Scaling asymptotics of heat kernels of line bundles

TL;DR

This paper derives a comprehensive asymptotic expansion for the heat kernel of the Kodaira Laplacian on line bundles over complex manifolds, extending classical results without requiring positivity of the line bundle.

Contribution

It provides the first complete asymptotic expansion of the heat kernel for general Hermitian holomorphic line bundles, generalizing Bergman kernel asymptotics without positivity assumptions.

Findings

Asymptotic expansion for the heat kernel along the diagonal

Generalization of Bergman/Szeg"o kernel asymptotics

Two different proofs provided for the main result

Abstract

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.