HEAT EQUATION ASYMPTOTICS IN KAEHLER GEOMETRY

Suppose we are given a differential operator P on a manifold M. Certain natural conditions on P guarantee the existence of the fundamental solution to the associated heat equation. It is well known that the coefficients in the asymptotic expansion of this solution give links between the spectrum of P and the geometry of M.;We consider the real Laplacian acting on forms of type (p,q) for a Kaehler manifold M and calculate the fourth term in the asymptotic expansion of the heat kernel. Several applications are derived, including a spectral criterion for M to be symmetric.