HOLOMORPHIC MORSE INEQUALITIES AND BERGMAN KERNELS
CONTENTS INTRODUCTION CONNECTIONS ON VECTOR BUNDLES HERMITIAN CONNECTION CHERN CONNECTION CONNECTIONS ON THE TANGENT BUNDLE LEVI–CIVITA CONNECTION BISMUT CONNECTION SPINC DIRAC OPERATOR CLI?ORD CONNECTION DIRAC OPERATOR AND LICHNEROWICZ FORMULA MODI?ED DIRAC OPERATOR ATIYAH–SINGER INDEX THEOREM LICHNEROWICZ FORMULA FOR ?E THE OPERATOR ? E + ? E,? BISMUT’S LICHNEROWICZ FORMULA FOR ?E BOCHNER–KODAIRA–NAKANO FORMULA BOCHNER–KODAIRA–NAKANO FORMULA WITH BOUNDARY TERM SPECTRAL GAP VANISHING THEOREM AND SPECTRAL GAP SPECTRAL GAP OF MODI?ED DIRAC OPERATORS ASYMPTOTIC OF THE HEAT KERNEL STATEMENT OF THE RESULT LOCALIZATION OF THE PROBLEM UNIFORM ESTIMATE ON THE HEAT KERNEL DEMAILLY’S HOLOMORPHIC MORSE INEQUALITIES BIBLIOGRAPHIC NOTES LINE BUNDLES, DIVISORS AND BLOWING-UP THE SIU–DEMAILLY CRITERION BIG LINE BUNDLES MOISHEZON MANIFOLDS THE SHI?MAN–JI–BONAVERO–TAKAYAMA CRITERION SINGULAR HERMITIAN METRICS ON LINE BUNDLES BONAVERO’S SINGULAR HOLOMORPHIC MORSE INEQUALITIES VOLUME OF BIG LINE BUNDLES SOME EXAMPLES OF MOISHEZON MANIFOLDS ALGEBRAIC MORSE INEQUALITIES L2-COHOMOLOGY AND HODGE THEORY ABSTRACT MORSE INEQUALITIES FOR THE L2-COHOMOLOGY THE FUNDAMENTAL ESTIMATE ASYMPTOTIC DISTRIBUTION OF EIGENVALUES MORSE INEQUALITIES FOR THE L2 COHOMOLOGY UNIFORMLY POSITIVE LINE BUNDLES SIU–DEMAILLY CRITERION FOR ISOLATED SINGULARITIES MORSE INEQUALITIES FOR Q-CONVEX MANIFOLDS MORSE INEQUALITIES FOR COVERINGS COVERING MANIFOLDS, VON NEUMANN DIMENSION HOLOMORPHIC MORSE INEQUALITIES NEAR DIAGONAL EXPANSION OF THE BERGMAN KERNEL DIAGONAL ASYMPTOTIC EXPANSION OF THE BERGMAN KERNEL SOBOLEV ESTIMATE ON THE RESOLVENT (? ?L T 2)?1 UNIFORM ESTIMATE ON THE BERGMAN KERNEL BERGMAN KERNEL OF L THE COE?CIENT B1: A PROOF OF THEOREM 4 1 2 O?-DIAGONAL EXPANSION OF THE BERGMAN KERNEL FROM HEAT KERNEL TO BERGMAN KERNEL UNIFORM ESTIMATE ON THE HEAT KERNEL AND THE BERGMAN KERNEL THE KODAIRA EMBEDDING THEOREM UNIVERSAL BUNDLES CONVERGENCE OF THE INDUCED FUBINI–STUDY METRICS CLASSICAL PROOF OF THE KODAIRA EMBEDDING THEOREM GRASSMANNIAN EMBEDDING STABILITY AND BERGMAN KERNEL EXTREMAL K¨AHLER METRICS SCALAR CURVATURE AND PROJECTIVE EMBEDDINGS GIESEKER STABILITY AND GRASSMANNIAN EMBEDDINGS DISTRIBUTION OF ZEROS OF RANDOM SECTIONS ORBIFOLD PROJECTIVE EMBEDDING THEOREM BASIC DE?NITIONS ON ORBIFOLDS COMPLEX ORBIFOLDS ASYMPTOTIC EXPANSION OF THE BERGMAN KERNEL PROJECTIVE EMBEDDING THEOREM THE ASYMPTOTIC OF THE ANALYTIC TORSION MELLIN TRANSFORMATION DE?NITION OF THE ANALYTIC TORSION ANOMALY FORMULA THE ASYMPTOTICS OF THE ANALYTIC TORSION ASYMPTOTIC ANOMALY FORMULA FOR THE L2-METRIC UNIFORM ASYMPTOTIC OF THE HEAT KERNEL EXPANSION ON NON-COMPACT MANIFOLDS COMPLETE HERMITIAN MANIFOLDS COVERING MANIFOLDS THE SHI?MAN–JI–BONAVERO–TAKAYAMA CRITERION REVISITED COMPACTI?CATION OF MANIFOLDS