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arXiv · 1406.0201

Scaling asymptotics of heat kernels of line bundles

Xiaonan Ma·George Marinescu·Steve Zelditch
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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/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.

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Xiaonan Ma, George Marinescu, Steve Zelditch. 2014-06-01. Scaling asymptotics of heat kernels of line bundles. https://doi.org/10.1090/conm%2F644

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