An inhomogeneous elliptic complex

Abstract

We define a 3 term sequence P of differential operators of mixed type; the first and third operators are 1st order while the second operator is 2nd order. P is always elliptic; it forms a complex if M is einstein. It was first discussed by Gasqui. P is related to similar complexes C and G discussed by 02 Calabi and Gasqui-Goldschmidt. The index and equivariant index of P vanish. In dimension 2, P=C⊗s where S is of Dirac type;C and-S determine the same equivariant index. We study the heat equation asymptotics of the operators of P; the associated Laplacians do not have scalar leading symbol.