%0 Journal Article
%T Scaling asymptotics of heat kernels of line bundles
%A Xiaonan Ma
%A George Marinescu
%A Steve Zelditch
%J Mathematics
%D 2014
%I arXiv
%X We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal. It is a generalization of the Bergman/Szeg\"o kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the Hadamard parametrix for the heat kernel on a principal bundle and the second based on the analytic localization of the Dirac-Dolbeault operator.
%U http://arxiv.org/abs/1406.0201v1