Asymptotic expansion of the off-diagonal Bergman kernel on compact K\"ahler manifolds

TL;DR

This paper calculates the first four coefficients of the off-diagonal Bergman kernel expansion on compact Kähler manifolds, revealing polynomial structure and the vanishing of a specific coefficient under certain conditions.

Contribution

It provides explicit formulas for the initial coefficients of the asymptotic expansion and shows their polynomial nature and homogeneity properties.

Findings

First four coefficients explicitly computed

Coefficient b_1 vanishes in K-frame

Coefficients are polynomials in curvature and derivatives

Abstract

We compute the first four coefficients of the asymptotic off-diagonal expansion of the Bergman kernel for the N-th power of a positive line bundle on a compact Kaehler manifold, and we show that the coefficient b_1 of the N^{-1/2} term vanishes when we use a K-frame. We also show that all the coefficients of the expansion are polynomials in the K-coordinates and the covariant derivatives of the curvature and are homogeneous with respect to the weight w.