Analysis, Complex Geometry, and Mathematical Physics

DOI10.1090/CONM/644zbMATH Open1322.53004arXiv1406.0201OpenAlexW3124082474MaRDI QIDQ2942152

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Publication date: 18 August 2015

Published in: (Search for Journal in Brave)

Abstract: We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${Box}_p^q$ 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 $exp(-u{Box}_p^q/p)(x,x)$ 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.

Full work available at URL: https://arxiv.org/abs/1406.0201

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