A closed formula for the asymptotic expansion of the Bergman kernel

Abstract

We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Englis and A. Loi.

25 pages, final version, Comm. Math. Phys

References in corpus (2)

• The SzegÖ kernel on an orbifold circle bundle
• On a formula of Gammelgaard for Berezin-Toeplitz quantization

Cited by in corpus (29)

• Geometry of Quantum Hall States: Gravitational Anomaly and Kinetic Coefficients
• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Random normal matrices, Bergman kernel and projective embeddings
• On canonical metrics on Cartan-Hartogs domains
• Asymptotic expansion of polyanalytic Bergman kernels
• Balanced metrics on some Hartogs type domains over bounded symmetric domains
• The Simanca metric admits a regular quantization
• On the third coefficient of TYZ expansion for radial scalar flat metrics
• On a formula of Gammelgaard for Berezin-Toeplitz quantization
• On a property of Bergman kernels when the Kähler potential is analytic
• Geometry and large N limits in Laughlin states
• Quantitative upper bounds for Bergman kernels associated to smooth Kähler potentials
• Off-diagonal asymptotic properties of Bergman kernels associated to analytic Kähler potentials
• Limits of balanced metrics on vector bundles and polarised manifolds
• Weyl invariant polynomial and deformation quantization on Kahler manifolds
• Abstract Bergman kernel expansion and its applications
• An invariant formula for a star product with separation of variables
• On the asymptotic behavior of Bergman kernels for positive line bundles
• The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle
• Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles
• Heat kernel coefficients on Kahler manifolds
• The first terms in the expansion of the Bergman kernel in higher degrees
• Asymptotic properties of Bergman kernels for potentials with Gevrey regularity
• On Homothetic Balanced Metrics
• Liouville perturbation theory for Laughlin state and Coulomb gas
• Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds
• Two conjectures on Ricci-flat Kaehler metrics
• Remarks on the canonical metrics on the Cartan-Hartogs domains
• The regular quantizations of certain holomorphic bundles