The second coefficient of the asymptotic expansion of the weighted Bergman kernel for $(0,q)$ forms on $\Complex^n$

TL;DR

This paper computes the trace of the second coefficient in the asymptotic expansion of the weighted Bergman kernel for (0,q) forms on complex n-space, under certain non-degeneracy conditions on the weight function.

Contribution

It provides an explicit calculation of the trace of the second coefficient in the asymptotic expansion for the weighted Bergman kernel on ,q forms, extending known results for the leading term.

Findings

Explicit formula for the trace of the second coefficient

Extension of asymptotic expansion results to higher-order terms

Insights into the structure of Bergman kernels under non-degenerate conditions

Abstract

Let $\phi\in C^\infty(\Complex^n)$ be a given real valued function. We assume that $\pr\ddbar\phi$ is non-degenerate of constant signature $(n_-,n_+)$ on $\Complex^n$. When $q=n_-$, it is well-known that the Bergman kernel for $(0,q)$ forms with respect to the $k$-th weight $e^{-2k\phi}$, $k>0$, admits a full asymptotic expansion in $k$. In this paper, we compute the trace of the second coefficient of the asymptotic expansion on the diagonal.