crucial property of the “action” , Sjif(A), ~ as the physicists would think of the function S m ~ is th at S m is nearly invariant under Q. In fact, (1 ) S m is invariant under the identity component Go of G and (2) Changes by an integer on the different components of G• It follows then th a t for every integer &, the function A —> e27rikSM(A) descends to A/G- So far so good, but now comes the supernatural part.

36

Raoul B o tt

Following sound physics folklore, Ed tells us to define an invariant Z £ (M ) out of these d ata simply by computing the integral Z g(M ) = f e^ i k S M(A)V A J a /G Q.E.D. In fact he does not stop there; he tells us also - and indeed, th at is how he came upon this recipe in the first place - th a t this hypothetical measure e2nikS(A)V (A ^ on A /Q can now be used to compute “expectation values of knots” in M . Namely, given an oriented knot k C M and a representation V of G, then every A E A defines a connection on the trivial bundle over M , so th at the closed curve h determines a corresponding holonomy element h(k) in G /A d G. The trace of h(k) relative to the representation V, is therefore a welldefined function on A, say x v (&), and its expectation value (*

*

*)

=

J

x v (k)e2*ihs^ A)V A / Z ? ( M )

should therefore yield invariants of the pair (M, k). I have run out of tim e to discuss these formulas in any detail here. But I hope you see how much work will be entailed in bringing these new invariants under control. But there is hope th a t they can indeed be tam ed, and even, th at suitably interpreted they lead to a complete classification of knots. The secret weapon of the physicists is of course the “higher loop” expansion procedures of Feynmann integrals. Thus to get a handle on an integral like Z ck (M ) = f e2*ikSM(A)T>(A), JA/g the physicist will start by computing an asym ptotic expansion of the right hand side as k —»> oo. The rationale for this expansion is derived from the principle of stationary phase for oscillatory integrals of this type over finite dimensional spaces. In th a t context it is then quite easy to see th at an oscillatory integral of the form:

[ e2nikl ^ v { x ) d x ,

Jm

On Invariants o f Manifolds

37

will tend to zero faster than any power of k as k oo; provided the sup­ port of (p avoids the critical points of / . Thus the asym ptotic expansion of JM e27rikf('x)dx is localized at the extrema of / . If the dimension of M is m, then every nondegenerate extremum, say at xq, of / contributes a term of the type: i

—— . i

equal to t. W hen this procedure is applied to Z (M ) one obtains a corresponding asym ptotic series, now indexed by the extrema of S m - And here the third crucial property of the Chern-Simons expression in dim 3 comes to the fore: th at is, if dim M — 3,then: (3)

S S(M ) \A = 0

the 2-form F (A ) = dA + A 2

vanishes. Thus the asym ptotic series in question is indexed by the flat connec­ tions on M , which in turn are classified by conjugacy classes of homomorphisms of 7Ti(M) into G. These remarks lead us to expect th at given M , and an isolated ho­ momorphism /i : 7Tq(M ) -* G, there should exist a well defined sequence of numerical invariants a^(M, /i),

t = 0, 1 , • • •

corresponding to the loop approxim ation to (***) at p. In attem pting to carry out the finite dimensional program in this infinite dimensional context, one encounters all the road blocks which over the years the field theorists have learned to overcome. Indeed the Hessian of S m at I1 turns out to be degenerate and the Fadeev Popof procedures have to be ap­ plied. For this purpose an auxiliary Riem ann structure, g, on M has to be chosen, and once this choice is made, a^(M, p) is seen to make sense, but as a very complicated integral involving the Green’s operator of the

Raoul B ott

38

Laplacian of g. At this stage one now has to show th at this integral is independent of the choice of g , and the physicists have developed for­ m al procedures to show this independence - called “B.R.S. invariance” . However it is only the recent work of Dror Bar-N atan (2), and Axelrod fc Singer (1 ) th at brings these questions into proper m athem atical form. Actually, Dror B ar-N atan’s work deals prim arily with the higher loop expansions for the expectation functions Xv (^) knots in S'3. This case is of course simplest because 7Ti(S3) = 0 and so one variable, the homomorphism of 7Ti to G, is eliminated. In any case, studying his invariants, and of course influenced by the work of Vasilief, Kontsevich and others, Dror Bar-N atan soon found how to separate the role of the group G and the representation V in the construction of the higher loop invariants ai(k). Indeed he constructed a universal com binatorial chain complex, - in fact, a Hopf algebra “A ” , in which a given pair (G, V) determ ine a cycle, and it is this algebra which therefore is now conjectured to be the appropriate com binatorial setting for a sufficient num ber of knot-invariants, to separate them. A few days ago I mentioned these m atters to my old student and friend, Tom Goodwillie. He thought for a while and then recalled an unpublished construction of his from years ago, m eant as a topologist’s approxim ation to the space of imbeddings of an interval in R n . He had proved convergence for n > 3(!). But when we wrote down his construc­ tion for n — 3, and then made a few com putations we found th at the E y te rm of his cosimplicial space was precisely the algebra A m entioned above! So let me end by writing down the scheme of Goodwillie’s “classifying space” , at least for the topologists among you, in the hope th a t it will prove to be the topological flesh and bones on which we can hang the V th loop “Feynm ann” invariants. Here it is:

BK:

*

C (l) -»■ C{2) ->

C(3) ••• ->

with G (r) denoting a “jet-version” of the configuration-space of r points in R 3. Thus at least the constituents G (r) of this space have a very geomet­ ric homotopy type, namely just the space of r distinct points in R 3. The “coattaching” maps are more difficult and involve subtle questions involv­ ing blowing up the diagonals in R 3 x • x R 3 - in short, precisely a deeper understanding of the dichotomy of “diagonal!” versus the “factors” .

On Invariants o f Manifolds

39

For us older and possibly more m ature m athem aticians, it is of course encouraging to find th a t our long and adventurous odyssey has brought us back home to old and familiar grounds. But I hope th at the youngsters among you, who, I know full well, love nothing more than to teach us new things, will not be disappointed by this turn of events. Indeed, via a construction of Kontsevich the Chern-Simons theory is at this mom ent indispensable in proving th at all the differentials in Goodwillie’s sequence vanish. And even if at some future tim e the topologists will be able to dis­ pense with this construction - the new bridges between analysis and topology which this theory has erected will surely always remain a beau­ tiful achievement of the m athem atics of this century.

R eferences (1) Axelroad & Singer, Chern-Simons perturbation theory, MIT Preprint, October 1991. (2) Dror B ar-N atan, Perturbative aspects of Chern-Simons topological quantum field theory. Ph.D. Thesis. Princeton Univ., June 1991. Dept, of M ath. (3) Dror B ar-N atan, On Vasiliev Knot Invariants, Harvard Preprint, Oct. 16, 1992.

R EM A R K S ON AN A LY TIC H Y PO E L L IPT IC IT Y OF db M ic ha el C h rist

1. I n tr o d u c tio n Let there be given a real analytic, pseudoconvex CR structure of finite type on a (small) open three-dimensional manifold M . Let db be the asso­ ciated Cauchy-Riemann operator, which m aps C°° functions to sections of a certain bundle B whose fibers have dimension one over C, and let db* be its adjoint, with respect to a nonvanishing real analytic density on M and a real analytic Herm itian inner product structure on B. Denote both the set of all functions real analytic in some open set U C M , and also the set of all real analytic sections of B over U , hy C w(U). db is said to be analytic hypoelliptic modulo its nullspace, in Q', if for any open set Q C for any distribution u such th at db§b*u E C W(Q), necessarily db*u E C w(Sl). It is a fundamental problem, suggested to this author by J. J. Kohn and to date unsolved, to determ ine for which such CR structures db is indeed analytic hypoelliptic modulo its nullspace. Recall th at a differential operator L is said to be analytic hypoellip­ tic in f2' if for any open set Q C SY, for any distribution u such th a t Lu E C w(fi), necessarily u E Cw(fi). db and dbdb* are never analytic hypoelliptic in this sense. It is of course also of interest to determine ju st which operators are analytic hypoelliptic. In more concrete terms, there are given, in a neighborhood of a point xo E 2£3, two real vector fields X , Y, having real analytic coefficients and linearly independent at every point, db is equal to X + iY . Fixing a third real vector field T linearly independent of X , Y, pseudoconvexity means th at the determ inant of the m atrix ( X y Y, T), com puted in some local coordinate system, is either everywhere nonnegative, or everywhere nonpositive. Finite type is the hypothesis of Horm ander, th at the Lie algebra generated by A, Y should span the tangent space to M3 at each point near x q . The CR structure is weakly pseudoconvex at x if the Lie bracket [X, Y] belongs to the span of X , Y at x, strictly pseudoconvex if A, Y, [X, Y] are linearly independent at x. Sometimes db is analytic hypoelliptic modulo its nullspace, and some­ times it is not; a necessary condition involves the following geometric notion. To any CR structure is associated a sm oothly varying field T of two-dimensional subspaces of the (real) three-dimensional tangent space, whose complexification is T 1,0 0 T 0,1; in other words the fiber Tx is the

42

Michael Christ

span of { X ( x ) , Y (x)}. 1 .1 . D e fin itio n s . A C°° curve 7 : (—£,£) for which 7 ' does not vanish is complex-tangential if for every s , 7'( 5) E 77(5)- 7 said £o 6e a weakly pseudoconvex, complex-tangential curve if in addition, 7 (s) is a weakly pseudoconvex point of the CR structure, for all s. 1 .2 . T h e o re m . Let there he given a real-analytic, pseudoconvex, threedimensional CR structure of finite type in an open set Q. I f there exists a weakly pseudoconvex, complex-tangential curve contained in Q, then db fails to he analytic hypoelliptic modulo its nullspace in Q. This is a special case of a conjecture of Treves [Tr2], which he for­ m ulated also for more general operators and in higher dimensions. It is expected th a t L = X 2 + Y 2 and db should share much the same regular­ ity properties. Indeed, retaining the finite type hypothesis but dropping pseudoconvexity, L — X 2 + Y 2 fails to be analytic hypoelliptic in Q whenever there exists a curve 7 in Q whose tangent vector is spanned by X, Y, and such th at X , Y , [X, Y] are linearly dependent, at each point of 7 . This, and Theorem 1.2, are proved in [CIO]. At one point it seemed plausible th at the necessary condition of The­ orem 1 .2 , th a t is, nonexistence of certain curves, should also be sufficient for analytic hypoellipticity. Although this has not yet been proved, the analysis of [C ll] now strongly suggests, to the contrary, th a t analytic hy­ poellipticity should fail for certain structures having only isolated weakly pseudoconvex points. Weakly pseudoconvex, complex-tangential curves have arisen in at least two other contexts. Noell [Nl] has shown th at if D (e C2 is pseudo­ convex and C u , and if there exist no such curves in d D , then any compact subset of dD which is locally a peak set for A°° (D ) is actually a peak set for A°°(D), while in the presence of such curves, local peak sets need not be peak sets. Montgomery [Mo] has given examples in which geodesics in sub-Riem annian geometry - curves which minimize distance between two points relative to a degenerate Riem annian m etric satisfying a finite type hypothesis - fail to satisfy the “geodesic equations” which are a formal consequence of variational calculations. The underlying geometric struc­ ture in these examples is almost identical to the models (1.4) below, and the geodesics in question are weakly pseudoconvex, complex-tangential curves. In the positive direction, there is one outstanding result: db is analytic hypoelliptic, modulo its nullspace, wherever the CR structure is strictly pseudoconvex. This result an d /o r closely related ones have been proved by Tartakoff [Tal],[Ta2], Treves [Tr2], Metivier [M2],[M3], and Geller [G]. Analytic hypoellipticity holds good in other instances, as well. Consider,

Rem arks on A nalytic H ypoellipticity o f db

43

with coordinates (#, y, t) on M3. (1.3)

X - dx ,

Y = dy + a(x, y)dt

where a is a homogeneous polynomial of some degree m —1 , such th a t dx a is nonnegative and vanishes only at x = y = 0. Since [X, Y] = dxa • dt , the CR structure is strictly pseudoconvex where (x,y) ^ 0, and weakly pseudoconvex along the curve {x — y = 0}; nonetheless, db is analytic hypoelliptic, modulo its nullspace. A variety of interesting partial results, for related problems, have been obtained by Derridj and Tartakoff; see [DT1],[DT2],[DT3] and the references therein. A better result is th a t of Grigis and Sjostrand [GS]: with X = dx and Y = dy + a (x , y)dt , the operator L = X 2 + Y 2 is hypoelliptic when dx a (x , y) = x k -f ym with k , m even positive integers (and a is analytic), and more generally when A(x,y) = dx a vanishes only at the origin and satisfies A(r1/*#, t*1/™*/) = rA (x,y). The first negative result for db was obtained in [CG]. Consider (1.4)

x = dT,

Y = dy + x m- 1dt

where m is an even, positive integer. Then db fails to be analytic hy­ poelliptic, modulo its nullspace, in any neighborhood of the origin, in the weakly pseudoconvex case m > 2. Earlier, Helffer, P ham The Lai and Robert had shown th a t for the same vector fields, L = X 2 + Y 2 fails to be analytic hypoelliptic in the (non-pseudoconvex) case m — 3. In these examples, any curve j ( s ) = (0,s,fo) is weakly pseudoconvex and complex-tangential; the two-dimensional locus of weakly pseudocon­ vex points is foliated by this one-parameter family of curves. In examples (1.3), when dx a vanishes only at x = y = 0, the set of weakly pseudo­ convex points is the single curve 7 (5) = (0, 0,s), which is not complextangential. More generally, when X — dx and Y — dy + a(x)dt with a(x) = x m + 0 (|# |m+1) near the origin, db fails to be analytic hypoelliptic m od­ ulo its nullspace near 0 [C3], for even integers m > 2 (and similarly for X 2 + y 2 [C5]). A consequence is the failure of analytic hypoel­ lipticity modulo the nullspace for 2; the weakly pseudocon­ vex locus is the single curve {x = t = 0}. In §5 a proof of the following special case of Theorem 1.2 is outlined. 1 .6. T h e o re m . 7/m , M are both sufficiently large then db is not analytic hypoelliptic, modulo its nullspace, for the CR structure (1.5). The proof is of a perturbative character, building on the results for the models (1.4). The hypothesis th at M should be large am ounts to requir­ ing the structure to be a sufficiently small perturbation of the models. A related question is th at of global analytic hypoellipticity, modulo the nullspace: given a compact three-dimensional CR m anifold M without boundary, analytic, pseudoconvex and of finite type, and given both th a t dbU = / E C UJ( M ), and th at u is orthogonal to the nullspace of db in L2(M ), does it follow th at u E C w ( M ) l S.-C. Chen has proved this (or a closely related result) to be true for R einhardt domains, so th at one has simultaneously negative results [C3] for the local problem and positive results for the global one, on the same domain. Chen has further proved global regularity for a large class of circular domains, and Derridj and Tartakoff [DT4] have been able to relax the sym m etry hypothesis to the existence of an analytic vector field transverse to the CR field of 2-planes and having certain favorable com m utation properties with db, db*. In [C12] it is shown th at global analytic hypoellipticity holds for a wide class of partial differential operators on compact manifolds, given the existence of a group of suitably transverse symmetries. Other reasons can be advanced in support of the hope th at global analytic hypoellipticity should always be valid. For instance, on a torus T n, convolution with any distribution K defines an operator m apping C UJ(T n) to C U}{T n), whereas such an operator is analytic pseudolocal if and only if K E C “ on T n minus the group identity element. In particular, any C°° hypoelliptic, constant-coefficient partial differential operator on a torus is globally analytic hypoelliptic, whereas such an operator is analytic hypoelliptic in the local sense if and only if it is

Rem arks on A nalytic H ypoellipticity o f db

45

elliptic. Thus global analytic hypoellipticity is a far weaker property th an its local analogue. These heuristics are, regrettably, misleading for the case of variable coefficients [C ll]. 1.7. T h e o re m . There exists a pseudoconvex, bounded domain Q C C2 with C ^ boundary such that db fails to be analytic hypoelliptic, modulo its nullspace, on dfl. Furthermore, the Szego projection does not map to C w(dSl). Although this has not yet been proved, the analysis of [C ll] suggests th at this should be the typical situation for weakly pseudoconvex do­ mains, and th at it can happen even for domains having only isolated weakly pseudoconvex points.

2. A n a ly tic h y p o e llip tic ity fo r m o re g e n e ra l o p e r a to r s Results known for operators other than db, its higher-dimensional ana­ logue and sums of squares of vector fields help to place our problem in better perspective. For simplicity we restrict attention to differential operators which have C u coefficients, act on scalar-valued functions, are C°° hypoelliptic, and whose principal symbols do not vanish identically at any point x E Q. Such an operator L is analytic hypoelliptic in £1 if and only if, for every open C ^ and every xq E there exists C < oo such th a t for any u E L 2(£l') satisfying Lu — 0 in Q', (2.1 )

|d “ ti(a;o)| < C'1+ |a ||a |l“ l||w||L2(n')

for all a .

