Scaling asymptotics of heat kernels of line bundles

Abstract

We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\square }_p^q$ 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 $\exp (-u{\square }_p^q/p)(x,x)$ along the diagonal. It is a generalization of the Bergman/Szegö 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.

arXiv commentDedicated to D.H. Phong on his 60th birthday. To appear in the Contemp. Math. volume in honor of Phong (Paul Feehand, ed.)

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