Asymptotic properties of Bergman kernels for potentials with Gevrey regularity

TL;DR

This paper investigates the asymptotic behavior of Bergman kernels for positive line bundles on Kähler manifolds with Gevrey regular potentials, showing improved expansion neighborhoods compared to smooth cases.

Contribution

It establishes new asymptotic expansion results for Bergman kernels with Gevrey class potentials, extending previous smooth potential results and providing bounds on Bergman coefficients.

Findings

Bergman kernels admit asymptotic expansions in shrinking neighborhoods for Gevrey potentials.

Derived upper bounds for Bergman coefficients involving factorial growth.

Results improve the size of neighborhoods where expansions are valid compared to smooth potentials.

Abstract

We study the asymptotic properties of the Bergman kernels associated to tensor powers of a positive line bundle on a compact K\"ahler manifold. We show that if the K\"ahler potential is in Gevrey class $G^a$ for some $a>1$, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^{-\frac12+\frac{1}{4a+4\varepsilon}}$ for every $\varepsilon>0$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $(\frac{\log k}{k})^{\frac12}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^m m!^{2a+2\varepsilon}$ for the Bergman coefficients $b_m(x, \bar y)$ in a fixed neighborhood by the method of \cite{BBS}. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal $x=y$ in our results.