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 $p\to \infty$, $B_p(x)$ has the asymptotic expansion $\sum_{j=0}^\infty b_j(x) p^{n-j}$. 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.