UC Irvine

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 14 These improve the earlier results in the subject for smooth potentials, where an expansion exists in a 7kappa; neighborhood of the diagonal. We obtain our results by finding upper bounds of the form Cmm2 for the Bergman coefficients bm(x,) which is an interesting problem on its own. We find such upper bounds using the method of [3]. 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 (ask→∞) neighborhood of the diagonal, which recovers a result of Berman [2] (see Remark 3.5 of [2] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e-kδ) $.