We refer to (2.1) as the Cauchy estim ates. If (2.1) is valid and if Lu — f E C w in a neighborhood of xq , then (by the assym ption on the principal symbol) the Cauchy-Kowalevsky theorem gives an analytic v such th at Lv — f near xq, s o th a t L(u —v) = 0 and the growth estim ates (2.1) then imply analyticity of u —v. The contrapositive of the converse implication may be proved by elementary reasoning. The elliptic case is simplest. If L is of order n and elliptic, then for any open sets U (s Uf, for any positive integer k , there is an inequality

\\u\\Hn+k(u)


for pseudoconvex, three-dimensional CR structures of finite type. It is of principal type, and is subelliptic with loss of less than one derivative, in a neighborhood of one half of its characteristic variety E (which is a line bundle over M3). Thus db is analytic hypoelliptic, hence is certainly so m odulo its nullspace, microlocally near th at half of E. The principal symbols for sums of squares operators X 2 + Y 2, and for dbdb* , are sums of squares of real-valued functions, hence vanish to order at least two. Such operators are said to have m ultiple characteristics, and their C°° and analytic regularity theories diverge markedly. The outstanding positive result for m ultiple characteristics was due in various formulations initially to Tartakoff and to Treves, and later to Metivier, Sjostrand and Geller in other cases and formulations. It states roughly th a t if L is of second order, subelliptic with loss of one deriva­ tive and if its principal symbol vanishes to order exactly two everywhere on its characteristic variety E, then L is analytic hypoelliptic, provided th a t E is a symplectic submanifold of the cotangent space. Examples are □5 on strictly pseudoconvex CR manifolds (of dimension strictly greater th an three), dbdb* on strictly pseudoconvex three-dimensional CR m ani­ folds, and certain, but not all, sums of squares of real vector fields. The result microlocalizes in the usual way, and there are now several distinct m ethods of proof. An example which satisfies all of these conditions ex­ cept for being symplectic, and indeed fails to be analytic hypoelliptic, is d2 x + x 2d 2 + d 2 in M3 [BG],[H1]. As already explained in §1, positive results do hold, for certain nonsymplectic operators with m ultiple characteristics. An easily treated ex­ ample is the following special case of the subtler result of Grigis and Sjostrand [GS]. 2.2. P r o p o s itio n . Let X = dx , Y = dy -f a(x, y)dt where a is a polyno­ mial homogeneous of degree m — 1 for some m > 2. Suppose that d a /d x vanishes only at (0,0). Then L = X 2 -f Y 2 is analytic hypoelliptic. Outline of proof We take for granted an array of facts belonging to the detailed C°° theory for such operators. First, there exists an operator P f ( x ) = f R3 K (u, v) f (v ) dv such th at L P = P L = /, acting for instance

Rem arks on A nalytic H ypoellipticity o f db

47

on Co°(M3), whose kernel K is C°° off of the diagonal, and may be chosen so as to satisfy certain point wise bounds of a type first obtained by Nagel, Stein and Wainger [NSW] and Sanchez-Calle [Sa]. K m ay be taken to be homogeneous: K ( r u , rv) — r 2 ~mK ( u , v) for all u ^ v E M3, r > 0. Likewise K may be taken to satisfy F l((x ,2/, f), (#', y', f')) = K ( ( x , y , t - t ' ) , (x1, yf, 0)). To show th a t L is analytic hypoelliptic, it suffices to prove K to be analytic with respect to u, at every point off of the diagonal. W rite u = (x ,y , t). Since L K = 0 off the diagonal, where L acts in either variable, K is analytic, with respect to it, off of the diagonal at all strictly pseudoconvex points, which is to say, where (x, y) ^ 0. f ( u ) = K ( u , 0) is analytic where x = y = because / satisfies an elliptic system of differential equations; not only is L f = 0 except at 0, b ut also the dilation sym m etry means th a t / is annihilated by a first-order differential operator, whose characteristic variety is disjoint from the characteristic variety of L in a neighborhood of {x = y = 0}. Differentiating with respect to t turns the first-order equation into one of second order, so th a t / is annihilated by a second-order elliptic system near x = y = 0, hence is analytic there. Combining this with the result for the strictly pseudoconvex points, u —» K ( u , 0) is analytic where u ^ 0, and then by the translation symmetry, u —►K ( u , (0, 0, s)) is analytic where t ^ s. The same reasoning, applied to i f as a function of (u, v) E M6, dem onstrates more generally th a t K is analytic where t ^ s and the first two coordinates of v are close to 0. u i—►K ( u , v ) is analytic wherever u ^ v and u is a strictly pseudo­ convex point, so it remains to treat the case where v = ( x , i yf i t /) with (a?', y') ± 0 and where u — (x , t/, t ) with (x , y) in a small neithgborhood of 0. It suffices to prove analyticity microlocally in a conic neighborhood of the characteristic variety of L, and since dt is elliptic in th a t region, it suffices to obtain the growth estim ate 3*7 = 0 (C * k k) for f ( u ) = K ( u , v). This would follow from a bound (2.3)

\ f (x , y, r ) | = 0 ( e x p ( - c |r |) ) as |r | -► oo,

where / denotes the partial Fourier transform with respect to the variable t. Setting w = (x, y) and g(w) = f ( w , r ) for a fixed r , g is annihilated by L t = dl + (dy + i a( w) r ) 2 in a fixed neighborhood U , at a positive distance from ( # ',t/) , of 0. Various elementary arguments yield a prelim inary bound g = 0 (1 ) in U, as |r | —►oo. Fixing smaller neighborhoods 0 E

48

Michael Christ

U" (s U' kmk, while the right side is 0 (kk eck). The factor eck on the right-hand side comes from exp (iT1/ ™ ^ ) , which grows since ^ M. More precisely, this analysis shows th at there exist solutions to Lu = 0 which do not belong to any Gevrey class Gs with s < m. This may be proved either by arguing th a t Gevrey regularity would imply a priori estim ates generalizing the Cauchy estim ates (2.1), with \ a \ ^ replaced by |a |sla l, or equivalently by forming a series f Tj( x , y , t ) and choosing the sequences Cj, Tj so as to obtain a convergent series whose sum does not belong to G s .

Rem arks on A nalytic H ypoellipticity o f db

51

Given the existence of a £ with the required property, the negative result for analytic hypoellipticity emerges directly from the underlying anisotropic dilation structure; the factor exp(irt) oscillates much more rapidly than the factor e x p (ir1/ m('y) grows. This is alm ost completely analogous to the rem ark several paragraphs above concerning homoge­ neous, elliptic operators with constant coefficients and the stronger form of the Cauchy estim ates, but the dilation groups are different, so the im­ plication of non-injectivity is different. There is however no fundam ental theorem of algebra to guarantee the existence of such £, here. Note th a t even if U is small, the growth properties of g(x) for large x come into play in estim ating the right-hand side. It is not essential th at g should be bounded, but for the argum ent to succeed, g(x) m ust grow less rapidly than exp(cxm) as \x\ —>■oo, for all c > 0. Now it happens th at there is a sharp dichotomy: for any given £ £ C, either every solution grows like exp(c#m), or there exists a solution which decays exponentially as \x\ -> oo, so in particular is in L 2. In term s of representation theory, then, the issue is whether c?7r(L) = L £ fails to be injective on a nonunitary representation 7r, corresponding to some £ E C \M . The remarkable m ethod used by Pham The Lai and Robert to prove, indirectly, the existence of such a £ was in part rather general, but relied at one step on the oddness of m. The equations L \ g = 0, L \ g = 0 are examples of nonlinear eigenvalue problems, in which the usual eigenvalue equation Ag = XBg is generalized to Y2k Xk A}~g — 0. There is a sub­ stantial theory devoted to this problem, but the general results [Ke],[FrS] require hypotheses not satisfied by L^, L £. In §4 we will outline a differ­ ent approach which applies to L £ and a family of related operators. There is the following heuristic connection between local and global an­ alytic hypoellipticity. Suppose th at the only obstruction to local analytic hypoellipticity were to arise from the functions f T and variants of them . Fix a compact CR manifold M w ithout boundary, satisfying the usual hypotheses. Then any non-analytic solution db*u of dbdb*u = 0 would be synthesized out of functions resembling db* /r , hence its high-frequency Fourier components would exhibit growth like ex p (cr1//my) along weakly pseudoconvex, complex-tangential curves (param etrized by y). But db*u m ust be globally bounded on the compact manifold M , so can not ex­ hibit such growth. Thus any obstruction to global analyticity should be different in nature.

Michael Christ

52 4. A n a ly sis o f th e m o d e ls Throughout this section the vector fields X = dt ,

Y = d y ~ X m- 1dt

are fixed, and m > 2 is assumed to be even. Set db = X + iY, db* — X —i Y , and

Lz = {i ~

{z~

°

^

+ {z~

2 6 c

The problem is to prove the existence of z and a solution g E L°°(M), not identically vanishing, of L zg = 0. One solution is exp(—z x + m ~ 1 x m), but it grows too rapidly. Two solutions are /*±oo

ft(x) =

/

e 2 ( z s -m - 1 s"') ds

JX For all z, / + remains bounded, in fact decays rapidly, as x —> +oo, while f ~ behaves similarly at —oo. Up to scalar m ultiples, these are the only solutions bounded at ±oo, respectively. Therefore given z, there exists a nontrivial globally bounded solution to L z , if and only if ff- are linearly dependent. This is equivalent to the vanishing of their W ronskian, which up to a constant factor equals

/

oo

e2(2, - m - l «~ )d s_

-OO In the strictly pseudoconvex case m = 2, N is a Gaussian ciex p (c2Z2) and assuredly has no zeroes. For m > 4, at least three elementary arguments establish the existence of zeroes of N . First, the asym ptotics of N for real z, coupled with simple upper bounds, show th a t N is an entire holomorphic function of order exactly m / ( m — 1) E (1,2), hence N has infinitely m any zeroes [CG],[C3]; this property of entire functions of finite order replaces the fundam ental theorem of algebra. Alternatively, for A E M, N ( i \ ) is real-valued, and as A —> oo satisfies for some exponent cr > 0 N ( i \ ) = c1A -,7e x p ( -c 2Am/(m- 1)) •cos(c3Am/(m- 1) + c 4) + 0 (A ~ 2 3 be an integer. I f P is a generic, homo­ geneous polynomial then either the set of all nonlinear eigenvalues of {P f i d / d x , (z — x m~ 1))} equals the set of all zeroes of an entire holomorphic function of order precisely m / ( m — 1); or there exists a nontrivial solution g of at least one of the two equations P { i d / d x , ± x ) g = 0 such that g{x) = 0 (\x\N ) as \x\ —» oo, f or some finite N . In the former case, constructing a family of solutions f T as in §3 leads to a contradiction with the Cauchy estim ates, hence implies th a t L = P ( X , Y ) is not analytic hypoelliptic. In the latter case, an auxiliary argum ent establishes the existence of solutions which are not even C °°. The definition of “generic” may be found in [C4],[C6]. The first step in the proof is the existence of solutions f f - to L z f = P ( i d j d x , (z —x m~ 1) ) f = 0, whose asym ptotic behavior as x —» d=oo may be com puted precisely, and which depend holomorphically on z E C. The W ronskian of an appropriate subcollection of these plays the role of N and is 0 (e x p (C |z |m/(m- 1))) as —> oo. The operator P ( i d / d x , —x) may be realized as the lim it, in an appropriate sense, of L z as z —» oo. If the Wronskian of the f f - does not grow like exp(czm/(m -1)) as M+ 3 z -> oo, T am indebted to R. Askey, G. G asper and D. Geller for this inform ation.

54

Michael Christ

then the functions become closer to being linearly dependent than they ought to be, and in the lim it z —> oo become dependent, resulting in a solution of P(i d / d x , x)g — 0 with m ild growth properties.

5. N o n -a n a ly tic ity in a p e r t u r b e d case In this section we outline the proof2 of Theorem 1.6; details are in [C7]. Fix X — dx , Y — dy — (x rn~ 1 + x t M )dt . By p we denote always an exponent which depends on m ,M and which may be taken to be as large as we please, by choosing m, M to be sufficiently large. The value of p may vary from one occurrence to the next, and m, M are always implicitly assumed to be large. One additional fact about the model case (1.4) will be required: there exists zo G C such th at N ( zq) = 0 but N'(zo) ^ 0. Indeed, N'(iX) satisfies asym ptotics sim ilar to those indicated in §4 for N(iX), and these imply th a t all but finitely many of the im aginary zeroes of N are simple. Fix for the remainder of the discussion such a z$. Set db — X + i Y . The formal adjoint, with respect to Lebesgue m ea­ sure, is —X + i Y + i M x t M~ l , but in order to simplify the exposition we will define dC — —X + iY; the necessary corrections are in [C7]. The aim is to construct a one-param eter family of solutions u T to dbdb*, so th at db*uT violate the Cauchy estim ates. (In fact dbdb*uT will not be zero, but will be small for large r.) These will take the form u T = eiTte - Tt2 e ^ ^ f ( x , y , t ) where 0 , / are to be determined; both of these depend on the large pa­ ram eter r G M+. As compared to the model case, the factor exp() plays the role of e x p (fr1/ mCy), / th at of the solution of the ordinary differential equation. The factor exp(—r t 2) is new, and serves to localize our solutions near t — 0. Where t ^ 0, the CR structure is strictly pseudoconvex, and hence our construction m ust fail, which in the model cases is reflected in the absence of zeroes for N when m — 2. W ithout the factor of exp (—r t 2), the procedure outlined below would still produce a good trial solution u r , but only in a small neighborhood of {t = 0} shrinking to {£ = 0} as r —» oo. In order to violate the Cauchy estim ates, solutions, or near-solutions, are required in a fixed open set. M ultiplication by cutoff functions would extend uT to a fixed dom ain — but would destroy the equation §bdb*uT = 0. If, however, u T is sufficiently small on the 2Since preparation of this article a more powerful and sim pler variant [CIO] has supplanted the argum ent outlined here.

55

Rem arks on A nalytic H ypoellipticity o f db

support of the gradient of such a cutoff function, then dbdb*ur is still approxim ately zero, and this suffices. Set A = r 1/m and xo = A-3 / 4, to = A-m / 4. r denotes always an element of [^, 1]. The fundam ental quantity in the construction is a function 0(y,t), depending also on r , related to 0 by dycj) = iXzo 4- 0,

0 (0, t) = 0.

It is convenient to ask th at 0 , 0 , / be holomorphic functions in certain domains: Set Dr = {(y , t ) G C2 : \t\ < r t 0, \y\ < rAm/8}, cjr = {x £ C : —r \ ~ l < 3S(a?) < 3rxo and

< r (A + r \ x\ m~ 1) ~ 1} ,

\$s(x)\

f lr — ujr x D r . For any function h ( y )t) holomorphic in D r , we write fy J h ( y , t ) = I h (u , t ) d u . Jo Set 0 (e, y, f) = Azo^ —iOx — (m ~ 1x m + 2~1x 2t M )(r + 2zrf — i J d t 0). Then 56ut = eiTt- Tt\ * ( V + E ) f where V —

o dx o e~^ and E — i Y — idy — i (xm~ 1 + x t M ).

Similarly db* is conjugated to T>* + E where Set C — V V *. is inverted by V - ' g i x , y, t) =

= —e ~ ^ d x e^.

f

A sim ilar form ula defines X*"1, and we set £ _1 = P ” 1 o V ~ l . Then V o V ~ 1 and £ o C ~ l equal the identity. We have (V + E){V+ + E ) = C + £,

E = V E + EV* + E 2.

Suppose for the m om ent th at 0 £ H ° ° ( Dr ) were given. To obtain a solution / to (£ + £ ) / = 0, define the initial approxim ation /o (» ,y ,t) = e " ^ ’^

f e ^ d s . J Xq

For j > 1 set /,- = ( - i ^ ' oc- W

o.

Fix large exponents p, q. Given r, set r' — r — \ ~ q.

