Bergman kernel, Heat kernel and constant scalar curvature

  • Xiaonan Ma (Ecole Polytechnique, Palaiseau, France)
A3 01 (Sophus-Lie room)


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.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail