  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/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.