Michael Christ

56

5.1. L e m m a . Assume that \\0\\H°°(Dr) — 0 (1 ). 7 /ra ,M are sufficiently large, then for all (x , y , £) E Tlr>, / or j > 1, ^)| < C^A- *7’^ e x p ( - r m - 1 xm)|. Thus the series / + = f j converges, and it can be shown to define a solution t o £ + £ i n Qr,. However, this solution is defined only, essentially, for $l(x) > 0. In the model case, we saw th at growth properties of the solutions of the ordinary differential equations were of the essence, and only a discrete set of £ led to adequate growth properties. The situation here is the same; for most #, our solution / + m ust be expected to be very large for x < 0 (assuming it exists there at all). Instead we repeat the construction for 3f(x) < 0, obtaining a solution / “ , and attem pt to choose 0 so th a t m atch across the hypersurface x = 0, defining together a single solution having good growth properties on both sides. C is a second-order ODE, so both and dx f ^ ought to be m atched at x — 0; so a second unknown function is needed. To th a t end, f ~ is defined by taking as the initial approxim ation fo = ( 1 + /?(!/, * ) ) e - *

T

eW 'vrtds

J —Xq where f3 E H ° ° ( D r ). If ||/?||tf°o = 0 (1 ), then the analogue of Lem m a 5.1 holds for (—x , y , t ) E ^ r' * The m atching equations f +(x = 0) = f ~ ( x — 0), dx f +(x = 0) = dx f ~ ( x — 0) am ount to a system of nonlinear integro-differential equa­ tions of infinite order and are solved by Newton’s m ethod. Define W(0, /?, y, t) = det (

) (0, y, t)

= ( f + - f~)(0,y,t)

and G(0,P){y,t) = ( ^ ) • We aim to solve the system Q(0 , (3) = 0 for the two unknowns 9, j3 E H ° ° . Assume always th at ||0|| < 1, ||/?|| < 1. 5.2. L e m m a . There exists a nonzero constant Co such that \\deW(0,

AO) - c0(l + f3)X~2A0\\H°°(Dr,)

< C \ ~ p \\A9\\tf°o(j)r') + CA_3 ||0|| • ||A0||

for all sufficiently large A, where r f = r — X~q, and p, q —> oo as m, M —> oo. In fact, co is a normalizing factor times N'(zo), so th a t a simple zero of N for the model case is needed for Newton’s m ethod to be applicable. The other first-order derivatives may also be estim ated:

Rem arks on A nalytic H ypoellipticity o f db

57

5.3. L e m m a . I f ||/?||#°o(£)r) + \\0\\H(Dr) ^ A Po then in D r>,

( a!?) where cq \

2

C2A- 2

0

ciA- 1

whfh cq, ci / 0, and \ \ n ( e , / i , A e t A m ( H ~ ® H ° ° K D r, ) < \ - p \\Ae\\ + \-pUAf3\\ where p —> oo as min(m, M ,po)

00•

Upper bounds may be given for the second derivatives of C? with respect to 0 , /?, in the same spirit. This is the crux of the construction. We are unable to invert the differential of Q, only its leading-order term . Even when m, M are small, 7Z will be small relative to A , provided th a t r — r' is not too small. But N ewton’s m ethod will converge relatively slowly since dQ will not be inverted exactly, necessitating a large num ber of iterations in order to make Q sufficiently close to zero. The assum ption th a t m, M are large perm its sufficiently m any iterations, for large r, th at Q may be m ade sufficiently close to zero for the Cauchy estim ates ultim ately to be violated. The upshot of Newton’s m ethod is as follows. 5.4. P r o p o s itio n . There exist 6 , f3 holomorphic in

and satisfying

OO

< A p , such that

\W(e,f3)(y,t)\ + \M(d,/3)(y,t)\ < C e - AP for all (?/, t ) 6 D, where p —> 00 as min(m, M ) -» 00. The construction may be thought of as an analogue of the m ethod of geometric optics. 6 satisfies a nonlinear aeiconal” equation, while the f j are determ ined successively by solving linear “transport” equations, which are determ ined by 6 . However, here it is necessary to solve the infinitely m any transport equations, before the eiconal equation is even known; we have an infinite coupled system. Fix such 0, /?, and let be the functions constructed in term s of 6^/3 by the procedure already outlined. Fix a sequence of cutoff functions

58

Michael Christ

r]k G C°°(M ) satisfying 0 for a: > 1 , such th at


t>}g ( x , y , t )

and u ( x , y , t ) = u(x,y,£) • ryx(fo- 1 f) ■Vk {A3/ 4#), where the new cutoff functions 77# E Cq°(M) are supported in (—1,1), are identically equal to one on [— A] and satisfy

• • l laxJ - r r ^ l k 0 < C3^ 3

for a ll° < i
0 and A < 00 such that for all sufficiently large r, for all k > 1 , \d*db*uT(o)\ > c \ mk. Also \\dZ,y,tdbdb*uT\\L2{{i) < C exp(—Ap) provided that |a | < A + 1Let To denote a sufficiently small conic neighborhood of th a t half of the characteristic variety of dbdb* in T *(M 3) on which db is not hypoel­ liptic. Denote by H s (To) the usual L2 Sobolev space of order s, but microlocalized to IV

Rem arks on A nalytic H ypoellipticity o f db

59

5.7. L e m m a . There exists c > 0 such that for all sufficiently large r,

\\db*uT\\HHro)> c x \ mX. The last sticking point is th at db&b*uT is not exactly zero, nor is it even real analytic, because of the imperfect m atching of Consequently there is no hope of using { uT} directly to contradict the Cauchy estimates. The same difficulty was encountered and surm ounted by Metivier in earlier work [Ml]. Building on model examples such as )2 E u c l id e a n ( Q u a n t u m ) F ie ld T h e o r y , P r in c e to n U n iver­ sity P ress, P r in c e to n , N J, 1974. [S] J. S jo stra n d , A n a l y t i c w a v e f r o n t se ts a n d operators w ith m u l t i p l e c h a r a c t e r i s ­ t i c s , H ok k aido M a th . J. 12 (1 9 8 3 ), 3 9 2-433. [Sm] H. S m ith , A calculus f o r t h r e e - d i m e n s i o n a l C R m a n i f o l d s of f i n i t e t y p e , J. F u n ctio n a l A n a ly sis (to a p p ea r). [St] E . M . S te in , A n exa m ple on th e H eisenberg group related to the L e w y o pera to r, In v en t. M a th . 6 9 (1 9 8 2 ), 2 0 9-216. [T al] D . Tartakoff, Local a na lytic hyp o ellip ticity f o r Q& on n o n - d e g e n e r a te C a uc hyR i e m a n n m a n i f o l d s , P ro c. N a t. A cad . Sci. U S A 7 5 (1 9 7 8 ), 3 0 2 7 -3 0 2 8 . [Ta 2]__ _______ , O n the local real a n a l y t i c i t y of so lu t i o n s to a n d the d - N e u m a n n p r o b l e m , A c ta M a th . 1 4 5 (1 9 8 0 ), 117-204. [Trp] J-M . T repreau, S u r I ’h yp o ellip ticite an a lytiq ue micr olocale des o p er a teu rs de type p r i n c ip a l , C om m . P D E 9 (1 9 8 4 ), 1119-1146. [Trl] F . T reves, A n a ly ti c - h y p o e l l i p t i c p a rtia l differen tia l eq ua tion s o f p rin c ip a l t y p e , C om m . P u re A p p l M ath . 2 4 (1 9 7 1 ), 537-570. [Tr 2] _______ , A n a l y t i c h y p o - e llip tic ity of a class of p s e u d o d i ff e r e n t i a l oper ators w ith double c h a r a c te ris tic s a n d a p plicatio n s to the d - N e u m a n n p r o b l e m , C om m . P a rtia l D ifferen tial E q u a tio n s 3 (1 9 7 8 ), 4 75-642. [P]

D epartm ent of M athem atics, UCLA, Los Angeles, Ca. 90024

FIN IT E T Y P E C O N D IT IO N S A N D SU BELLIPTIC ESTIM ATES

J ohn P. D ’A ngelo Introduction The present paper has two purposes. The prim ary purpose is to survey some work on finite-type conditions th a t arose because of K ohn’s work on subelliptic estimates. We also introduce a class of weakly pseudo­ convex domains for which it is possible to understand K ohn’s ideals of subelliptic m ultipliers in a rather concrete and precise manner. Although these domains are rather special, for them it is possible to see clearly the relationship between analysis and com m utative algebra anticipated by K ohn’s work. We discuss the geometry of the boundary of a dom ain in complex Eu­ clidean space and its relationship to the function theory on the domain. There are many relevant geometric conditions; here we emphasize those finite-type conditions th a t arose (directly or indirectly) as consequences of the work of Kohn. These notions apply when one considers subelliptic estim ates for the 3-Neumann problem; the author believes th a t they will be useful in m any other analytic problems. We consider several different generalizations of strong pseudoconvexity for a point on a real hypersur­ face. The language of com m utative algebra unifies these ideas. The only new result in this paper is Proposition 5, where we apply K ohn’s algorithm for finding subelliptic multipliers to an interesting class of domains. We call these domains Tegular coordinate dom ains’ because of an analogy with the use of regular systems of coordinates for ideals of germs of holomorphic functions. For these domains we compute the Kohn ideals of subelliptic m ultipliers, even though the Levi form is not generally diagonalizable. We obtain a value for the param eter epsilon in such estim ates in term s of the given exponents. This paper is not intended to be complete. Our discussion of subel­ liptic estim ates is restricted to (0, l)-form s. We do not consider im por­ tan t recent results about estim ates in function spaces other than L 2 and Sobolev spaces. Most im portant there are m any theorems known for strongly pseudoconvex domains th at generalize to appropriate classes of weakly pseudoconvex domains. The final forms of such theorems will involve different finite-type conditions. Thus there is no single notion of finite-type applicable to all problems. The author acknowledges Dave C atlin for helpful criticisms of a prelim inary version of this article.

64

John P. D ’Angelo

1. T he Levi form We begin by considering a dom ain Q C Cn whose boundary is a sm ooth real hypersurface M . The complexified tangent bundle CT M contains an integrable subbundle T 1,0M whose local sections are complex vector fields of type (1,0). We denote by rj a purely im aginary non-vanishing 1-form th at annihilates both T 1,0M and its conjugate bundle. In term s of a local defining function r for M , we may put rj = ^ (d — d) r. The Levi form A is the Herm itian form on T 1,0M defined (up to a multiple) by _ A (L,K) = (r,,\L,K ]). (1) The hypersurface (or the domain on one side of it) is called strongly pseudoconvex when A is positive semi-definite, and is called weakly pseudoconvex when A is semi-definite but not definite. We note im mediately th a t strong pseudoconvexity is a non-degeneracy condition: if A is pos­ itive definite at a point p E M , then it is positive definite in a neigh­ borhood. Furtherm ore strong pseudoconvexity is ‘finitely determ ined’: if M ' is another hypersurface containing p, and M ' osculates M to second order there, then M ' is also strongly pseudoconvex at p . We seek gen­ eralizations of strong pseudoconvexity th at have applications in analytic problems; we also want these two properties to remain valid. For later use we compute the Levi form for domains in Cn+1 defined locally by the equation N

r{z,~z) = 2E e (z 0) + ^

| / fc (z)|2 < 0.

(2)

k —1

Here the functions /& are holomorphic near the origin, vanish there, and depend only on the variables zi, 22) •••> The domain defined by (2) is pseudoconvex. Its Levi form near the origin has a nice expression:

(A«) =

= ( 5/)* (5/)-

(3)

It follows immediately from (3) th at the origin will be a weakly pseudo­ convex point if and only if the rank of (d f ) (as a m apping on Cn) is less th an full there. It is a point of finite-type in the sense of this paper if and only if the germs of the functions fk define a trivial variety. See the last paragraph of Section 2. This simple example allows us to glimpse the role of com m utative algebra in later discussions.

2. F in ite ty p e conditions and su b ellip tic estim ates

Finite T ype Conditions and Subelliptic Estim ates

65

The study of the inhomogeneous Cauchy-Riemann equations motivates much of this work. Suppose th at Fl CC Cn is sm oothly bounded and pseudoconvex. Let a be a differential (O-l)-form a with square integrable coefficients th a t satisfies d a = 0. We wish to solve the inhomogeneous Cauchy-Riemann equations du — a and discuss the regularity of the solu­ tion. See [K1-K5]. K ohn’s solution of the 9-Neumann problem constructs the particular solution u — d N a th at is orthogonal to the holomorphic functions. Suppose th at the d-Neum ann operator is pseudo-local; th at is, N a m ust be sm ooth wherever a is smooth. Then u — d N a is also sm ooth wherever a is. The pseudolocality property of N follows from certain a priori estim ates called subelliptic estimates. The finite-type condition emphasized here arises from K ohn’s program of seeking neces­ sary and sufficient conditions for subelliptic estimates. D e fin itio n 1. Suppose th at Q CC Cn is sm oothly bounded and pseu­ doconvex. Let p E Ft be any point in the closure of the domain. The d-Neumann problem satisfies a subelliptic estim ate at p on (0, l)-form s if there is a neighborhood U 9 p and positive constants C, e such th at

IIMIIe < c ( | | ^ | | 2 + | | r ^ | | 2 + | H | 2)

(4)

for every (0, l)-form th at is smooth, compactly supported in [/, and in the domain of the operators on the right hand side. In (4) ||||||e denotes the tangential Sobolev norm of order e. Also, the statem ent th at be in the dom ain of d is a boundary condition on its components. We abbreviate the right side by Q (0, 0). A basic theorem of Kohn-Nirenberg [KN] implies th at pseudolocality for A on (0, l)-form s follows from the estim ate (4). In fact, this estim ate has, in term s of local Sobolev norms, the following consequences: a E H s => N a E H s+2e] a e H s = > d * N a e H s+e.

(5)

Therefore it is significant to determine when there is a subelliptic estim ate at a given point p E Ft. Observe th at the set of points for which an estim ate of the form (4) holds m ust be an open subset of the closed domain. For interior points, the estim ate (4) is elliptic, and holds with 6 = 1. At strongly pseudoconvex-boundary points, the estim ate holds for e = C atlin has found necessary and sufficient conditions for a subelliptic estim ate of some order to hold. See Theorem 4. The precise value of the param eter e is not known in general. To gain some feeling for the estim ates, we recall the basic formula for Q (,) on (0, l)-form s. We assume th a t r is a defining function for bFl.

66

John P. D ’Angelo

L e m m a 1 . The quadratic form Q satisfies

Q(4>,4>) =

E / \(^h Sdv+ E / — II^ IIt+

f ^{A) dS +

+ /X >‘

2 dV

n *=1

IH I2 •

(6)

bn This formula reveals an asym m etry between the barred and unbarred derivatives. Observe also th a t the integral of the Levi form appears. This term is non-negative when Cl is pseudoconvex. Since an estim ate holds at strongly pseudoconvex points, and no estim ate holds at points where the Levi form vanishes identically, we see th at the existence of an estim ate is a non-degeneracy condition of the type we seek. We consider the case of (0, l)-form s in this paper and recall some of the history. The first result was in two dimensions. Let us say th at a (1.0) vector field L on a real hypersurface M is of finite-type at p if there is an integer k such th at (rp [... [L\ , L 2] , Lk])p / 0. In this definition each L j equals either L or L . The type of L at p is the smallest such integer. It is evident th at the type of every non-vanishing (1,0) vector field at p equals two precisely when M is strongly pseudoconvex there. For hypersurfaces in C 2, Tp1,0M is one dimensional; it follows easily th at the type of each non-zero (1,0) vector field L at p is the same. Kohn proved in [K3], a paper dedicated to Spencer on the occasion of his 60th birthday, the sufficiency condition in the following result. Greiner [Gr] established the converse. The precise relationship between the type of the vector field and the value of epsilon is due to Rothschild-Stein. [RS]. T h e o r e m 1 . Suppose that Q CC C 2 is pseudoconvex, and that p £ M — bQ. Let L he a non-vanishing (1,0) vector field at p. I f L is of finite-type m at p, then there is a subelliptic estimate. The estimate holds for e = but for no larger value. Conversely, if there is a subelliptic estimate for some e, then every such vector field is of finite-type. Kohn then conjectured th at a necessary and sufficient condition for subellipticity on pseudoconvex domains in higher dimensions would be th at every (1,0) vector field is of finite-type. This turns out to be false, because this latter condition is not an open condition. The following simple example reveals this: E x a m p le . The polynomial r(z,~z) — 2Re(zs) + defines a pseudoconvex hypersurface containing the origin, and each non-vanishing (1.0) vector field is of type 4 or 6 there. Let Lj — —rj for j = 1,2 be the usual commuting basis for (1,0) vector fields. Let V be the complex analytic 1-dimensional variety given by z \ — z \ and Z3 = 0. The vector 3z ^

field L = 2^-L i + L 2 is then sm ooth along V , but it cannot be smoothly

Finite Type Conditions and Subelliptic Estim ates

67

extended to a neighborhood of the origin. This difficulty occurs because V is not a norm al variety. We cannot detect the existence of this complexanalytic curve by studying the types of smooth vector fields. From this example the author realized th at the finite-type condition should consider the contact with all holomorphic curves, including those with singularities. This led to the following definition of point of finitetype. [D2] We discuss also a recent refinement of this idea. Let M be the germ at p of a real hypersurface in C n , and let r be a defining function. (The resulting notion will not depend on the choice of defining function). We denote by vp (g) = v (g) the order of vanishing of the function g —g (p) at the point p . Consider the collection of non-constant germs of holomorphic m appings z : (C ,0) —>■ (C n ,p). We say th at M is of finite-type at p if there is a constant T so th at v (z*r) < T v (z) for every such holomorphic mapping. The infimum of all such constants is called the type of M at p, and is w ritten A 1 (M ,p). Observe th at A 1 (M, p) — sup^

).

This num ber is therefore a

measurem ent of the maxim um order of contact of complex analytic one­ dimensional curves with M at p. More generally, if J is an ideal in some local ring at p, we define T (J) = sup^ inf^

• Here the infimum is

taken over the local ring. Thus T (J) = A 1 (M,p) when J is the ideal in the ring of germs of sm ooth functions at p of functions vanishing on M . One of the m ain points in [D2] is to relate this num ber to T (I) where I is an ideal in the ring O of germs of holomorphic functions. We now give a refined equivalent definition of point of finite-type. This uses the idea of scheme, but no results from the theory of schemes. The idea is the following. For each integer k , we consider the hypersurface Mk defined by the Ar-th order Taylor polynomial j k iPr. Thus Mk osculates M at p to or­ der k. We consider jk,Pr as an H erm itian form on the finite-dimensional vector space of holomorphic polynomials of degree at m ost k. After diagonalizing this form, we can write jk,pr = ||-^p||2 — 1 1 1 1 2, where F p }GP are holomorphic vector-valued polynomials. This construction leads to a family of ideals of holomorphic polynomials, generated by the compo­ nents of F p —UGP, where U is unitary. We call this family I (U, A, p ). The point of this construction is th at the complex-analytic varieties defined by V (I (17, Ar,p )) lie in Mk, and conversely, any irreducible complex-analytic variety lying in Mk m ust be a subvariety of one of these. Thus the order of contact of complex-analytic varieties with M can be completely ana­ lyzed by letting the degree of osculation tend to infinity. For each k we have the estim ate

supuT (I (U, k,p)) + 1 < A 1 (Mk,p) < 2suPuT (I ( U , k , p ) ) .

