Bergman kernels and equilibrium measures for line bundles over projective manifolds

By Robert J. Berman. 2007

122 citations recorded by the source corpus.

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/010ac311bd4c895c8a6c8640003a8309500fb672

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Growth of balls of holomorphic sections and energy at equilibrium (2008, methodology): “Since this result plays a crucial role in what follows, we will sketch its proof for the convenience of the reader, and refer to [6] for the complete proof—a slightly more involved one in fact since Fujita’s theorem is not used there but rather given a direct proof by analytic means.”
Monte Carlo with determinantal point processes (2016, background): “Although DPPs on higher-dimensional supports have attracted attention in complex geometry [Berman, 2009a,b, 2013, 2014], in statistics [Lavancier et al., 2015, Møller et al., 2015], and in physics [Torquato et al., 2008, Scardicchio et al., 2009], it seems no CLT has been established yet when d > 3.”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, unknown): “We will be very brief about this well-known notion, and refer e.g. to [ Bou04 , § 2.8], [ BBGZ13 , § 1], [ Ber09 ] or [ BD12 ] for more details and recent results.”
A direct approach to Bergman kernel asymptotics for positive line bundles (2008, methodology): “11) After this paper was written this observation was used in [2] to obtain an asymptotic expansion of Kk(x, y) on a certain subset of X(0) for any Hermitian line bundle (L, φ) over a projective manifold X .”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “(cf. [4, Theorem 1.5] for the case K = X ).”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, methodology): “We use the idea of Berman [3] (see also [31] for a similar idea in the context of real Monge–Ampère equations on domains in Rn) who proved C1,γloc (M\EnK(α)) regularity for all 0 < γ < 1.”