Leading terms in the heat invariants for the Laplacians of the De Rham signature, and spin complexes.

. Let D be a vector bundle-valued differential operator with positive definite leading symbol on a compact, Riemannian manifold. Asymptotic expansions of the kernel function and L 2 trace of the heat operator e \ , t ? 0, naturally lead to sequences of homogeneous local and global scalar invariants an (x; D), an (D) j R an (x; D), n 2 N. Within each homogeneity class of local invariants, there is a filtration by degree; the lowest-degree terms in an (x; D) are linear, those in an (D) quadratic. Information about such leading terms has been crucial in the work of Osgood, Phillips, and Sarnak, of Brooks, Chang, Perry, and Yang, and of Melrose on compactness problems for isospectral sets of metrics and domains, modulo gauge equivalence, in dimensions two and three. We specialize our earlier general results to give the leading terms in the heat invariants produced by the Laplacians of the de Rham, signature, and spin complexes. A main technical point is a calculation of fiber ...