(7)

68

John P. D ’Angelo

For pseudoconvex Mk we have an equality A 1 (Mfc, p) = 2suP u T (I ( U , k , p ) ) .

(8)

The construction assigns a family of ideals of germs of holomorphic func­ tions to each point on a sm ooth hypersurface. The ideals depend on the degree of osculation and on a unitary m atrix. The varieties of the ideals have high order of contact with the hypersurface. As in the theory of schemes, the ideals themselves are more im portant than the sets they define. Even when all the varieties are trivial, essentially the finite-type case, we obtain numerical inform ation on the geometry of the hyper­ surface by com puting numerical invariants of the ideals. In particular the codimension D (I) = dim c ( O / I ) is particularly useful. The author used its upper semicontinuity properties and its relationship with T (I) to establish the following theorem. T h e o r e m 2 . Let M be a smooth real hypersurface in C n . Suppose that po £ M and that A 1 (M ,po) < oo. Then there is a neighborhood of po on which A 1 (M,p) < 2 (A 1 (M ,p 0))n_1. (9) I f M is pseudoconvex, then we can improve (9) to the following sharp bound:

(10) Furthermore, there is an integer ko that ‘f initely determines’ the type. That is, whenever k > ko, A 1 (M,po) = A 1 (Mk,po); also V (I (U, k,po)) = {po} for every unitary U , and in fact 2su p u T (I (U, k,po)) < k . This theorem establishes the two fundam ental non-degeneracy proper­ ties of finite-type. Observe th at the ideals I (U,k,p) all equal the m axim al ideal in the strongly pseudoconvex case. The generalization to points of finite-type parallels the generalization from the m axim al ideal to ideals prim ary to it. In case the hypersurface is real-analytic, we do not need to osculate the hypersurface by algebraic hypersurfaces. We can define the ideals I (U,p) directly. Finite type for a real analytic hypersurface M is then equivalent to the statem ent th a t there is no complex-analytic variety of positive dimension passing through p and lying in M . This statem ent is also equivalent to the statem ent th a t there is a constant k such th at D ( /( C /,p ) ) < k VC/. See [Dl] for these results. For domains defined by (2) we do not need to consider unitary matrices. For such domains the origin is a point of finite-type if and only if the functions fk together define a trivial variety.

3. T he M eth od s of K ohn and C atlin

Finite Type Conditions and Subelliptic Estim ates

69

We return to the discussion of subelliptic estimates. We assume th a t Q is a sm oothly bounded pseudoconvex domain. In [K5] Kohn first developed the m ethod of subelliptic multipliers. Let x be a boundary point of Q. The idea is to consider the set of all germs of functions / such th at d

(11)

+

Here both constants may depend on / . Let Jx denote the collection of all such germs at x\ its elements are called subelliptic m ultipliers. It is easy to see th a t the defining equation is a subelliptic m ultiplier, with 6 = 1 , essentially because the estim ate is elliptic in the interior. It is considerably harder to prove th at the determ inant of the Levi form is also a m ultiplier, with e = From this starting point, Kohn gives an algorithmic procedure for constructing new multipliers, for which the value of epsilon is typically smaller. [K5]. We summarize the procedure after stating some properties of the multipliers. P r o p o s itio n 1 . Let x be a boundary point of the psuedoconvex domain Q. Then the collection of subelliptic multipliers Jx on (0,1 )-forms is a radical ideal. In particular, g E J, \f\N < |tf|

=> / G J.

(12)

We also have the estimate

WMWl will be chosen according to the needs of the problem. After this choice is properly made, one employs,

Finite T ype Conditions and Subelliptic Estim ates

71

as a substitute for Lemma 1, the inequality

/

n

n

E

t,j= i

n

^ z . l j d i a j d V + ^ 2 ||L ^afc[|2 < C Q ( a , a )

(17)

j,fe=i

where |$ | < 1. Here Lj are vector fields of type (0,1) on Cn . There could be also a term on the left side involving the boundary integral of the Levi form, but such a term does not need to be used in this approach to the estim ates. Instead, one needs to choose $ with a large Hessian. One step in C atlin 5s proof is the following reduction: Theorem 5. Suppose that Q CC Cn is a pseudoconvex domain defined by Q = {r < 0}, and that p £ bf2. Let U be a neighborhood of p. Sup­ pose finally that for all S > 0 there is a smooth real-valued function satisfying the following properties: | $o on u E «*V > C^ T on U n { - S < r < 0}. (18) *\j = 1 Then there is a subelliptic estimate of order e at p. Theorem 5 reduces the problem to constructing such bounded sm ooth plurisubharm onic functions whose Hessians are at least as large as S~2e. C atlin accomplishes this by a considerable refinement of the ideas in [D2], [DF], and [K5]. We briefly discuss the necessity result. An early example of the au­ thor showed th a t one cannot in general choose epsilon as large as the reciprocal of the order of contact. [D5,D1]. The result is very simple. The function p —> A 1 (bQ,p) is not in general upper semicontinuous, so its reciprocal is not lower semicontinuous. Definition 1 reveals th at, if there is a subelliptic estim ate of order epsilon at one point, then there also is one at nearby points. C atlin has extended this example to show th at the param eter value cannot be determ ined by inform ation based at one point alone. [Cl]. Nevertheless Theorem 2 on openness shows th at the condition of finite-type does propagate to nearby points. This sugyn —2 gests th at one can always choose epsilon as large as e = (Ai(bn p))™-1 * ^ more precise conjecture is th a t we may always choose epsilon as large as e= p) • T k e denom inator is the 'm ultiplicity5 of the point, defined by the author to be B (M, p) = lim supfc_).002 su p ^D (I (U, k,p)) . The function p —> B (M, p) is upper semicontinuous.

(19)

72

John P. D ’Angelo

4. R egular coordinate dom ains We continue this paper by discussing the best value of epsilon th a t we can get from the m ethods of Kohn, for a class of domains we call Tegular coordinate dom ains’. We use this term because the Weierstrass poly­ nomials involved exhibit the usual coordinates as a regular system of coordinates for an ideal. [Gu]. There are several reasons for considering the Kohn ideals of subelliptic m ultipliers. One is th a t they can be defined on CR manifolds w ithout reference to an embedding into C n . This has applications to the boundary operator Furtherm ore it remains an open problem whether, in the sm ooth case, K ohn’s condition on subellip­ tic multipliers is equivalent to finite-type. By understanding subelliptic multipliers better, even in the very special case considered here, one hopes to obtain the deeper understanding required to solve this problem. P ut n

r ( z , z ) = 2 Re ( z 0) + ^ 2 \ f j (z)\2 3-

(20)

1

where z = (zo, Z i,..., zn) and the f j are Weierstrass polynomials of the following form:

f 2 (z) = z ? ’ + . . . . /3 (z) = z™3 + • • • .

(21)

More precisely,each f j is a Weierstrass polynomial in zj whose coeffi­ cients depend only on (zi, ...,Z j_i). We assume th at rrij > 2for all j. Our earlier com putation shows th at the Levi form is the m atrix product (d f )* (Of). The especially nice property following from (21) is th at the m atrix ( df ) is lower triangular, so it is easy to compute determ inants. We assume th at Q is a sm oothly bounded pseudoconvex dom ain for which (20) is a defining equation in some neighborhood of the origin. The origin is a point of finite-type. Although the num ber A 1 (M ,p) cannot be com puted from the given inform ation alone, we can compute the m ultiplicity easily. From standard com m utative algebra [Dl] it follows n

th a t B (M, 0) = 2 Yi m j ' By work °f fhe author, we have (in dimension 3-1

n + 1) th at A 1 (M, 0) < B (M, 0) < ^A l ^

’1° )^ ■

(22)

As a consequence of (22) the type satisfies

(

n

\ n

]q m d

j =i

/

n

< a 1 (M, 0) < 2 J J mj 3=i

(23)

Finite T ype Conditions and Subelliptic Estim ates

73

for these domains; thus the origin is a point of finite-type, but we can­ not compute the m axim um order of contact from the inform ation given. Simple examples show th at each extreme equality in (23) is possible. The following lem m a is not stated in [K5], but follows from the work there; see the section in [K5] called ‘some special dom ains’. L e m m a 2 . (Holomorphic version of Proposition 3). Suppose that (20) is the defining equation near the origin for a pseudoconvex domain. Then the rows of (Of) are allowable rows; we may take e = - in (16). K ohn’s algorithm thus reduces to the situation where the starting collection of allowable rows is given by {9 /j} , and we m ay stay within the category of holomorphic functions. This inform ation could be useful for general holomorphic functions; for m aps defined by (2 1 ) things are much simpler yet. By following K ohn’s procedure we determ ine an epsilon for which a subelliptic estim ate holds at the origin. The result is not best possible. The author and C atlin believe th at the m axim um value of epsilon in this case is equal (generically) to 2 j-jm . In simple cases it is larger. One interesting aspect of this example is th at calculation of the value of epsilon is independent of everything except the exponents. Although the order of contact depends on the rest of the term s in the Weierstrass polynomials, our calculations do not. In order to facilitate the understanding of the com putation, we first perform it in the simplest cases. We suppose th at (20) is the defining equation, and we assume th at mj > 2. The simplest case is when n = 1. P ut m = m i. Then z rf l~ 1 is a subelliptic m ultiplier with e = \ according to Lemma 2. Applying Proposition 1 we see th at z\ is a m ultiplier with e= I Applying Proposition 2 to differentiate and then Proposition 4, reveals th at 1 is a subelliptic m ultiplier with

The case in more variables is considerably more difficult. To simplify notation we put E j — L. We write D E j to denote and D k E j for the derivative of order k. For these domains we never need to know the derivatives of E j with respect to any other variables. The determ inant of the Levi form is the product E i E 2 . - . E n . The technique of proof involves “peeling off” the E k until we obtain E \ as a subelliptic multiplier. Since this is a (constant times) a power of a coordinate function, we then take radicals to discover th at z\ is a m ultiplier, and hence th a t (1 , 0, • • •, 0) is an allowable row. Working modulo the first variable, we obtain a similar problem in fewer variables. We begin with the case n — 2. The starting point is then th at E 1 E 2 is a subelliptic m ultiplier. Its gradient is an allowable row. Taking de­ term inants again reveals th a t E \ D E 2 is a multiplier. We repeat this

74

John P. D ’Angelo

process m 2 — 1 times, obtaining E™2 as a m ultiplier, because D rri2~ 1E 2 is a non-zero constant. Up to this point we have m ultiplied the originial epsilon by 2“ (m2_1). Taking radicals shows th at E \ is a m ultiplier and m ultiplies the epsilon by a factor of Therefore z™1 - 1 is a m ultiplier with the original epsilon now muliplied by ^ 2 ~ ( rri2~1\ Taking radicals again shows th at z\ is a m ultiplier, and applying Proposition 2 reveals th at (1,0) is an allowable row. These last two steps m ultiply epsilon by mi1_ 1 Since z\ is a multiplier, and the set of multipliers is an ideal, we m ay now set z\ equal to zero. Using the same notation we see th at E 2 is now (a constant times) z^™2 1\ Hence, we take radicals and one gradient to obtain the allowable row (0,1). The final value of epsilon is ± — !— i — !— 2 m 2 —1 2 m i —1 m 2 2

(25)

Now we set up some notation to handle any num ber of dimensions. For k = n + 1, n, n — 1,..., 2 we define Au by A n+1 = 1 , A n = ^ mn , and 1 1 mk —1 ^ = — (26) m/e z9 The idea is th at the process of peeling off the last factor from the product E i E 2 ---Ek multiplies epsilon by the value Ak- We begin with E \ E 2 ^-En. As in the case n — 2 we can “peel off” E n to see th at E \ E 2 ...En- i is a m ultiplier. Now we use its gradient in the penultim ate row and take determ inants to obtain E i . . . E n- 2D E n - \ E n as a m ultiplier. Again “peel off” the E n . Use the gradient again in the penultim ate row. Take determ inants. Apply this process until E \ . . . E n - 2 is a m ultiplier. Now use its gradient in row n — 2. Eventually we “peel off” everything but E\ . As above we take radicals and gradients to obtain th a t (1,0, ...,0) is an allowable row. Working modulo z\ we obtain a similar situation in n — 1 variables. We obtain the following result. P ro p o sitio n 5. Let Ll be a pseudoconvex domain for which (20) is a defining equation in some neighborhood of the origin. Then the n

3=1

^

1

3

n

3=2

The reader should observe using (26) how complicated the term is.

5. O ther uses of fin ite-typ e conditions

Aj

Finite T ype Conditions and Subelliptic Estim ates

75

Other finite-type conditions arise in various analytic problems. We con­ sider but a few of these. Many of them involve preventing complex ana­ lytic objects of various dimensions from osculating the boundary to high order. In particular one can define numbers A g (M , p ) , A-?eg (M , p ) that measure the contact of (/-dimensional complex-analytic varieties or m an­ ifolds with a hypersurface. [D4,D1]. Related numbers arise in C atlin’s proof of subelliptic estim ates for (0, 0M and T 0,1M are tangent to N. Finally we consider briefly another finiteness condition th at involves multiplicities of holomorphic ideals. Suppose th at / : (M,p) is a sm ooth CR diffeomorphism between (germs of) real-analytic real hy­ persurfaces. A regularity theorem of Baouendi-Jacobowitz-Treves [BJT] states th at such a m apping m ust be itself real analytic in case the germ (M, p) is ‘essentially finite’ This finiteness condition is th at a certain ideal of holomorphic functions constructed from the defining function m ust de­ fine a trivial variety. Assume th at p is the origin and write the power series defining the target hypersurface germ as r(z,z) =

= Re ^ c*oZa + ^ 2 ^ 2 ci l/?l>iM>i = Re (h (2)) + ^ hp (z) J & + Im (h (z)) 3> (z, z) . (28) P Here the second term in the last line is independent of h, the functions hp are holomorphic, and $ is defined by the equation. The germ (M, 0) is essentially finite if the ideal {h,hp )

(29)

(including all the hp) in O defines the trivial variety consisting of the origin alone. Note th at we may always take h to be a coordinate func­ tion. Any numerical invariant such as D ,T of a holomorphic ideal can be applied to give a m easurement of the extent of ‘essential finiteness’. Baouendi-Rothschild [BR1,BR2] have given many uses of this concept and its generalization to formal power series germs. See [D1,D7] for the relationship between these ideals and the ideals used to consider points of finite-type. The author finds it amazing th at two deep theorems, subellipticity for the 9-Neum ann problem and the real analyticity of CR diffeomorphisms, each involve checking whether certain ideals of germs of holomorphic func­ tions define trivial varieties. This conclusion evinces a deep connection

Finite Type Conditions and Subelliptic Estim ates

77

between partial differential equations and com m utative algebra th a t was anticipated by the work of Kohn.

R eferences [BR1] Baouendi, M.S. and Rothschild, L.P., Germs of CR m aps between real analytic hypersurfaces, Inventiones M athem aticae 93(1988), 481-500. [BR2]____ , Geometric properties of mappings between hypersurfaces in complex space, Journal of Differential Geometry 31(1990), 473-499. [BR3]____ , M inimality and the extension of functions from generic m an­ ifolds, Pp. 1-13 in Proceedings of Symposia in Pure M athem atics, Vol. 52, P art 3, Several Complex Variables and Complex Geometry, (1991) American M ath. Society, Providence, Rhode Island. [BR4]____ , Cauchy Riemann functions on manifolds of higher codimen­ sion in complex space, Inventiones M athem aticae 101(1990), 45-56. [BJT] Baouendi, M.S., Jacobowitz, H., and Treves, F., On the analyticity of CR mappings, Annals of M ath. 122(1985), 365-400. [BT] Baouendi, M. S. and Treves, F. , About the holomorphic extension of CR functions on real hypersurfaces in complex space, Duke M ath J. 51(1984), 77-107. [Cl] Catlin, D., Necessary conditions for subellipticity of the 5-Neumann problem, Annals of M ath 117(1983), 147-171. [C2]____ , Boundary invariants of pseudoconvex domains, Annals of M ath. 120(1984), 529-586. [C3] , Subelliptic estim ates for the 3-Neum ann problem on pseudo­ convex domains, Annals of M ath 126(1987), 131-191. [DI] D ’Angelo, J. P., Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, 1992. [D2] , Real hyper surf aces, orders of contact, and applications, Annals of M ath 115(1982), 615-637. [D3] , Intersection theory and the 5-Neumann problem, Proc. of Symposia in Pure M ath. 41(1984), 51-58. [D4] , Finite type conditions for real hypersurfaces, in Lecture Notes in M ath. 1268, Springer, Berlin, 1987. [D5]_____, Subelliptic estim ates and failure of semi-continuity for orders of contact, Duke M ath. J., Vol. 47(1980), 955-957. [D6] ____ , Finite type and the intersections of real and complex vari­ eties, Pp. 103-117 in Proceedings of Symposia in Pure M athem atics, Vol. 52, P art 3, Several Complex Variables and Complex Geometry, (1991) American M ath. Society, Providence, Rhode Island. [D7] , The notion of formal essential finiteness for sm ooth real hy­ persurfaces, Indiana Univ. M ath. J. 36(1987), 897-903.

