Off-diagonal asymptotic properties of Bergman kernels associated to analytic K\"ahler potentials

TL;DR

This paper establishes new off-diagonal asymptotic expansions for Bergman kernels associated with positive line bundles on compact K"ahler manifolds, especially for real analytic potentials, improving previous smooth potential results.

Contribution

It provides sharper off-diagonal asymptotics for Bergman kernels with real analytic potentials and introduces bounds on Bergman coefficients using advanced methods.

Findings

Asymptotic expansion in a neighborhood of size $k^{-1/4}$ for real analytic potentials.

Improved bounds on Bergman coefficients with factorial growth.

Explicit formulas for Bergman kernels in special curvature cases.

Abstract

We prove a new off-diagonal asymptotic 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 real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^{-\frac14}$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $k^{-\frac12}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^m m!^{2}$ for the Bergman coefficients $b_m(x, \bar y)$, which is an interesting problem on its own. We find such upper bounds using the method of Berman-Berndtsson-Sj\"ostrand. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal $x=y$ in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as $k \to \infty$) neighborhood of the diagonal, which recovers a result of Berman [Ber] (see Remark 3.5 of [Ber] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e^{-k \delta} )$.