A closed formula for the asymptotic expansion of the Bergman kernel

TL;DR

This paper derives a graph-theoretic closed-form expression for the coefficients in the asymptotic expansion of the Bergman kernel, linking combinatorics and complex geometry.

Contribution

It introduces a novel combinatorial formula for the expansion coefficients using characteristic polynomials of directed graphs.

Findings

Provides a closed-form formula for Bergman kernel coefficients

Connects graph theory with complex geometric invariants

Offers a combinatorial interpretation of recursive formulas

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.