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[DF] Diederich K., and Fornaess, J.E ., Pseudoconvex domains with real analytic boundary, Annals of M ath (2) 107(1978), 371-384. [Gr] Greiner, P., Subellipticity estimates of the 3-Neum ann problem, J. Differential Geometry 9 (1974), 239-260. [Gu] Gunning, R.C., Lectures on complex analytic varieties: the local param eterization theorem, Princeton Univ. Press, Princeton, New Jersey, 1970. [H] Hormander, L. An introduction to complex analysis in several vari­ ables, Van Nostrand, Princeton 1966. [Kl] Kohn, J., Harmonic integrals on strongly pseudoconvex manifolds I,II, Annals of M ath (2) (1963), 112-148; ibid 79 (1964), 450-472. [K2]____ , Boundary behavior of d on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom., Vol 6 (1972), 523-542. [K3] ____ , Global regularity for d on weakly pseudoconvex manifolds, Transactions of the American M ath Society 181(1973), 273-292. [K 4] , A survey of the 3-Neum ann problem, Proceedings Symposia Pure M ath, No. 41, 137-145, Providence, RI, 1984. [K5] ____, Subellipticity of the 3-Neum ann problem on pseudoconvex domains: Sufficient conditions, Acta M ath. 142(1979), 79-122. [KN] Kohn, J. and Nirenberg, L. Non-coercive boundary value problems, Comm. Pure Appl. M ath 18(1965), 443-492. [RS] Rothschild, L. P. , and Stein, E. M. , Hypoelliptic operators and nilpotent Lie groups, Acta M ath. 137(1976), 247-320. [T] Trepreau, J. M., Sur le prolongement holomorphe des fonctions CR definies sur une hypersurface reele de classe C2 dans C n , Inventiones M athem aticae 83(1986), 583-592. [Tu] Tumanov, A. E. , Extension of CR-Functions into a wedge,M ath. USSR Sbornik Vol. 70(1991), No. 2, 385- 398. D epartm ent of M athem atics, University of Illinois, Urbana, IL 61801

C H A R A C T ER IZA T IO N OF CERTAIN H O LO M O RPH IC G EO DESIC CYCLES O N H E R M IT IA N LOCALLY SY M M E T R IC M A N IFO LDS OF TH E N O N C O M P A C T T Y P E P h ilippe E y ssid ie u x a n d N g a im in g M ok

C ontents Introduction 1.

Local approxim ation of almost-geodesic complex submanifolds on bounded sym m etric domains

2.

The gap phenomenon on a product sym m etric dom ain as a conse­ quence of the uniqueness of Kahler-Einstein metrics

3.

The exponential sequence and a gap phenomenon on the Siegel upper half-plane

4.

An optim al Arakelov inequality on Chern numbers for compact complex surfaces of quotients of B 2 x B 2

5.

C haracterization of certain holomorphic geodesic cycles on quotients of B 2 x B 2 modelled on ( B 2 x B 2, B 2;S) References

Introduction In the study of quotients of bounded sym m etric domains a m ain them e is th a t of rigidity. In this article we are interested in compact complex sub­ manifolds of quotients of bounded sym metric domains. In this respect the compact holomorphic tot ally-geodesic cycles, which are themselves quotients of bounded sym m etric domains, are rigid except for trivial de­ form ations (as can be obtained in the case of product domains). Here we explore the question of characterizing compact holomorphic tot allygeodesic cycles of quotients 0,/T of bounded sym m etric domains by a dis­ crete properly discontinuous subgroup T of Aut(Q) and study a stronger

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rigidity phenomenon for such geodesic cycles, which we term the gap phenomenon. Our point of departure is the study of sufficiently pinched compact complex submanifolds of ft/r. Given a bounded symmetric dom ain ft, there are only a finite number of isomorphism classes of totally-geodesic complex submanifolds E, which are themselves bounded sym m etric do­ ft/r be an immersed compact complex submanifold such mains. Let S th at the second fundam ental form is everywhere of norm < e. By an appli­ cation of Bishop’s Theorem of subconvergence of complex-analytic subva­ rieties, we show th at when the pinching constant e is sufficiently small, S can be locally approxim ated by a unique isomorphism class of of totallygeodesic complex submanifold of ft, represented by some z : E ^ ft. We will say th a t S is modelled on (ft, E; i) or simply on (ft, E). We say th a t the gap phenomenon holds for (ft, E) if there exists an e such th at for every discrete subgroup T C A ut(ft) acting without fixed points, any compact e-pinched complex submanifold S ^ ft/r modelled on (ft, E) is necessarily totally geodesic. In this regard Mikhail Gromov has suggested th a t a sufficiently pinched compact complex submanifold of ft/r should be totally-geodesic. This am ounts to saying th at the gap phenomenon holds for every bounded sym metric domain ft and every totally-geodesic bounded symmetric subdom ain E C ft. While we do not have supporting evidence th a t the gap phenomenon holds for all (ft, E), our investigation confirms th at it indeed holds in certain special cases. The significance of the gap phenomenon is th at, as opposed to most pinching theorems in Riem annian geometry, we do not impose any bound on the volume of the submanifold S. As a m atter of fact, if we impose an upper bound on the volume of 5, the gap phenomenon is an easy consequence of the rigidity of compact totally-geodesic holomorphic cycles (modulo trivial deformations), at least if we fix T and assume th a t ft/r is compact (c/. (1.5)). From this point of view the gap phenomenon may be regarded as a strengthened version of rigidity of compact totally-geodesic holomorphic cycles on ft/r. A second m otivation for our investigation is to find optimal inequal­ ities on Chern numbers for S ^ ft/r. Here an inequality on Chern numbers is said to be optim al if and only if equality is attained when S is itself totally geodesic and modelled on some (ft, E) for some bounded sym m etric subdom ain E. From this perspective there is first of all an inequality on Chern numbers, due to Arakelov, for the special case of compact holomorphic curves in quotients of the Siegel upper half-plane, where equality is attained precisely for compact totally-geodesic holomor­ phic curves of m axim al Gauss curvature. We are interested to generalize such inequalities on Chern numbers for compact complex submanifolds on bounded symmetric domains.

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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The two perspectives are related to each other in the proof of the following pinching theorem for the Siegel upper half-plane 7in • T h e o re m . (Mok[Mok2]) Normalize the canonical m etric on Tin so th at the m axim um holomorphic sectional curvature is -1. Let C c—>>X — l~Ln / T be an immersed compact holomorphic curve on X verifying the curvature condition —( 1 H

V

4 n)

I < Gauss curvatures of C < —1

Then, C X is totally geodesic in X . The Siegel upper half-plane is here interpreted as the param eter space of polarized abelian varieties. As a consequence, the curve C can be in­ terpreted as the base space of a variation of polarized Hodge structures of weight one. The immersion C X gives rise to a linear represen­ tation $ : 7Ti(C) —» Ki (X) S p (n ,R ). Let L r be the associated real 2n-dimensional locally constant bundle over C and L c = C be its complexification. The variation of polarized Hodge structures leads to a theory of harmonic forms and to Hodge decomposition. In Mok [Mok2] the proof was geometric and uses the Gauss-Bonnet formula. Here we give a variation of the proof in term s of harmonic forms. The proof consists of of an analytic and a topological part. The analytic part con­ sists of a vanishing theorem arising from a Bochner formula. We prove th at for a curve satisfying the pinching hypothesis of the theorem, a certain component of the Hodge decomposition of H 1 (C, L c) is always zero. The topological part is on the other hand a non-vanishing theo­ rem, which asserts th at for any immersed compact curve C X , th at particular component (which is H 1 (C, L c )1,1 in standard notations) is non-zero unless C is totally-geodesic and of m axim um Gaussian curva­ ture. The topological part comes from a calculation of an Euler-Poincare characteristic. Let V denote the universal seminegative holomorphic vec­ tor bundle over C . The non-vanishing theorem follows from Arakelov’s inequality —deg(F) < n(g(C) — 1), where g(C) denotes the genus of the compact curve C, and the assertion th at equality is attained if and only if C is totally-geodesic and of m axim um Gauss curvature. The Siegel upper half-plane %n consists of n-by-n symmetric matrices with positivedefinite im gainary part. The subdom ain of diagonal m atrices is an n-fold Cartesian product %n of the upper half-plane H. The pinching theorem above then verifies the gap phenomenon for i) where i : H Hn is the composition of the diagonal embedding S : H %n with the inclusion %n C TinThis new form ulation of the proof of the pinching theorem above opens up an approach to verify the gap phenomenon. According to an idea of Pierre Deligne’s we interpret bounded sym m etric domains Q as param eter

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space for polarized Hodge structures with some additional structure on an underlying real vector space W . More precisely, let p be a representa­ tion of Aut(Q) in the linear group of a real finite-dimensional vector space W . The flat vector bundle Q x W on Q is endowed with a com patible action of Aut(Q) by the formula g • (#, w) = (g • x, p(g) • w). Zucker [Zu] constructs Aut(f2)- equivariant Hodge structures on this flat vector bun­ dle, which form the basis of our interpretation. This flat vector bundle can be descended to 0, /T and pulled back to any m-dimensional com­ pact complex submanifold S of X = Pt/T. Denote by W c the resulting complex local system on S. Following Gromov [Gro2] by proving asym p­ totic vanishing theorems for H r (S, W c ) for r / m, S not necessarily pinched, with respect to a tower of unramified coverings Sk, we conclude th at the asym ptotic dimension of each component H m (Sk> W c ) p,q, nor­ malized by Volume(Sfc), is com putable in term s of Euler-Poincare char­ acteristics of holomorphic vector bundles over S. We obtain in particular a universal inequality vPfq(S) > 0 where vp>q(S) is up to sign the EulerPoincare characteristic pertaining to the particular Hodge component H m ( Sk ) W c ) p,q- We will call such an inequality on Chern numbers an Arakelov inequality. In certain special situations there exist Hodge com­ ponents for which v ( S 0) = 0 (v meaning vP}q) when S 0 is to t ally-geodesic, modelled on (fl, E), say. We will then call the inequality v(S) > 0 an op­ tim al Arakelov inequality on Chern numbers for the pair (Q, E). Given an optim al Arakelov inequality v(S) > 0 we can look for Bochner-Kodaira formulas to explain the vanishing of v(S). Such Bochner-Kodaira for­ mulas can then be used to prove the vanishing v(S) = 0 for sufficiently pinched compact holomorphic cycles S modelled on (Q ,E ). Finally to verify the gap phenomenon for (0 ,E ) it is sufficient to prove the topo­ logical non-vanishing theorem J'(S') > 0 for all compact S which are not totally geodesic and modelled on (Q ,£ ). By the Chern-Weil theory of characteristic forms, v(S) is represented by an integral of an (m, m)-form vs- To prove the nonvanishing theorem desired it suffices to show th a t the inequality v(S) > 0 is local in the sense th at vs > 0 for all germs S of complex submanifolds and th at vs = 0 implies th a t S is totally geodesic. As in the case of curves on quotients of the Siegel upper half-plane, this will lead to a verification of the gap phenomenon for (Q ,E ). We have thus reduced the gap phenomenon to finding local optim al Arakelov in­ equalities with accompanying Bochner-Kodaira formulas. In this context it is also natural to formulate the gap phenomenon for period domains of Griffiths as param eter spaces for polarized Hodge structures. The gap phenomenon is then defined for (T ),E ;i), where V is a period do­ m ain of Griffiths, E is a bounded symmetric domain, and i : E ^ V is a totally-geodesic period m ap (where V is equipped with a canonical pseudo-Kahler metric).

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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The article is organized as follows. In §1 we prove a local approxim a­ tion theorem in order to formulate the gap phenomenon. In §2 we give the first examples when the gap phenomenon holds in the situation of a product dom ain Qn = £2 x ••• x Q, showing th a t the gap phenomenon holds for (Qn , fi; J), S denoting the diagonal embedding, as a consequence of the uniqueness of Kahler-Einstein metrics. In §3 we give a proof of the pinch­ ing theorem of Mok [Mok2] for the Siegel upper half-plane %n by making use of harmonic forms in place of the Gauss-Bonnet formula. The result­ ing verification of the gap phenomenon for (7in ^7i ] i) then strengthens the gap phenomenon for (A n , A; S) as obtained from the elementary m ethod of §2. In §4 we obtain an optim al Arakelov inequality and an accompa­ nying Bochner-Kodaira formula for (B 2 x B 2, B 2; J) by using the adjoint representation. This inequality is a special case of optim al Arakelov in­ equalities obtained by Eyssidieux [Eysl] in the context of Hodge theory. In §5 we show th a t the optim al Arakelov inequality for (B 2 x jB2, B 2 ; J) is local. Together with the Bochner-Kodaira formula this gives an elaborate proof th a t the gap phenomenon holds for ( B 2 x B 2 , B 2;S). While this is a special case of results in §2 the optim al Arakelov inequality obtained in [Eysl] applies also to the situation of (T>,B2;i) for some Griffith do­ m ain V and may serve as a basis for verifying the gap phenomenon for ('» , B 2;i). This article is expository in nature. Although the most natural con­ text of our investigation is th a t of Hodge theory, our purpose here is to illustrate the principle behind the approach in some simple classical cases. In §3 for the case of curves on the Siegel upper half-plane we will give a direct approach by using the geometric description of the Siegel upper half-plane as the param eter space for polarized abelian varieties. The equivalent form ulation in term s of polarized Hodge structure of weight 1 m otivates then a general form ulation of our approach in term s of Hodge theory. In §4 we will state facts from Hodge theory necessary for the general approach and explain how this can be applied to the special case of B 2 x B 2 by regarding each individual factor B 2 as the base space for polarized Hodge structures of weight 2 arising from the adjoint represen­ tation of SU(2,1). In the conference held in March 1992 in Princeton University in honor of the sixtieth birthdays of Professor Robert Gunning and Professor Joseph J. Kohn, the second author gave a talk in relation to the differen­ tial geometry of Kuga families of abelian varieties. This article is in part an outgrowth from the circle of ideas expounded in the talk, especially the idea th at elliptic operators and eigensection equations are useful in such a context. We would like to thank Mikhail Gromov for discussions in relation to the gap phenomenon and Wing-Keung To for comments concerning asym ptotic vanishing theorems for cohomology groups in the

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context of Hodge theory.

1. Local approxim ation of alm ost-geod esic com p lex subm anifolds on bounded sym m etric dom ains (1.1) Let Q be a bounded sym m etric domain, T be a torsion-free discrete subgroup of holomorphic isometries and S C f ^ /r = X be an immersed compact complex submanifold such th at the second fundam ental form is everywhere of norm < e. We say th at S is e-pinched. We are going to approxim ate S locally by totally-geodesic complex submanifolds. For an Cl fixed, there exists a finite set of totally-geodesic submanifolds {S i, ..., S m) such th at any totally-geodesic complex submanifold S C Cl is of the form Cl for a fixed k (cf. (1.2), (1.3)). We will then say th at the e-pinched complex submanifold S is modelled on (fi.Efc). P r o p o sitio n 1 . Let Cl be a bounded symmetric dom ain of dimension n and let B be a geodesic ball of radius 1 (with respect to the KahlerEinstein m etric with Einstein constant -1) centered at x 0 E Cl. Let p < n be a positive integer and {V*} be a sequence of connected p-dimensional complex-analytic submanifolds of B containing x Q such th at the second fundam ental forms cr; = ctv^cl are pointwise of norm < e; with et- —>■ 0. Then, some subsequence of {Vf} converges to a totally-geodesic complex submanifold. Proof: By Bishop’s theorem on subconvergence of complex-analytic subvarieties, in order to extract a convergent subsequence of {Vi}, whose lim it is to be denoted by V, it is enough to prove th a t {Vi} is of uni­ formly bounded volume on any relatively compact dom ain of B. For the rest of this paragraph we will change notations and use {Vi} to denote an extracted convergent subsequence. We will further prove th a t the lim it V is unreduced. For any sm ooth point x of V, there is an open neighbor­ hood Ux of x in B with the following property: V H Ux is the graph of some vector-valued holomorphic functions ( / i , ..., f p) over a p-dimensional polydisk D\ ViC\Ux is the graph of some vector-valued holomorphic func­ tions (/{••,, /p^) for i sufficiently large such th a t converges uniformly to fk for 1 < k < p. By expressing the second fundam ental forms which is irreducible since Vi are connected. It remains to prove th a t V is unreduced. It suffices to show th at V is of m ultiplicity 1 at x Q. This will follow if for r > 0 and i sufficiently large, Vol(Vi DB( r) ) < |V o l(f?^ (r)), where B pE (r) is the p-dimensional Euclidean ball of radius r. But this is

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Philippe Eyssidieux and N gaiming M ok

again obvious from uniform boundedness of Riem annian sectional curva­ tures and the standard volume-comparison theorem. (1 .2) We consider now the structure of the set S of all totally-geodesic complex submanifolds of a bounded symmetric domain. Let 0 be a bounded symmetric dom ain and E C ^ be a totally-geodesic complex submanifold. Since involutions on a Riem annian sym m etric manifolds can be defined by reflection along geodesics, it follows th a t E is invariant under the involution ix of Q at any point x £ E. Thus, the complex submanifold E C kl is itself a bounded sym m etric domain. Regarding the structure of S we only need the following simple fact. P r o p o sitio n 2. Denoting A ut0(f2) by G, we say th at two totallygeodesic complex submanifolds E, E' C are G-equivalent if and only if E ' = 7 (E) for some 7 £ G = A ut0(G). Then, there are only a finite num ber of equivalence classes of totally-geodesic complex submanifolds E c fi. Proof: Proposition 2 is in fact a special case of the statem ent th a t for any Riem annian symmetric manifold X — G / K with G semisimple, there are only a finite number of equivalence classes of totally-geodesic subm ani­ folds. W rite g resp. k for the Lie algebras of G resp. K . Denote by 0 the C artan involution on g with fixed point set k. Any totally-geodesic submanifold S of X is itself symmetric with respect to the m etric induced from X . To classify totally-geodesic submanifolds of X it is equivalent to classify embeddings (h ,r) 1, so th at we can apply [(1.1), Proposition 1]. This is not essential since one can always argue by lifting to the universal cov­ ering. By Proposition 2, there are only a finite num ber of equivalence classes of totally-geodesic complex submanifolds on Q, represented by Efc ^ Q, 1 < k < m. By Proposition 1, given any 8 > 0 there exists an e > 0 such th a t the following holds: For any e-pinched complex subm an­ ifold S X = f2 /r of injectivity radius > 1 there is a covering of S by open sets Ui such th a t Ut is an irreducible component of S' D where Bt is a geodesic ball of radius ^ on X . (S O Bi is itself the union of a finite num ber of e-pinched submanifolds which m ay intersect each other).

