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

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 $(M,\omega )$. 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 $L^2$ 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.