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 partial derivative-Neumannn boundary conditions defined by projection operators, R-+(eo), which give subelliptic Fredholm problems for the Spin(C)-Dirac operator, partial derivative(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 (partial derivative(eo)(+), R-+(eo)) equals the relative index, on the boundary, between Re-+(eo) and the Calderon projector, P-+(eo). 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 (X-0, J(0)) and (X-1, J(1)) be strictly pseudoconvex, almost complex manifolds, with phi : bX(1) -> bX(0), a contact diffeomorphism. Let S-0, S-1 denote generalized Szego projectors on bX(0), bX(1), respectively, and R-0(eo), R-1(eo), the subelliptic boundary conditions they define. If (X) over bar (1) is the manifold X, with its orientation reversed, then the glued manifold X = X-0 II phi (X) over bar (1) has a canonical Spin(C)-structure and Dirac operator, partial derivative(eo)(X). Applying these results and those of our previous papers we obtain a formula for the relative index, R-Ind(S-0, phi*S-1), (1) R-Ind(S-0, phi* S-1) = Ind(partial derivative(e)(X)) - Ind(partial derivative(e)(X0), R-0(e)) + Ind(partial derivative(e)(X1), R-1(e)). For the special case that X-0 and X-1 are strictly pseudoconvex complex manifolds and S-0 and S-1 are the classical Szego projectors defined by the complex structures this formula implies that (2) R-Ind(S-0, phi*S-1) = Ind (partial derivative(e)(X)) - X-O' (X-0) + X-O' (X-1),which is essentially the formula conjectured by Atiyah and Weinstein; see [37]. We show that, for the case of embeddable CR-structures on a compact, contact 3-manifold, this formula specializes to show that the boundedness conjecture for relative indices from [7] reduces to a conjecture of Stipsicz concerning the Euler numbers and signatures of Stein surfaces with a given contact boundary; see [35].