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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The covering {Ui} has furtherm ore the property th at over Bi there exists a totally-geodesic complex submanifold Ei with the property th a t E* is of Hausdorff distance < J from Ut. Given a point x € Cl, the set of germs of p-dimensional complex totally-geodesic submanifolds passing through x is param etrized by a compact subset E of the G rassm annian Gr(Tx (Cl),p) of complex p-planes on the holomorphic tangent space Tx (X). By [(1.2), Proposition 2] E C Gr(Tx (Cl),p) is a finite union of -orbits, where K x C G is the isotropy subgroup at x and acts on Gr(Tx (X) , p) by isometries. It follows easily th at for 8 > 0 sufficiently small (and hence for e > 0 sufficiently small), the totally-geodesic complex submanifolds Ei can all be identified as open pieces of the same totally-geodesic complex submanifold E = E& C Cl. We will say th a t the e-pinched complex submanifold S X is modelled on E C Cl. The preceding discussion can be form ulated as P r o p o s itio n 3. Let Cl CC be a bounded symmetric domain. Fix x 0 £ Cl and let B(r) C Cl denote the geodesic ball (with respect to the Bergman metric) of radius r and centered at x 0. For 8 > 0 sufficiently small (8 < S0) there exists e > 0 such th at the following holds: For any e-pinched connected complex submanifold V C B ( x 0; 1), x 0 £ V , there exists a unique equivalence class of totally-geodesic complex submanifold on to be represented by i : E c-» Cl, and a totally-geodesic complex submanifold S C B( 1) modelled on (f2,E;z) such th at the Hausdorff distance between E D B ( ^ ) and V fl B ( \ ) is less than 8. Proposition 3 allows us to define the gap phenomenon, as follows. D e fin itio n 1. Let Cl CC be a bounded sym m etric dom ain and i : E Cl be a totally-geodesic complex submanifold. We say th at the gap phenomenon holds for (Q ,E ;i) if and only if there exists e < t{80) (£ as in Proposition 3) for which the following holds: For any torsion-free discrete group T C Aut(Q) of autom orphism s and any epinched immersed compact complex submanifold S Cl/T modelled on (Cl, E; i), S is necessarily totally geodesic. (1.4) In most pinching theorems in Riem annian geometry one consid­ ers compact Riem annian manifolds whose volumes are by hypothesis bounded, such as in Gromov [Gro] and M in-O o/R uh [MR], or else there is an a-priori bound on the volume as a consequence of hypotheses on the curvature, such as in the ^-Sphere Theorem (c/. Klingenberg [Kli]) or in Ros [Ros]. We want to explain in our situation th a t with the additional hypoth­ esis of a bound on the volume of the submanifold S, the gap phenomenon for a fixed and compact X = Cl/T will follow from the local rigidity

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(modulo deformation obtained from geodesic translations) of compact totally-geodesic holomorphic cycles. Fix any positive real num ber M and a positive integer p < dim e & and consider the space T>m of pdimensional complex submanifolds S of volume not exceeding M . Then T>m is a union of a finite num ber of compact irreducible branches of the Douady space of X . We claim th at there exists a constant cm > 0 such th at any cm -pinched complex submanifold [S] E P m is totally geodesic. We argue by contradiction. Since T>m is compact the absence of such an cm would imply by Bishop’s theorem on subconvergence of complex subvarieties th at there exists a sequence of distinct {5 i} of non-geodesic e2-pinched complex submanifolds of X in Dm , e* -> 0, which converges as a subvariety to some S E P m - By the argument of [(1.1),Proposition 1] S is necessarily totally geodesic and reduced. Since the submanifolds Si are distinct and they converge to S it fol­ lows th a t S is not an isolated point in P m - Passing to a subsequence we may assume th at Si belongs to the same irreducible branch P °M . There is a canonical holomorphic m ap from the germ of P°M at [S] to the germ of the semi-universal Kuranishi deform ation space /C of S at [S]. We write o for the point correponding to [S] and write the canonical m ap as i : (P°m i 0) We claim th at i is constant. Otherwise there exists a holomorphic 1-paramenter family {S* : t E C, |£| < 1} of complex submanifolds of X centered at S 0 = S such th a t the com­ plex structures are not identical on any neighborhood of o. Suppose for the tim e being th a t the infinitesimal deformation of the abstract fam ­ ily {5 *} at t = 0 is non-trivial, i.e., given by a non-trivial element in 9 E H 1(5, Ts). W rite T x \ s for the restriction of the holomorphic tangent bundle of J to S' and N $\ x f°r the holomorphic norm al bundle of S in X . The infinitesimal deform ation at o of {St} as a family of complex submanifolds in X is given by an element p E H°(S, N s \x ) - From the short exact sequence 0 —» Ts T x \ s —> X s \ x —> 0 we obtain a connect­ ing homomorphism £ : H ° (5, N s \ x ) —> ^ { S , Ts) such th a t 0 = 8(p). As S C X is totally geodesic, Ts C T x |s are H erm itian locally homogeneous holomorphic vector bundles on S', so th a t T x \ s — Ts 0 N s \ x holomorphically and isometrically. It follows th at H ° ( S j T x \ s ) H°{S, N s \ x ) is surjective and hence th at the image of the connecting hom om orphism 8 : H°(S, N s \ x ) H l {S,Ts) is trivial, so th a t 9 = 8(p) = 0, contradict­ ing with the assum ption 9 / 0. In general the family {S'*} of complex manifolds may be infinitesimally rigid at o up to exactly order k. We have then to consider the (k + 1 ) —st infinitesimal deform ation 9k+i and higher order tangent m aps between the Douady space and the Kuranishi space. A modification of the same argument will lead to a contradition to the non-vanishing of 9k+1 We have thus proved th at i : ( P ^ i °)

°) is constant. It means

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th at in deforming S C X as a complex submanifold the complex structure does not change for small deformations. From the uniqueness of harmonic m aps up to geodesic translations (H artm an [Har]) it follows th a t small deformations of S Q — S are obtained by geodesic translations. This contradicts with the existence of non-geodesic Si converging to S and proves by argument by contradiction th at there indeed exists an cm > 0 such th a t any ejyf-pmched p-dimensional complex submanifold of X of volume < M is necessarily totally geodesic. We can thus conclude th at, with X = fl/r fixed and compact and under the additional hypothesis of a bound on the volume of the subm an­ ifold S ^ X , the gap phenomenon for (Q, E) on X follows readily from Bishop’s theorem and the local rigidity (modulo deformations obtained by geodesic translations) of totally-geodesic holomorphic cycles S 0 C X . The gap phenomenon for (ft,£ ), if verified, can thus be regarded as a strengthened version of local rigidity for compact totally-geodesic holo­ morphic cycles modelled on (Q, £ ). In the next section we will verify one very simple case of the gap phenomenon in the case when Q is a product dom ain with isomorphic direct factors.

2. T he gap phenom enon on a product sym m etric dom ain as a consequence of th e uniqueness of K ahler-E instein m etrics (2.1) In this section we prove a very simple case of the gap phenomenon, resulting in the following theorem. T h e o r e m 1 . Let Q be an bounded symmetric dom ain and fln = x ••• x Q be the n-fold Cartesian product of Q. Let S : Q —> Qn be the diagonal embedding defined by S(x) — (a?,..., x). Then, the gap phenomenon holds for (Qn , Q; J). This result serves two purposes. On the one hand, it gives the first instance where one has a pinching theorem on quotients of bounded sym­ m etric domains (hence of negative Ricci curvature) without assumptions on the volume of the complex submanifolds. On the other hand, it serves as a m otivation for the form ulation of an effective pinching theorem on the Siegel upper half-plane by suggesting the correct growth order of the pinching constants (as a function of n). In Theorem 1 we consider immersed compact complex submanifolds S 2 and T C A ut(Q )n be a discrete group of holo­ morphic isometries acting without fixed points. W rite X for Qn / r . Let S ^ X be an n-dimensional compact immersed complex submanifold of X and write S ^ Qn for the lifting to universal covering spaces. Let Q,i denote the i—th direct factor of Qn . Suppose for every i, 1 < i < n, the i —th canonical projection : S —» Cli is unramified. Then, S ^ Qn is totally geodesic and is equivalent to the diagonal J(f2n ) of Qn . Proof: Denote by gi be the canonical Kahler-Einstein m etric on Qi with Einstein constant -1. Since T C A ut(Q )n , the restriction of gi to S is in­ variant under the action of T and defines in general a positive semi-definite sym m etric 2-tensor hi on S. Since in our situation by assum ption the canonical projection Vi : S —» Pti is unramified, hi is in fact a Riem annian m etric on S. As S is of complex dimension n — dim e the Riem anninan m etric hi on S is Kahler-Einstein. By the uniqueness of Kahler-Einstein metrics as a consequence of the Schwarz lem m a (Yau [Y]) we conclude th a t hi — hj for 1 < i, j < n. Fix a point x — (x\, ...,£ n) E S. There exists an open neigborhood U of x in S such th at the canonical projection m ap i/i : U —> m aps U biholomorphically onto an open neigborhood XJ% of Xi in Thus, for 1 < z, j < n the m ap Vji := Vj o v f 1 : Ui —>• Uj is a biholomorphism such th at Vji(gi) = g j . The holomorphic local isometry Vji : Ui —> Uj extends via the exponential m ap to a global holomorphic isometry i)ji : Q* —> klj. This implies readily th a t S £ln is embed­ ded and th at it is the image of Qi under the m ap (id, z>2i , In particular, S is equivalent to the diagonal S(Qn ) of f2n . Proof of Theorem 1: Theorem 1 follows readily from Proposition 1. In fact, if S ^ X = Qn / r is modelled on (Qn ,Q; S) and is e-pinched for a sufficiently small constant e the canonical projection m ap Vi : S —> is unramified. R e m a rk s : Since the Ahlfors-Schwarz lem m a of Yau [Y] holds for com­ plete Kahler manifolds, Theorem 1 remains valid when the compactness assum ption on S ^ X = Qn/ T (implicit in the form ulation of the gap phenomenon) is replaced by the assum ption th a t 5 X is closed and hence a complete submanifold. In fact, in the notations of the proof of Proposition 1, the Kahler metric g\g = 9i\s + J" 9n\s is complete. If S ^ X is modelled on (fin , Q; S) and if the pinching constant e in [(1 .1 ), Proposition 1] is sufficiently small, all the gi\§ are equivalent, so th a t the Kahler-Einstein metrics gi\si " m>9n\s are complete since their sum g\§ is complete. The arguments of Proposition 1 then applies to show th at S ^ kln is totally geodesic.

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(2.2) In the special case when Q is of constant holomorphic sectional curvature, i.e., for D = B m the complex unit ball, Proposition 1 implies the following effective result T h e o r e m 2 . Let (i?m)n = B m x • • • x B m be the n-fold Cartesian prod­ uct of the m-dimensional complex unit ball and T be a discrete group of holomorphic isometries on ( B m )n acting without fixed points. Nor­ malize the Kahler-Einstein gi m etric on each direct factor B™ so th at (B™,gi) has constant holomorphic sectional curvatures equal to — n. Let S X := ( B m )n / T be an immersed m-dimensional compact complex submanifold whose holomorphic sectional curvatures are pinched by < holomorphic sectional curvatures < —1 . Then, S is totally-gedesic and modelled on (( Bm)n , B m ]£). Proof: Let S ^ (B m )n be the lifting of S X to the universal covering spaces. We calculate holomorphic sectional curvatures at a point x E S . W rite r] = (771,..., Tjn ) according to the decomposition of T ^ ° ( ( B m )n ) . W rite Qi for the curvature tensor of (B™ >gi) and 0 for th a t on the product dom ain (( Bm )n , g). W rite R for the curvature tensor on S. Denote by || • || norms m easured with respect to the m etrics gi and g. From the equation of Gauss holomorphic sectional curvatures satisfy Rr)f)T)fj ^

^ ^

fji] 'hit fji) —

^

^ ^ Ill’ll ^ •

Since ||t;|| = 1 we have ^ | | ^ i ||2 = 1. The m inim um for Y2\\rh\\4 (subject only to the condition ||r;|| = 1 ) is attained when all ||t7z|| are the same and thus equal to Thus we have always < ( ~ n )(n ) = —1. To prove Theorem 2 it suffices to show th a t under the curvature assum ption for each x E S and for each 77 E T ^ ° ( S ) ) we have m ^ 0 for each 77*-, so th at the projection m aps S —>■B™ are unramified. We argue by contradiction. If one of the 7 7 say gn , is zero the sum X^i 2 let X be C n~ l x C 1 which is A n / ( r n_1 x T7). Define F : C —> C n~2 x C' by F ( x ) = (a?, ..., x; f ( x ) ) and let Cn X be the graph of F . A straightforward calculation then gives the estim ate — ( 1 + — j < Gauss curvatures of Cn < —1

V

n)

for some positive constant A. Cn is obviously not totally geodesic for any n > 2.

3. T he exp on en tial sequence and a gap phenom enon on th e Siegel upper half-plane (3.1) In this section we are going to verify the gap phenomenon for (Tin ,'H]i) for the Siegel upper half-plane Tin as a generalization of the gap phenomenon for (A n , A ]S) in the case of the polydisk. The Siegel upper half-plane Tin consists of all complex sym m etric n-by-n m atrices r with Im (r) > 0. It is a Cayley transform of a bounded sym m etric domain. In w hat follows we will choose the canonical Kahler-Einstein m etric g on Tin such th at the m inim al holomorphic sectional curvature of (Tin , g) is -2

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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and the m axim al holomorphic sectional curvature is —^ ( c f (3.2)). This m eans th a t the Kahler form is —idd logdet(Im r), which is dr A d f in the special case of n — 1, r — x iy. W ith this convention we form ulate the theorem cited in the introduction in the following equivalent form. T h e o r e m 1 .Normalize the canonicalmetric on H n so th a t the m axi­ m um holomorphic sectionalcurvature is — Let C X = %n/T be an immersed compact holomorphic curve on X verifying the curvature condition 1 \ 2 1 + — < Gauss curvatures of C < ---An J n Then, C ^ X is totally geodesic in X . In Mok [Mok2], Theorem 1 was proved using an infinitesimal GaussBonnet formula (cf. (3.2) for explanation). Here we will give a varia­ tion of the proof by considering instead elliptic differential equations on C c—>• X . Our approach consists of finding, in case C is non-geodesic, a non-trivial solution 77 to a certain elliptic differential equation on C. This part is topological and hinges on the Riemann-Roch formula. We then deduce Theorem 1 by showing th at such a solution cannot possibly exist when C verifies the pinching conditions in Theorem 1 by proving a Bochner-Kodaira formula for 77. Here we give a direct proof by interpret­ ing the Siegel upper half-plane as a param eter space for polarized abelian varieties. By interpreting the Siegel upper half-plane equivalently as the param eter space of polarized variations of Hodge structures of weight 1 (cf. Barth-Peters-Van de Ven [BPV, §14, p .36 — 39]) the proof can be form ulated in term s of Hodge theory and harmonic forms ( c f (5.7)). In the event X is an arithm etic quotient and non-compact, the first au­ thor [Eysl] showed th a t Theorem 1 remains valid for holomorphic curves C 0) corre­ sponds to a principally polarized abelian variety C n / L r , where the lattice L t is generated as an abelian group by the standard basis {e?} of C n and

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by the n column vectors of r . The variation of the uniformizing com­ plex Euclidean spaces VT = C n constitutes the universal (seminegative) homogeneous holomorphic vector bundle (V, h) over %n . We denote by V the associated locally free sheaf of germs of local holomorphic sections of V . The variation of the abelian groups of lattice points L r co n sti­ tutes a locally constant subsheaf L C V. Tensoring over R we obtain Lr = L R «—>• V. A local section of L r is then a real linear combina­ tion of lattice points. We have the following short exact sequence which we will call the exponential sequence. 0 —y L r —> V —y V /L r —y 0.

