  We study the asymptotic properties of the Bergman kernels associated to
tensor powers of a positive line bundle on a compact K\"ahler manifold. We show
that if the K\"ahler 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 $k^{-\frac12+\frac{1}{4a+4\varepsilon}}$ for
every $\varepsilon>0$. These improve the earlier results in the subject for
smooth potentials, where an expansion exists in a $(\frac{\log
k}{k})^{\frac12}$ neighborhood of the diagonal. We obtain our results by
finding upper bounds of the form $C^m m!^{2a+2\varepsilon}$ 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.
