Subelliptic SpinC Dirac operators, III The Atiyah-Weinstein conjecture

In this paper we extend the results obtained in [9, 10] to manifolds with SpinC-structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there are modified ∂̄-Neumann boundary conditions defined by projection operators, Reo + , which give subelliptic Fredholm problems for the SpinC-Dirac operator, ð eo + . We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of (ð + ,Reo + ) equals the relative index, on the boundary, between Reo + and the Calderon projector, Peo + . Using the relative index formalism, and in particular, the comparison operator, T eo + , introduced in [9, 10], we prove a trace formula for the relative index that generalizes the classical formula for the index of an elliptic operator. Let (X0, J0) and (X1, J1) be strictly pseudoconvex, almost complex manifolds, with φ : bX1 → bX0, a contact diffeomorphism. Let S0,S1 denote generalized Szegő projectors on bX0, bX1, respectively, and Reo 0 , Reo 1 , the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 ∐φ X1 has a canonical SpinC-structure and