(*)

The fundam ental group T C Sp(n, R ) acts on L r (but not necessarily on L) and on V so th at the exponential sequence descends to X and by restriction to C. From now on we will use the notations in (*) to denote instead the corresponding exponential sequence on C. We are interested in the real vector space H°(C, V /L r) , the real di­ mension of which will be denoted by h(C). We give here an interpretation of H ° ( C , V /L r) in term s of holomorphic functions defined on some im ­ mersed holomorphic curves on 7f n . Let C —>• C , C > %n be a regular covering m ap corresponding to the kernel of the canonical hom om orphism 4> : 7Ti(C) —> n i ( X ) = T. Then, Im() acts on C as deck transform ations. A section of H°(C, V /L r) is equivalently a vector-valued holomorphic function F : C —>• C n satisfying a system of com patibility relations, as follows. For any 7 G T write 7 r = (A yr + R7 )(C 7r + D7 )- 1 . Denote a point on C by f and its image in H n by r . Then, the holomorphic m ap F : C —>■C n corresponds to an element of LT°(C, V /L r) if and only if for any 7 E Im(4>) there exist real vectors P1 , Q1 G= R n such th at F ( j f ) = ([C7r + D7]*)- 1 F ( f ) + [7 r /„]

(1)

We will call the last column a period of F . Two holomorphic m aps F1, F ' : C —>• C n define the same element in H ° ( C , V /L r) if and only if ( F - F ' ) ( r ) = [r /„ ][$ ]

(2)

for some P , Q G R n . If T C ^ ( ^ j Z) then we have a polarized family of abelian varieties A = V / L z defined over C. In this case H ° ( C , A ) corresponds to the subset of those F : C —>• C n with integral periods, and F , F' : C -7 C n are identified if and only if (2) holds with F , Q e z n. (3.3) A non-vanishing theorem for h(C). We proceed to prove a non­ vanishing theorem for h(C) whenever C satisfies the pinching hypothesis

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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in [(3.1), Theorem 1 ] but is not totally geodesic. (A simple modifica­ tion actually shows th a t h ( C ) > 0 for any compact holomorphic curve C ^ X unless C is totally geodesic and of constant curvature —^ .) We consider the long exact sequence associated to the exponential sequence. Under the pinching hypothesis we have Ff°(C, £ r ) = 0, because the non­ vanishing would produce a constant sub-local system which would imply th a t the universal covering m ap C —>■Hn factors through a Siegel subdo­ m ain %n (r < n) and th a t the Gaussian curvature is less than — while H 2( C , L ji ) = 0 by duality since the standard symplectic pairing on L r yields a self-duality L r = L*R . On the other hand the holomorphic vec­ tor bundle V, equipped with a canonical H erm itian-Einstein metric, is of seminegative curvature in the sense of Griffiths. The pinching hypothesis on the Gauss curvature implies th a t V is of strictly negative curvature over the compact holomorphic curve C, so th at H ° ( C , V ) = 0. We ob­ tain therefore from the long exact sequence associated to the exponential sequence over C the exactness of H ' i C , V) -* H \ C , V / L r ) -> 0. (1) By considering the Euler-Poincare characteristic we have 0 -> H°{C, V / L r ) -»■ H \ C , L r )

(2 )

t i m R H 1( C , L R ) = 4 n { g - \ )

where g — g(C) is the genus of the compact Riem ann surface C. On the other hand by Riemann-Roch we obtain dim e H ^ C , V) = - X(V) = n(g - 1 ) - deg(V).

(3)

(det(U ), det(h)) is a Herm itian-Einstein bundle on %n with Einstein con­ stant —1 while the m axim al holomorphic sectional curvature of (/Hn , g ) is — By the Gauss-equation the Gauss curvature of (C,g\c) is every­ where < — By considering first Chern forms and integrating over C we obtain d eg (^ ) > Q ) ( 2 - 2 g ) = - n { g - 1), (4) and th a t equality holds if and only if C is totally geodesic and is of constant Gaussian curvature — P utting (2), (3) and (4) into (1) we have the inequality d im R

H°{C,

V /L r )

> %n(g

-

1)

+

2 d eg(U )

>

0

(5)

with strict inequality unless the curve (C, g\c) is totally geodesic and of constant Gaussian curvature — This yields a non-vanishing theorem for H°(C, V / L r ) . (3.4) A vanishing theorem for h{C). For C X satisfying the hypoth­ esis of Theorem 1 there is on the other hand a vanishing theorem for

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h(C) obtained in [Mok2,§4] from geometric considerations. On the to­ tal space 7r : Tin x C n —> Tin there is a family of homogeneous metrics {vt} degenerating to 7t*uj, uj denoting the canonical Einstein m etric on 7in (cf. [Mok2, §3]). Let / E fL°(C, V /L r). By considering the graph of a lifting of / we obtain a one-parameter family of Herm itian metrics {£*} on X degenerating to the restriction of the canonical m etric of X on C. If / is non-trivial we obtain a contradiction by expanding the Gauss-Bonnet integral over C with respect to {£*} in term s of t. More precisely there is a tensor rj{f) E C°°(C, V 0 Q c) which vanishes if and only if F : C —> C n (as in (3.2)) is a horizontal section, i.e., a real linear com bination of lattice points. From the t 2-term in the expansion of the Gauss-Bonnet integrand we derive the vanishing of r](f) and thus h(C). This contradiction to h ( C ) 7^ 0 obtained above means th at a non-geodesic compact holomorphic curve C verifying the pinching hypothesis of [(3.1), Theorem 1] cannot possibly exist. Here we give a proof of Theorem 1 by showing th at in the geodesic case of Gaussian curvature — 77 satisfies the eigensection equation d*drj = —77, while in general 77 satisfies a similar equation with a zeroorder perturbation term arising from the second fundam ental form (and not involving its covariant derivatives). On the Siegel upper-half plane Tin we have a canonical isomorphism of the holomorphic tangent bundle T with S 2V . We will choose the homogeneous H erm itian m etric h on V such th at (T, g) = ( S2V , S 2h) as H erm itian holomorphic vector bundles. This is reflected by the fact th a t elements of Tin are sym m etric matrices. The canonical isomorphism S 2V = T defines in an obvious way a canonical bundle embedding 5 : y* rp* ^ y identify V with V * by contraction with the canonical H erm itian m etric. There is a canonical bundle homomorphism $ : S 2V ® { S 2V ® V ) - + S 2V 0 V defined by $ ( a 0 77) = 5 (77(0:)). We write $ ( 0 , 77) for 3>(d 0 77). Then, any 77 = rj(f) of a local section / E H°(U, V /L r) over an open subset U of Tin satisfies the first-order differential equation (Mok-To [MT, (2.4)]) • • • > a n > 0. la ^ 2 = n. We have

< - ^ | a n | 2||r/||2-

(11)

Suppose the holomorphic sectional curvature in (%n ,9 )is —- (l + ^ ) , 0 < a < 1, we have (cf. [(2.2), proof of Theorem 2]) K | 4 + ( n - 1) ( ^ ^ ) implying

>n(l+ £ ) ,

(12)

(as in [Mok2, (4.3), Proposition 1]) .9

1 j(n — \)a

S s v S r L


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which follows from y/a + y/b > 2. W ith this we have proved th a t for any / E H°(C, V /L r) , ry(/) = 0, contradicting with the non-vanishing theorem h{ C) ^ 0 obtained in (3.2) by the Riemann-Roch Theorem. W ith this we have completed the proof of [(3.1), Theorem 1] and verified the gap phenomenon for ('Hn , H ; i) on the Siegel upper half-plane H n • In the case of arithm etic quotients X = f i n / T Eyssidieux [Eysl] showed th a t Theorem 1 continues to hold for C ^ X of finite volume. The proof makes use of Mumford compactiflcations and the SL(2)-orbit Theorem of Schmidt.

4. An optim al A rakelov inequality on Chern num bers for com pact com p lex surfaces of q uotients of B 2 x B 2 (4.1) Starting from the gap phenomenon on the Siegel upper half-plane f i n obtained in §3, our perspective is th at the same line of thought, for­ m ulated in term s of variations of Hodge structures, will lead to the gap phenomenon for compact complex surfaces on quotients of bounded sym­ m etric domains or of period domains. In this section as an illustration we consider complex surfaces on quotients of B 2 x B 2. While the results of §2 already implies the gap phenomenon for ( B 2 x R 2,R 2;£) we will de­ rive in this section and the next a new proof, giving a m ethod which can probably be generalized. We will obtain along the way Chern-number in­ equalities on B 2 x B 2 characterizing certain holomorphic geodesic cycles, a result which is of independent interest. Recall th at in §3 for curves C on quotients of %n we obtain a non­ vanishing theorem h ( C ) ^ 0 from the Riemann-Roch form ula and from the Gauss equation ([(3.2), eqn. 4), which yielded d e g (F |c ) > —n(g(C) — 1 ), an inequality due to Arakelov. Our point of departure is to obtain new inequalites on Chern numbers of complex submanifolds S of quotients X of bounded sym m etric dom ain by interpreting the latter as param eter spaces for polarized Hodge structures, following an idea due to Deligne. We refer the reader back to the Introduction of this article for an infor­ m al definition of what we call an Arakelov inequality. We are interested in those which are optimal in the sense th at equality holds for certain holomorphic geodesic cycles. For the purpose of applying such inequali­ ties to verify the gap phenomenon in the same way as in §3 for the case of (f i n ^ f i ; i) we will need to show th at such inequalities are local in the sense th at they are obtained from integrals of nonnegative (m, m)-forms vs over 5, m — dim e S, such th at the vanishing of vs over an open set implies th at S is totally geodesic.

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In this section we will obtain an optim al Arakelov inequality on B 2 x B 2 by considering adjoint representations of S U ( 2 , 1) x S U ( 2,1) on its Lie algebra. Let T be a torsion-free cocompact arithm etic group of auto­ morphisms of 5 2 x 5 2 and define X — (B 2 x B 2)/T. We will sometimes denote by B 2; i = 1, 2; the i —th direct factor of B 2 x B 2. Denote by 7r : B 2 x B 2 -4 X the canonical projection. W rite T x for the holomorphic tangent bundle on X . We define two rank-2 holomorphic vector subbun­ dles Ti resp. T 2 C T x consisting of vectors tangent to 7r(B2 x W ) resp. 7r({xi} x R 2), X{ being arbitrary points on B 2. In what follows we will prove a special case of Arakelov inequalities obtained in Eyssidieux [Eysl]. Furtherm ore, we will verify th a t the inequality is optim al and lo­ cal in this special case, leading to a characterization theorem for certain totally geodesic holomorphic cycles. T h e o r e m 1. Let S ° 4 X be an immersed compact complex surface. Then, on S we have

c2(S) > l(c?( 7i|5) + c?(:r2|s)) Furtherm ore, equality is attained in the special case when S is totally geodesic and modelled on (B 2 x R 2,R 2; X and denote T2|s simply by Ti. (4.2) A Chern-number inequality of Gromov’s on immersed compact com­ plex surfaces. As an introduction we start by recalling a special case of some result of Grom ov’s ([Gro]) giving Chern-number inequalities on im ­ mersed compact complex surfaces i : S ^4 X . These inequalities arise from ordinary de R ham cohomology. The inequalites are also valid when X = ( B 2 x B 2) / T is replaced by any quotient of a bounded sym m etric dom ain by a torsion-free cocompact lattice. P r o p o s it io n 1. (from Gromov [Gro, (1.3)]) Let i : S' 4 immersed compact complex surface. Then, we have

(i) c2(S) > 0, and (ii) ( - l ) r c/i(ft£)-Todd(T5 ) > 0, r = 0,1,2;

I

be an

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where is the trivial line bundle, is the holomorphic cotangent bundle of S, = K s is the canonical line bundle, ch(-) denotes the Chern character and Todd(-) denotes the Todd class. Proof (outline): The crux of Grom ov’s argument was to show th a t there are no f 2 harmonic p-forms for p / 2 on some covering space S S. Let E C 7Ti(S) be the kernel of the canonical m ap 7Ti(S) —>• ^ i { X ) and denote by S —> S the regular covering space corresponding to the norm al subgroup E C tti(S'). (Write tt : X —> X for the canonical projection for the universal covering X = B 2 x B 2. If S C X is embedded, then S can be identified with a connected component of 1r_ 1 (5).) By the assum ption th at 7Ti(X) = T is arithm etic we know th at there is a tower of unramified coverings X n —> X corresponding to subgroups Tn C T of finite index such th a t n r n = {id}. We have correspondingly a tower of unramified coverings S n —> S with ri7ri(S'n ) = E. X is Kahler-hyperbolic in the sense of Gromov ([Gro, para. (0.3), p .265]). More precisely, the Kahler form Q on X corresponding to the Kahler-Einstein m etric is of the form da, where a is a sm ooth 1-form on X which is bounded with respect to the Kahler-Einstein metric. Denote also by i : S X the immersion arising form i : S ^ X . By restriction we have i*& = d(z*a) on 5, so th at the argument of Gromov ([Gro, Theorem (1.4.A), p.274]) yields the vanishing of the space of L 2 harmonic p-forms on S', provided th at p ^ 2. Equivalently this means th at dim c H 2(Sn , C) h m 1|7T ~iF(5) q \ : 7Ti T(5„)J c T T n-s-oo

=

0

f o r

p^

2

>

Ti(Sn)

where [7Ti(5) : 7Ti(5n )] stands for the index of 7 in 7Ti(5). As a consequence, the cohomology is asym ptotically concentrated in real di­ mension 2, i.e., di mc H 2(Sn , C) c2(S) = hm > 0. n-*oo [7Ti (5) : 7Ti (5„)J From Hodge decomposition harmonic forms split into sums of harmonic forms of different bidegrees. The inequalities (ii) then follow from the Riemann-Roch formula. (4.3) Locally constant vector bundles and harmonic forms. The proof of Proposition 1 can be generalized to the situation of variations of polar­ ized Hodge structures. In our situation we are going to interpret B 2 x B 2 as the param eter space for polarized Hodge structures of weight 2 aris­ ing from the adjoint representation of G — S U ( 2 , 1) x S U ( 2 , 1 ). W rite p : G —> GL(g) for the adjoint representation of G on its Lie algebra g. This gives rise to a representation of T = tti(X) on g and consequently

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a locally constant sheaf to be denoted by Vad• Denote by Vc the complexification of Vad• We will make no notational distiction between the locally constant sheaf Vc and the underlying locally constant complex vector bundle. We will denote the corresponding objects on the univer­ sal covering space B 2 x B 2 by the same symbols. By a generalization of Gromov [Gro,Theorem (1.4. A), p. 274] to the context of Hodge theory one can prove the analogue of [(4.2), Proposition 1] for the locally constant sheaf i * V c on S . The analogue of [(4.2), Proposition 1, statem ent (ii)] will then yield inequalities on Chern numbers which turn out to be sharp. We proceed to formulate this analogy. The Killing form on g descends to a symmetric non-degenerate bilinear form B on Vad which is indefinite. At each point x in B 2 x B 2 we have the C artan decomposition g — kx (&mx into an orthogonal direct sum, where kx is the Lie algebra of the isotropy subgroup at x and m x is identified with the real tangent space at x. We have B x \kx < 0 and B x \mx > 0. In what follows the superscript C will denote the complexification of a real vector space. W rite gc = k£ 0 ra+ 0 m ~ , where mj? = m + 0 m~ is the decomposition of the complexified tangent space m J? into subspaces of type (1 ,0) and type (0,1). At each point x £ B 2 x B 2 let Cx be the Weil operator on g ° which is id on k£ and —id on m £\ Define bx (v) = —B ( C v , v ) . Then, bx is a Herm itian bilinear form on pc , identified with Vc,x > the fiber of Vc —> B 2 x B 2 over x. The family of H erm itian bilinear forms bx then defines a Herm itian metric h on Vc over B 2 x B 2. Furtherm ore it is invariant under the action of G on Vc over B 2 x B 2 defined by the adjoint representation p. This means th a t h descends to a H erm itian m etric, to be denoted by the same symbol, on the locally constant complex vector bundle Vc on X . Since Vc is locally constant over X there is a canonical flat connection d. We note th at the H erm itian m etric h is not parallel with respect to d. Nonetheless we still have P r o p o s itio n 2 . Let i : S X be an immersed compact complex surface in X and let S —> S be the covering space corresponding to E C Tri(S'), where £ is the kernel of the canonical m ap ffi(S) —>■ 7Ti(X). Denote also by Vc the induced locally constant bundle over S. Then, every L 2 harmonic p-form on S with values in Vc is trivial provided th at p ^ 2. Here L 2 norms are measured with respect to the Kahler-Einstein m et­ ric on B 2 x B 2 and the Herm itian m etric h on Vc over B 2 x B 2. (4.4) A brief description of the underlying variation of Hodge structures on V c. The proof of Proposition 2 follows easily from the proof of Gro­ mov ([Gro, Theorem (1.4.A), p .2 7 4 //.]. In what follows we will give a brief description of the variation of Hodge structures underlying Vc and state the necessary relevant and standard properties which make the gen­ eralization possible. This description of the Hodge structures will also be