FILLING STRONGLY PSEUDOCONCAVE ENDS THE COMPACTI?CATION THEOREM WEAK LEFSCHETZ THEOREMS KERNEL CALCULUS ON CN ASYMPTOTIC EXPANSION OF TOEPLITZ OPERATORS A CRITERION FOR TOEPLITZ OPERATORS ALGEBRA OF TOEPLITZ OPERATORS TOEPLITZ OPERATORS ON NON-COMPACT MANIFOLDS BERGMAN KERNELS OF MODI?ED DIRAC OPERATORS TOEPLITZ OPERATORS ON SYMPLECTIC MANIFOLDS BERGMAN KERNEL: MIXED CURVATURE CASE GENERALIZED BERGMAN KERNEL NEAR DIAGONAL ASYMPTOTIC EXPANSION SYMPLECTIC KODAIRA EMBEDDING THEOREM INTRODUCTION LET X BE A COMPACT COMPLEX MANIFOLD AND L BE A HOLOMORPHIC LINE BUNDLE ON X WE DENOTE BY HQ(X, L) THE QTH COHOMOLOGY GROUP OF THE SHEAF OF HOLOMORPHIC SECTIONS OF L ON X MANY IMPORTANT RESULTS IN ALGEBRAIC AND COMPLEX GEOMETRY ARE DERIVED BY COMBINING A VANISHING PROPERTY WITH AN INDEX THEOREM, OR FROM THE ASYMPTOTIC RESULTS ON THE TENSOR POWERS LP WHEN P ?? ONE OF THE MOST FAMOUS EXAMPLES IS THE KODAIRA–SERRE VANISHING THEOREM WHICH ASSERTS THAT IF L IS POSITIVE, THEN HQ(X, LP) VANISH FOR Q ?1 AND LARGE P THE KEY REMARK IS THAT THE SPECTRUM OF THE KODAIRA–LAPLACE OPERATOR ?P ACTING ON (0, Q)-FORMS, Q ?1, WITH VALUES IN THE TENSOR POWERS LP, SHIFTS TO THE RIGHT LINEARLY IN THE TENSOR POWER P AS A CONSEQUENCE THE KERNEL OF ?P IS TRIVIAL ON FORMS OF HIGHER DEGREE AND THE VANISHING THEOREM FOLLOWS BY THE HODGE THEORY AND THE DOLBEAULT ISOMORPHISM MOREOVER, THE RIEMANN–ROCH–HIRZEBRUCH THEOREM IMPLIES THAT LP HAS A LOT OF HOLOMORPHIC SECTIONS ON X FOR LARGE P, WHICH INDEED EMBED THE MANIFOLD X IN A PROJECTIVE SPACE AN IMPORTANT GENERALIZATION WHICH WE WILL EMPHASIZE IS THE ASYMPTOTIC HOLO- MORPHIC MORSE INEQUALITIES OF DEMAILLY THEY GIVE ASYMPTOTIC BOUNDS ON THE MORSE SUMS OF THE ?-BETTI NUMBERS DIM HQ(X, LP) IN TERMS OF CERTAIN INTEGRALS OF THE CURVATURE FORM OF L THE HOLOMORPHIC MORSE INEQUALITIES PROVIDE A USEFUL TOOL IN COMPLEX GEOMETRY THEY ARE AGAIN BASED ON THE ASYMPTOTIC SPECTRAL BEHAVIOR OF THE KODAIRA–LAPLACE OPERATOR ?P FOR LARGE P THE APPLICATIONS OF THESE VANISHING THEOREMS AND HOLOMORPHIC MORSE INEQUAL- ITIES ARE NUMEROUS LET US MENTION HERE ONLY THE KODAIRA EMBEDDING THEOREM, THE CLASSICAL LEFSCHETZ HYPERPLANE THEOREM FOR PROJECTIVE MANIFOLDS, THE COMPUTATION OF THE ASYMPTOTICS OF THE RAY-SINGER ANALYTIC TORSION BY BISMUT AND VASSEROT, AS WELL AS THE SOLUTION OF THE GRAUERT–RIEMENSCHNEIDER CONJECTURE BY SIU AND DEMAILLY OR THE COMPACTI?CATION OF COMPLETE K¨AHLER MANIFOLDS OF NEGATIVE RICCI CURVATURE BY NADEL AND TSUJI DONALDSON’S WORK ON THE EXISTENCE OF SYMPLECTIC SUBMANIFOLDS WAS INSPIRED BY THE SAME CIRCLE OF IDEAS THE HOLOMORPHIC MORSE INEQUALITIES ARE GLOBAL STATEMENTS WHICH CAN BE DE- DUCED FROM LOCAL INFORMATION SUCH AS THE BEHAVIOR OF THE HEAT OR BERGMAN KERNELS IN THIS RE?NED FORM WE CAN ESTABLISH THE ASYMPTOTIC EXPANSION OF THE BERGMAN KER- NEL ASSOCIATED TO LP AS P ??, WHICH