Off-diagonal asymptotic properties of Bergman kernels associated to analytic Kähler potentials

Abstract

We prove a new off-diagonal asymptotic 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 real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^{-\frac{1}{4}}$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $k^{-\frac{1}{2}}$ 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-Sjöstrand. 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 $O(e^{-k\delta })$.

Cited by in corpus (1)

• The Bergman kernel in constant curvature