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We study the asymptotic properties of the Bergman kernels associated totensor powers of a positive line bundle on a compact K\"ahler manifold. We showthat if the K\"ahler potential is in Gevrey class $G^a$ for some $a>1$, thenthe Bergman kernel accepts a complete asymptotic expansion in a neighborhood ofthe diagonal of shrinking size $k^{-\frac12+\frac{1}{4a+4\varepsilon}}$ forevery $\varepsilon>0$. These improve the earlier results in the subject forsmooth potentials, where an expansion exists in a $(\frac{\logk}{k})^{\frac12}$ neighborhood of the diagonal. We obtain our results byfinding upper bounds of the form $C^m m!^{2a+2\varepsilon}$ for the Bergmancoefficients $b_m(x, \bar y)$ in a fixed neighborhood by the method of\cite{BBS}. We also show that sharpening these upper bounds would improve therate of shrinking neighborhoods of the diagonal $x=y$ in our results.

Asymptotic properties of Bergman kernels for potentials with Gevrey regularity