Pointwise Weyl Law for Partial Bergman Kernels

Algebraic and Analytic Microlocal Analysis - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-030-01588-6_13 ◽

2018 ◽

pp. 589-634 ◽

Cited By ~ 2

Author(s):

Steve Zelditch ◽

Peng Zhou

Keyword(s):

Bergman Kernels ◽

Weyl Law

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Weyl law for semi-classical resonances with randomly perturbed potentials

Mémoires de la Société mathématique de France ◽

10.24033/msmf.446 ◽

2014 ◽

Vol 1 ◽

pp. 1-144 ◽

Cited By ~ 3

Author(s):

Johannes Sjöstrand

Keyword(s):

Weyl Law

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BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

International Journal of Mathematics ◽

10.1142/s0129167x10005933 ◽

2010 ◽

Vol 21 (01) ◽

pp. 77-115 ◽

Cited By ~ 2

Author(s):

ROBERT J. BERMAN

Keyword(s):

Toeplitz Operators ◽

Equilibrium Measure ◽

Markov Property ◽

General Setting ◽

Space Structure ◽

Tensor Power ◽

Bergman Kernels ◽

Equilibrium Measures ◽

Holomorphic Sections ◽

Positive Line Bundle

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

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