HAVE HAD A TREMENDOUS IMPACT ON RESEARCH IN 2? RLN + O(PN) (1) INTRODUCTION 2? RLN > 0 , (3) (IN PARTICULAR, IF L IS SEMI-POSITIVE AND POSITIVE AT AT LEAST ONE POINT), THERE ARE A LOT OF SECTIONS IN H0(X, LP), WHICH BY TAKING QUOTIENTS DELIVER N INDEPENDENT MEROMORPHIC FUNCTIONS, I E , X IS MOISHEZON IN SECTION 2 4 WE PRESENT AN ALGEBRAIC REFORMULATION OF THE HOLOMORPHIC MORSE INEQUALITIES IN CHAPTER 3 WE PROVE THE MORSE INEQUALITIES FOR THE DOLBEAULT L2-COHO- MOLOGY SPACES FOR A NON-COMPACT MANIFOLD SATISFYING THE FUNDAMENTAL ESTIMATE (POINCAR´E INEQUALITY) AT IN?NITY USING THIS MORE ABSTRACT FORMULATION OF THE MORSE INEQUALITIES, WE CAN ?ND A LOWER BOUND FOR THE GROWTH OF THE HOLOMORPHIC SECTION 2? RL DE?NES A K¨AHLER FORM ON X, WHERE RL IS THE CURVATURE OF THE HOLOMORPHIC HERMITIAN CONNECTION ?L ON (L, HL) IN THE REST OF THE INTRODUCTION WE DENOTE BY GT X THE ASSOCIATED K¨AHLER METRIC TO ? ON TX WE ALSO LET E BE A HOLOMORPHIC VECTOR BUNDLE ON X WITH A HERMITIAN METRIC HE SINCE L IS POSITIVE, THE KODAIRA–SERRE VANISHING THEOREM SHOWS THAT HQ(X, LP ?E) = |SP I (X)|2, IF E = C (6) THE BERGMAN KERNEL HAS BEEN STUDIED BY TIAN, YAU, BOUCHE, RUAN, CATLIN, ZEL- DITCH, LU, WANG, AND MANY OTHERS, IN VARIOUS GENERALITIES, ESTABLISHING THE ASYM- PTOTIC EXPANSION FOR HIGH POWERS OF L MOREOVER, IT WAS DISCOVERED THAT THE CO- E?CIENTS IN THE ASYMPTOTIC EXPANSION ENCODE GEOMETRIC INFORMATION ABOUT THE UNDERLYING COMPLEX PROJECTIVE MANIFOLDS OUR APPROACH TO THE STUDY OF THE ASYMPTOTIC EXPANSION CONTINUES THE METHOD APPLIED IN CHAPTER 1 WE TREAT BOTH THE DIRAC OPERATOR AND THE KODAIRA LAPLACIAN IN THE SAME TIME BY MEANS OF THE MODI?ED DIRAC OPERATOR THE KEY POINT OF OUR METHOD IS THAT THE SPECTRUM SPEC(?P) OF ?P (OR OF THE HALF OF THE SQUARE OF THE INTRODUCTION BR(X)PN?R, (8) WHERE BR(X) ?END(E)X ARE SMOOTH COE?CIENTS, WHICH ARE POLYNOMIALS IN RT X, RE AND THEIR DERIVATIVES WITH ORDER ?2R ?2 MOREOVER B0 = IDE, B1 = 4? 2RE(WJ, WJ) + 2RX IDE , (9) WHERE RX IS THE SCALAR CURVATURE OF (TX, GT X) AND {WJ}N J=1 IS AN ORTHONORMAL BASIS OF T (1,0)X IN THE CASE OF TRIVIAL BUNDLE E THE TERM B1 WAS CALCULATED BY LU AND USED BY DONALDSON IN HIS WORK ON THE EXISTENCE OF K¨AHLER METRICS WITH CONSTANT SCALAR CURVATURE WE ALSO ?ND THE FULL O?-DIAGONAL EXPANSION OF THE BERGMAN KERNEL PP(X, X?) WITH THE HELP OF THE HEAT KERNEL IN CHAPTER 5, WE STUDY IN DETAIL THE METRIC ASPECT OF THE KODAIRA MAP AS AN APPLICATION OF THE ASYMPTOTIC EXPANSION OF THE BERGMAN KERNEL FIRST, WE PRESENT AN ANALYTIC PROOF OF THE KODAIRA EMBEDDING THEOREM FOLLOWING AN ORIGINAL IDEA OF BOUCHE, AND WE STUDY THE CONVERGENCE OF THE INDUCED FUBINI–STUDY METRIC THEN THE KODAIRA MAP ?P : X ??P(H0(X, LP)?), DE?NED BY ?P(X) = {S ?H0(X, LP) : S(X) = 0} FOR X ?X, IS AN EMBEDDING FOR P LARGE ENOUGH AND FOR ANY L ?N, THERE EXISTS CL > 0 SUCH THAT