Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles

Abstract

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kähler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

Bergman kernel and Kähler tensor calculus

“…4.5], we proved that the stabilization graph of a semistable (pointed) graph G is strong if and only if G is strong. So the corollary follows from (32) and (33).…”

The Second Coefficient of the Asymptotic Expansion of the Weighted Bergman Kernel for (0,q) Forms on $\mathbb{C^{n}}$

“…When M is complete and L is uniformly positive on M with √ −1R K * M and ∂Θ bounded below, where R K * M is the curvature of the bundle of (n, 0) forms, Ma-Marinescu [15] obtained the asymptotic expansion of the Bergman kernel for q = n − = 0. More generally, if M is any complex manifold and (q) k has O(k −n 0 ) small spectral gap on an open set D ⋐ M (see Definition 1.5 in [9], for the precise meaning of O(k −n 0 ) small spectral gap), then it is known by a recent result (see Theorem 1.6 in [9]) that the Bergman kernel admits a full asymptotic expansion in k on D. The coefficients of these expansions turned out to be deeply related to various problem in complex geometry (see e.g. [3], [4], [5]).…”

“…We write [9], for the precise meaning of O(k −n 0 ) small spectral gap). From this observation and Theorem 4.12, Theorem 4.14 in [9], we deduce the following…”

Bergman Kernel Asymptotics and a Pure Analytic Proof of the Kodaira Embedding Theorem

“…Let P k,M \Σ be the associated Bergman projection on M \ Σ and let P k,M \Σ (x) be the associated Bergman kernel function. In [8], we showed that…”

“…In this paper, we survey recent results in [8] about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem. …”