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 − 1 2 + 1 4a+4ε for every ε > 0. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a ( log k k )1 2 neighborhood of the diagonal. We obtain our results by finding upper bounds of the form C m m! 2a+2ε for the Bergman coefficients bm(x, ȳ) in a fixed neighborhood by the method of [BeBeSj08]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results.