Bergman kernel, Heat kernel and constant scalar curvature

Let $L$ be a positive holomorphic line bundle on a compact Kähler manifold $X$. The Bergman kernel is $B_p(x)=\sum _i|s_i{|}^2$ with $s_i$ an orthonormal basis of $H^0(X,L^p)$. Theorem (Zelditch) : There exist soom\tm function $b_j$ on $X$ such that as $p\to \mathrm{\infty }$, $B_p(x)$ has the asymptotic expansion $\sum _{j=0}^{\mathrm{\infty }}{b}_j(x)p^{n-j}$. It plays a very important role in Donaldson's work on the relation of constant scalar curvature and the balance condition for the projective embedding.

In this talk, I will explain how to get Zelditch's Theorem from the asymptotic expansion of the heat kernel, and its generalization to the symplectic case.