Subelliptic Spinℂ Dirac operators, III the Atiyah-Weinstein conjecture

Abstract

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, T+eo, which give subelliptic Predholm problems for the Spinℂ-Dirac operator, ∂̃+eo. We introduce a generalization of Predholm pairs to the "tame" category. In this context, we show that the index of the graph closure of (∂̃+eo, R +eo) equals the relative index, on the boundary, between R+eo and the Calderón projector, P+eo. Using the relative index formalism, and in particular, the comparison operator, τ+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 Szego projectors on bX0, bX1, respectively, and R 0eo, R1eo, the subelliptic boundary conditions they define. If X̄1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 II φ X̄1 has a canonical Spinℂ-structure and Dirac operator, ∂̃Xeo. Applying these results and those of our previous papers we obtain a formula for the relative index, R-Ind(S0, φ* S1), (1) R-Ind(S0, φ*S1) = Ind(∂̃Xex) - Ind(∂̃X0e, R0e) + Ind(∂̃X1e, R1e). For the special case that X0 and X1 are strictly pseudoconvex complex manifolds and So and Si are the classical Szego projectors defined by the complex structures this formula implies that (2) R-M(S0, φ*S1) = Ind(∂̃Xe) - χ′Ο(X0) + χ′Ο(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].