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Let be a compact complex manifold and be a holomorphic line bundle on . We denote by () the th cohomology group of the sheaf of holomorphic sections of on .

The first aim of this chapter is to provide the background material on differential geometry for the whole book. Then, in the last two sections, we present a heat kernel proof of Demailly’s holomorphic Morse inequalities, Theorem 1.7.1.

In this chapter we start some basic facts on analytic and complex geometry (divisors, blowing-up, big line bundles), we prove the theorem of Siegel-Remmert-Thimm, that the field of meromorphic functions on a connected compact complex manifold is an algebraic field of transcendence degree less than the dimension of the manifold. Then we study in more detail Moishezon manifolds and their relation to projective manifolds. In particular we prove that a Moishezon manifold is projective if and only if it carries a Kähler metric. We end the section 2.2 by giving the solution of the Grauert-Riemenschneider conjecture as application of the holomorphic Morse inequalities from Theorem 1.7.1.

We start by the Hodge theory on non-compact Hermitian manifolds in Section 3.1. In Section 3.2, we prove holomorphic Morse inequalities for the -cohomology in a quite general context, namely, when the fundamental estimate (3.2.2) holds. This gives a fairly general method which may be applied in many situations. The main idea, going back to Witten, is to show that the spectral spaces of the Laplacian, corresponding to small eigenvalues, inject in the spectral spaces of the Laplacian with Dirichlet boundary conditions on a smooth relatively domain, The asymptotic of the latter operator is calculated in Theorem 3.2.9. For a compact manifold we recover of course Theorem 1.7.1.

In this chapter, we establish the asymptotic expansion of the Bergman kernel associated to high tensor powers of a positive line bundle on a compact complex manifold. Thanks to the spectral gap property of the Kodaira Laplacian, Theorem 1.5.5, we can use the finite propagation speed of solutions of hyperbolic equations, (Theorem D.2.1), to localize our problem to a problem on ℝ. Comparing with Section 1.6, the key point here is that we need to extend the connection of the line bundle such that its curvature becomes uniformly positive on ℝ. Then we still have the spectral gap property on ℝ. Thus we can instead study the Bergman kernel on ℝ (cf. (4.1.27)), and use various resolvent representations (4.1.59), (4.2.22) of the Bergman kernel on ℝ. We conclude our results by employing functional analysis resolvent techniques.

In this chapter we present some applications of the asymptotic expansion of the Bergman kernel.

We show in Section 6.1 that the asymptotic expansion of the Bergman kernel still holds on compact sets of certain non-compact complete manifolds. In this way we can obtain another proof of some of the holomorphic Morse inequalities. As a corollary, we re-prove the Shiffman-Ji-Bonavero-Takayama criterion for Moishezon manifolds in Section 6.2.

We show in this chapter how the asymptotic expansion of the Bergman kernel implies the semi-classical properties of Toeplitz operators acting on high tensor powers of a positive line bundle over a compact manifold. In particular we obtain a construction of a star-product (a deformation quantization) using this technique. Moreover, our approach works with some modifications on non-compact and symplectic manifolds.

In this chapter, we study the asymptotic expansion of the Bergman kernel associated to modified Dirac operators and renormalized Bochner Laplacians on symplectic manifolds. We will also explain some applications of the asymptotic expansion in the symplectic case. One is, for example, the extension of the Berezin-Toeplitz quantization studied in Chapter 7. We also find Donaldson’s Hermitian scalar curvature as the second coefficient of the expansion.