Heat kernel asymptotics for Kodaira Laplacians of high power of line bundle over complex manifolds

2023/11/05 by Wang, Huan, Zhu, Weixia
#32W05 #35J05 #35P15 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.02548

Abstract

This paper presents a simple method to prove the heat kernel asymptotics for the Kodaira Laplacian with respect to the high power of a holomorphic Hermitian line bundle (L,h^L) over a possibly non-compact Hermitian manifold (M,ω). As a consequence, we give a direct proof of the holomorphic Morse inequalities on covering manifolds. Furthermore, we generalize it to the vector bundle via the L² Le Potier isomorphism and provide an algebraic version of the holomorphic Morse inequalities. The approach used in this work employs a scaling technique and is applicable to M regardless of its compactness.

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