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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used in (4.5) to obtain Chern-number inequalities. The following applies to any bounded symm etric dom ain X 0 = G / K in place of B 2 x B 2. Here we adopt notations in conformity with the usual practice in Hodge theory and note th a t such notations are not necessarily consistent with those used in §3. The description th at follows is informal and presupposes some rudim entary knowledge about bounded symmetric domains (cf. for instance Mok [Mokl, Chap. 3-5]). The complexified C artan decomposition Vb,* = m - © kx © m+ := H f © H ? 0 H ° 2 gives rise to a sm ooth decomposition of Vc — H 20 0 H 11 0 H 02 over B 2 x B 2 into sm ooth complex vector bundles. H 20 = T can be en­ dowed the holomorphic structure of the holomorphic cotangent bundle by contracting with the Kahler m etric and as such it is holomorphically embedded in Vc- We can see this as follows. Let X c be the compact dual of X Q and X 0 C C X c be the Borel embedding. Then the local system Vc and the complex C artan decomposition V c )X — 9c — ra” 0 kx 0 r a j can be defined over X c. Write Vc = H 20 0 H 11 0 H 02 for the corresponding sm ooth global decomposition over X c. To say th at H 20 C Vc is a holo­ m orphic subbundle it is equivalent to show th at H 20 is invariant under the action of the complex Lie group G c of biholomorphisms of X c. Write X c = G c / P x as a complex homogeneous space where Px is the Borel sub­ group corresponding to the complex Lie subalgebra px — ra“ 0 kJ?. The infinitesimal action of Px on Vc at x is given by the adjoint action, i.e., by Lie brackets. But from the theory of bounded sym m etric domains we have = 0 ; [ k f , m ~ ] C m~; and C kf (*) ( cf for instance MokfMokl, (3.1), pp.51-52]). The first two relations then imply th at H 20 C Vc is a holomorphic subbundle. Recall th at d is the canonical flat connection on Vc- In what fol­ lows we explain why H 20 C Vc is not a d-parallel subbundle and how the discrepancy from being d-parallel is measured. Denote by flP)q the bundle of (p , g)-covectors. The obstruction on H 20 to being d-parallel with respect to d is measured by the second fundam ental form cr of H 20 in Vc- ■ H 11 0 H 02 is a sm ooth bundle m ap. cr is non­ trivial since in fact [ra j, m~\ — k£ The last relation in (*) implies th at the image of a lies in H 11. The second fundam ental form a corresponds equivalently to a sm ooth bundle m ap V ' : H 20 —> H 11 0 Q1,0, which is called a Gauss-M anin connection. V ' is in fact a parallel bundle m ap between homogeneous holmorphic vector bundles which happens in this case to be injective. In the same vein, (*) together with the relationship [ k £, k^} C k ^ means th at H 20 0 H 11 C Vc is a holomorphic subbundle

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Philippe Eyssidieux and N gaiming M ok

and th at the obstruction to its being d-parallel is given by a Gauss-Manin connection H 11 —» H 02®Q,1,0. The description thus far can be recaptured by describing the locally constant vector bundle Vc over Q as arising from a variation of Hodge structures, as follows (c/. Zucker [Zu]). The (1,0) part of the flat connection d on Vc, to be denoted by V, decomposes according to V = d' + V ;, where d'(C°°{Hpq)) c r f F ^ O V' (C°°(Hpq)) c r ( F " w

1-0); ® O 1)0).

Here C°°(-) denotes the sheaf of germs of sm ooth sections of the bundles concerned. The (0,1 ) part of the flat connection d on Vc, to be denoted by V, decomposes according to V = d" + V", where d"{C°°(Hpq)) C C°°(Hpq 0 f20,1); V"{C°°(Hpq)) C C°°(HP+1^ _1 (8)Q0)1). The ^''-operator endows H pq with holomorphic structures. Furtherm ore, the sm ooth complex vector bundle m orphism V ' : H pq —» H p~ 1,q~^1Q1,0 is holomorphic. This morphism is called the Gauss-Manin connection. P u ttin g D " — d!' + V ' and D f — d' + V " and denoting by the same symbols the corresponding exterior differential operators on Vc-valued differential forms, the de R ham complex (Vc, d) is biflltered by 0

fi"(Vfc)=

« n (Vb)p ’a ;

P + Q -n + 2

0

n n (Vc)P'Q =

H p q ®Q,r’°

p + r = P ,q + s = Q

for 0 < n < 2d im cA 0 such th at d

" (c “ ( n n (Vc)p,° ) ) c c°°(fin+1 (Vb)p 'Q+1);

Df (C00(fin (Vrc ) i>,(3)) c C°°(fin+1(Vc)P+1’Q)-

(*)

Denote also by d the exterior de R ham operators and decompose ac­ cordingly d — D' + D " . Then, from d2 — 0 and (*) it follows readily D ' 2 = D //2 = 0. W ith this formalism the classical Hodge-theoretic iden­ tities can be proved and the simple Lefschetz-type argument of G rom ov’s can be adapted to yield ([4.3], Proposition 1), i.e, the non-existence of L 2 harmonic p-forms on S with values in Vc for p / 2 . (4.5) An index formula. P ut £5 ’^ = C°° (S, Qln (Vc )p>®- As an analogue to [(4.2), Proposition 1] we consider the complex

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

105

arising from the de R ham complex (V c, d). We have the following in­ equality.

holds for the Euler-Poincare characteristic Proof: The Kahler identities imply th at A d >> = for the Laplacians A d if = D " D"* + D ,/*D,/ and = dd* + d*d. As a consequence har­ monic forms with values in Vc over S decompose according to the (P, Q)bigrading. Let S n S be a tower of unramified coverings as in [(5.4), Proposition 2]. The vanishing theorem in [(4.2), proof of Proposition 1] then yields

( - i ) px(£s'*;D") = ( - i y lim

n —>■oc

= Hm dim n —too

cH2(Sn,Vc) [7Ti(5) : 71-1 (S,

(4.6) Proof of Theorem 1. Consider for p — 3 the complex (£5’*; D") as defined in (4.5). There is another complex ( 1)

corresponding to

0

h

20 0 n 1’0 4

0

h

20 0 fi1’1 4

h

20 0 n 1'2

0

H 11 0 Q2’0 4 H 11 0 Q2’1 4 H 11 0 ft2’2

0.

(2)

The complex ( £ j’*; dn) has the same index as th at of (£ f’*; D") since the Gauss-M anin connection V ' is of order zero. We have thus 0 < - x ( £ l ’*;D”) = - x ( 4 ’*;d")

= x(S, H 11 ft2) - x{S, H 20 ft1)

(3)

where ft1 = ft.s denotes the cotangent bundle and ft2 = K s denotes the canonical line bundle on S'. Denote by T the holomorphic tangent bundle on X . Then, T = 0 T2 and H 20 = T* = T* 0 T2*, H 11 = E nd (T i) 0 E n d (T 2 ) = (T* 0 Ti) 0 (T2* 0 X2). We have thus over S H 20 ® fts = (77 ® fts) 0 (T2* ® f ts )

Philippe Eyssidieux and N gaiming M ok

106 H 11 0 Q2 = {T* 0

0 K s ) 0 (T* 0 T2 0 K s )

(4)

Suppose S' = S 0 is totally geodesic and modelled on (B 2 x B 2, B 2\S). Then, T i \ s 0 — T2|s o = T5 , the holomorphic tangent bundle over S. Then for i = 1 , 2, we have T5 0 A"5 = = fl1, so th a t the two holomorphic vector bundles are isomorphic and we obtain

x(£l:*;D") = x(So,Hn 0 ft2) - X{S0, H20 0 ft1) = 0.

(5)

In general, by Serre duality we have x( S, H n 0 ft2) = x( S, ( H n Y ) = x ( S, T* 0 Tx) + X (S, T ? 0 T2).

(6)

For a cohomology class rj in 77* (S, R ) we denote by rj4 the component in H 4(S, R ). We identify furtherm ore H 4(S, R ) with R by evaluating on the fundam ental class [S] determ ined by the orientation defined by the complex structure. We have x{S, H 20 0 f t1) - X{S, H 1 1 0 ft2) = [((cA(2T 0 Qs ) - ch(T? 0 Tx)) +{ch{T; 0 Qs) - ch{T2* 0 T2))) • Todd(S)]4.

(7)

For i — 1,2 we have ch{T* 0 Ti) = 4 + [c2(Ti) - 4c2(7i)];

(8)

ch{T* 0 fts ) = 4 - 2ci(7;) - 2c1(S) + [c2(7i) - 2ca(7i)]+ +[c2(5) - 2c2(5)] + Cl(Ti) ■Cl(5);

(9)

Todd(S) = 1 + l Cl(5) + 4 [ c2(5) + c2(S)].

(10)

Substituting (8), (9) and (10) into (7) and using the fact th at c2(Ti) = 3c2(Tj) we obtain (ci(Ti) + 4 ( T 2)) [cxiTx) + Cl(T2) + 2Cl(S)} ■Cl{S) {cl(Tx) + c2(T2) ) - 2 [ c 2( S ) - 2 c 2( S) ]- cx( Tx )- cx( S) - cx( T 2)-cx(S) > 0, which simplifies to

4 c2( S ) > ^ ( c2(Tx) + c2(T2)), proving [(4.1), Theorem 1], as desired.

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

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5. Characterization of com pact holomorphic geodesic cycles on quotients of B 2 x B 2 m odelled on ( B 2 x B 2, B 2;5) (5.1) In this section we will show th at the Arakelov inequality on Chern numbers obtained in §4 for (B 2 x B 2 , B 2 ; 8 ) is local. First we set up some notations. Let gi) i = 1 , 2, be Kahler-Einstein metrics on B 2 with the same Einstein constant. Denote by g — g\ + the Kahler-Einstein m etric on B 2 x B 2 and by hi the H erm itian metric on the canonical line bundle Ki on B 2 thus obtained. We will normalize gi so th at ( B 2 ,gi) is of constant holomorphic sectional curvature —2. We also consider (A"z-,h z) as a H erm itian holomorphic line bundle on B 2 x B 2 in the obvious way. We have

Theorem 1. Let U C B 2 x B 2 be an open subset and S C U be a 2dimensional complex submanifold on U . Then, in term s of Chern forms arising from the Kahler m etric g\s on S and the H erm itian metrics hi on K i , we have the pointwise inequality M-S'.tfls) > 1 (c?(A 'i,/ii)|5 + c ? (/i2,/i2)|s) between (2,2)-forms on S. Furtherm ore, equality holds everywhere on U if and only if S is totally geodesic and (U, S ) is modelled on ( B 2 x J52, 8 ( B 2)) for the diagonal embedding 6 . In this section we will only be considering characteristic forms on S C B 2 x B 2. The notation g will stand for g\s while ci(K i,h i) will m ean the restriction of first Chern forms on S. The inequality in Theo­ rem 1 is an inequality between real differential 4-forms on the underlying real 4-manifold S. The Arakelov inequality on Chern numbers in §4 for compact complex surfaces on quotients of B 2 x B 2 is then a consequence of Theorem 1 obtained by integrating real 4-forms. In other words, The­ orem 1 asserts th a t the Arakelov inequality in §4 for (B 2 x B 2 , B 2 ; 8 ) is local in nature. While the argument in §4 via harmonic forms is the origin of Theorem 1, the proof of the latter is completely independent of Hodge-theoretic considerations. It consists of a straight-forward verifica­ tion using the Chern-Weil theory of characteristic forms. (5.2) Preliminaries on the second Chern form. Let (M ,g) be a Kahler manifold with Kahler form lo. From the Chern-Weil theory of character­ istic forms the second Chern class C2 (M ) is represented by the (2 ,2) form c2 (M ,g)

±

Y a,(3,7,6

Tcfosdza A dzp A dz< A dz5,

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Philippe Eyssidieux and Ngaim ing M ok

where

T«fhS= X ,11,(7, E 9Xd9 ^ R x f ap If at a point x

E

E

R ^ -

t

(1)

A,/i, |2- | < a 2tp 2 > |2 < - ( l | a i l l 2IIA II2 + I M 2IIA II2)

(3)

Philippe Eyssidieux and N gaiming M ok

110

Ignoring the term s R apadc^Rpappi Raj 3ap an

4 ( I M I 4 +

| | a 2 ||4) 2 +

4 ( I K ||2||A ||2 +

4 (H A ||4 +

||A II4) 2 +

|| a 2 ||2 | | A | | 2) 2 .

(4)

W riting w ithout loss of generality ||a i ||2 = 0.5 + A, 0 < A < 0.5, ||/?i||2 = 0.5 ± 0 < p < 0.5, we have ||a 1 ||4 + ||a 2||4 = ( | | a 1 ||2 + ||a 2||2) 2 - 2| | a ( | | . | | a 2||2 = 1- 2(0.5 + A)(0.5 - A) = 1 - 2(0.25 - A2) = 0.5 + 2A2;

(5)

||A ||4 + ||A ||4 = 0.5 + 2p2;

(6)

I M 2IIAII2 + I M I W = (0.5 + A)(0.5 ±fx) + (0.5 - A)(0.5 t / j ) = 0.5 ± 2Ap.

(7)

Thus, l | | i ?||2 > (0.5 + 2A2)2 + (0.5 + 2/x2)2 + (0.5 ± 2An ) 2 > 0.75 + 2A2 + 4A2 + 2 n 2 + 4 / ± 2 \ / i + 4 A V .

(8)

By our choice of coordinates the Ricci tensor is diagonalized, i.e., R ap = 0. W rite R aa = A, Rpp = B . We have —2||i?ic||2 + A' 2 = - 2 (A 2 + B 2) + {A 4- B ) 2 = - ( A 2 + B 2) + 2 A B = - { A - B ) 2.

(9)

Since A — B = {Raaaa + R aapp) ~ {Rpppp + Rppaa) = Raaaot ~ Rpppp, we have by (8) -2 ||i?* c[|2 + A 2 = - ( 2 ( |K ||4 + I M I 4) - 2 ( ||A ||4 + IIAII4))2 = - 4 ((0.5 + 2A2) - (0.5 + 2p 2))2 = —16(A2 - /i2)2 = —16(A4 - 2A2//2 + n 4).

(10)

We conclude thus from (7) and (9) th at l ( \ \ R \ \ 2 - 2 \ \ R i c \\2 + K 2)

= (0.75 + 2A2 + 4A4 + 2\i 2 + 4p4 ± 2Ap + 4 A V ) - 4(A4 - 2A2 p 2 + p 4) = {0.75 + 2X2 + 2p2 + I 2 \ 2p 2 ± 2 \ p ) .

(11)

Holomorphic Geodesic Cycles on Locally Sym m etric Manifolds

111

We summarize the results obtained so far as the following lemma. L e m m a 1 . Let U C B 2 x B 2 be an open subset and S C U be a 2dimensional complex submanifold. Let x E S and (a, /3) be an orthonor­ m al basis at x such th a t R ap — 0. W ithout loss of generality suppose a = ( a i , a 2), /? = (A ,/?2) with ||a x||2 = 0.5 + A, ||/?i||2 = 0 .5 ± /u , 0 < A, p < 0.5. Then, we have the inequality Ld2 C2 ( S , g ) >

—n

(0.75 + 2A2 +

2f i 2

+ 12A2//2 i

2X/ x)

.

(5.4) Verification of the inequality at a point with vanishing second fu n­ damental fo r m . Denote by 0* the curvature (1,1) form of (K i , hf). Iden­ tifying 0* with the corresponding H erm itian m atrix we have
|2) < 9||a,-||2 • ||A ||2-

(2)

Thus, c \{ K l , h l ) + c\{ K l , h l ) / )2 ^2

= =^

0/ ,2

(det(01) + det(02)) < ^ 2 (H^lll2 ' II^H2 + IMI' ' H^ll2)

((0-5 + A)(0'5 ± A + (°'5 - A)(°'5 T M = ^ ( ° ' 5 ± 2A^ ' ^

At a point x E S where the second fundam ental form vanishes we have thus V2 (S, >

Ld2

g) =

C2(S, g ) - ± (cliKx

,h1)+ c\{K l , hi))

(0.75 + 2A2 + 2n 2 + 1 2 A V ± 2A/x - 1.5(0.5 ± 2Aj*)) to2

> ^ ( 2 A 2 + 2^ 2 + 1 2 A V T A/i) .

(4)

If the sign attached to X/x is + , we have obviously v 2 (S,g) > 0. If the sign is —, then the inequality v 2 (S,g) > 0 is a consequence of 2A2 + 2fi2 + 1 2 A V > Xfx,

(5)

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Philippe Eyssidieux and Ngaiming M ok

which follows readily from A2 -f p 2 > 2Xp. We have thus proved the inequality at a point x E S under the assum ption th at the second funda­ m ental form vanishes. Under the same assum ption cr(x) — 0 we examine when we have equality ^ ( S , g)(x) — 0. First of all by (5) and (6) we conclude th a t A = p = 0. In other words, we have an orthonorm al basis {a,/?} of T*’°(S) with a = ( a i , a 2), P = (P i,p 2 ) such th at Ha,!!2 = ||a 2||2 = ||/?i||2 = ||/?i||2 = Furtherm ore from [(5.3),eqn.(3)] we conclude th a t {S^g){x) = 0 implies < oq,/?i > = < c*2,/?2 > = 0. The description of {a,/?} implies readily that, taking x to be the origin, To1 ,0(5) consists of (77, $ ( 77)) where 4> is a unitary transform ation. Thus, up to a holomorphic isometry of B 2 x B 2 we have proved th at a(x) = 0 and V2 {S,g)(x) = 0 implies th a t S is tangential to the diagonal S (B 2). (5.5) Contribution of the second fundamental form. To complete the proof of the inequality (5.4) we have to consider the contribution of the second fundam ental form a to 02(6*,