Asymptotic properties of Bergman kernels for potentials with Gevrey regularity

Abstract

We study the asymptotic properties of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler 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 $k^{-\frac{1}{2}+\frac{1}{4a+4\epsilon }}$ for every $\epsilon >0$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $(\frac{\log k}{k}{)}^{\frac{1}{2}}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^{m}m{!}^{2a+2\epsilon }$ 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.

arXiv admin note: text overlap with arXiv:1705.09281