Jean Pierre DEMAILLY Universite de Grenoble I

Jean-Pierre Demailly

Description

Niveau: Supérieur, Doctorat, Bac+8
HOLOMORPHIC MORSE INEQUALITIES Jean-Pierre DEMAILLY, Universite de Grenoble I Series of Lectures given at the AMS Summer Institute held in Santa Cruz, California, July 1989. 1. Introduction Let M be a compact C∞ manifold, dimR M = m, and h a Morse function, i.e. a function such that all critical points are non degenerate. The standard Morse inequalities relate the Betti numbers bq = dimHqDR(M,R) and the numbers sq = _ critical points of index q , where the index of a critical point is the number of negative eigenvalues of the Hessian form (∂2h/∂xi∂xj). Specifically, the following “strong Morse inequalities” hold : (1.1) bq ? bq?1 + · · ·+ (?1)qb0 6 sq ? sq?1 + · · ·+ (?1)qs0 for each integer q > 0. As a consequence, one recovers the “weak Morse inequali- ties” bq 6 sq and the expression of the Euler-Poincare characteristic (1.2) ?(M) = b0 ? b1 + · · ·+ (?1)mbm = s0 ? s1 + · · ·+ (?1)msm . The purpose of these lectures is to explain what are the complex analogues of these inequalities for ∂?cohomology groups with values in holomorphic line (or vector) bundles, and to present a few applications.

• hermitian connection


• complex vector


• atiyah-bott-patodi's proof


• compact c∞


• morse inequalities


• connection corresponding


• over abelian


• ?? ?


• line bundles

Sujets

• Demailly
• Bouche
• Hermitian connection
• Vector space

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 11
• Langue: English

Extrait

HOLOMORPHIC MORSE INEQUALITIES

Jean-Pierre DEMAILLY leIenobinU,sreve´tirGed

Series of Lectures given at the AMS Summer Institute held California, July 1989.

1. Introduction

in

Santa

Cruz,

LetMbe a compactC ∞ manifold, dim R M=m, andha Morse function, i.e.points are non degenerate. The standard Morsea function such that all critical inequalities relate the Betti numbersb q = dimH R q D (MR) and the numbers s q = # critical points of indexq  where the index of a critical point is the number of negative eigenvalues of the Hessian form (∂ 2 h∂x i ∂x j ). Specifically, the following “strong Morse inequalities” hold : (11)b q −b q−1 +  + (−1) q b 0 6s q −s q−1 +  + (−1) q s 0 for each integerq>0. As a consequence, one recovers the “weak Morse inequali- ties”b q 6s q tsiretcarahce´raicisnorpseehxenatdoincer-PeEulofth (12)χ(M) =b 0 −b 1 +  + (−1) m b m =s 0 −s 1 +  + (−1) m s m  The purpose of these lectures is to explain what are the complex analogues of these inequalities for∂−groups with values in holomorphic line (orcohomology vector) bundles, and to present a few applications.

LetXbe a compact complex manifold,n= dim C XandE Fholomorphic vector bundles overXwith

rankE= 1rankF=r  We denote hereh q (F) = dimH q (XO(F))

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Assume thatEis endowed with aC ∞ hermitian metric and denote byc(E) its curvature form. Thenic(E) is a real (11)−form onX(cf.§2). Finally, consider theq−index sets ( E) = ( x∈X;ic(E) x hasqvitagieeavneseul esl n u e v g a )  X q n−qpositive eigen X(6q E) = [ X(j E) 16j6q

Observe thatX(q E) andX(6q E) are open subsets ofX. Main Theorem. —The sequence of tensor powersE k ⊗Fsatisfy the following asymptotic estimates ask→+∞:

(1.3) Weak Morse inequalities: h q (E k ⊗F)6 k r n n ! Z X(qE (−1) q  2 i c π (E)  n +o(k n ) ) (1.4) Strong Morse inequalities: 06 X j6q (−1) q−j h j (E k ⊗F)6 k r n n ! Z X(6qE) (−1) q  2 c π i (E)  n +o(k n )

(1.5) Asymptotic Riemann-Roch formula: χ(E k ⊗F) = n r k n ! Z X  2 c π i (E)  n +o(k n )

Observe that (1.5) is in fact a weak consequence of the Hirzebruch-Riemann- Roch formula.

The above theorem was first proved in [De 2] in 1985. It was largely motivated by Siu’s solution of the Grauert-Riemenschneider conjecture ([Siu 2], 1984), which we will reprove below as a special case of a stronger statement. The basic tool is a spectral theorem which describes the eigenvalue distribution of the complex Laplace-Beltrami operators. The original proof of [De 2] was based partly on Siu’s techniques and partly on an extension of Witten’s analytic proof [Wi, 1982] for the standard Morse inequalities. Somewhat later Bismut [Bi] and quite recently Getzler [Ge 3] gave new proofs, both relying on an analysis of the heat kernel in the spirit of Atiyah-Bott-Patodi’s proof of the Atiyah-Singer index theorem [A-B-P]. Although the basic idea is simple, Bismut used deep results arising from probability theory (the Malliavin calculus), while Getzler relied on his supersymmetric symbolic calculus for spin pseudodifferential operators [Ge 1].

We will try to present here a simple heat equation proof, based essentially on Mehler’s formula and elementary asymptotic estimates (cf.§3 and§4). The next sections deal with various generalizations and applications :

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