Subelliptic $\mathrm{Spin}_{\mathbb{C}}$ Dirac operators, III. The Atiyah–Weinstein conjecture | Annals of Mathematics

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, eo+, which give subelliptic Fredholm problems for the Spinℂ-Dirac operator, ðeo+. We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of (ðeo+,eo+) equals the relative index, on the boundary, between eo+ and the Calderón projector, eo+. Using the relative index formalism, and in particular, the comparison operator, 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 0,1 denote generalized Szegő projectors on bX0,bX1, respectively, and eo0, eo1, the subelliptic boundary conditions they define. If X1⎯⎯⎯⎯ is the manifold X1 with its orientation reversed, then the glued manifold X=X0⨿ϕX1⎯⎯⎯⎯ has a canonical Spinℂ-structure and Dirac operator, ðeoX. Applying these results and those of our previous papers we obtain a formula for the relative index, R−Ind(0,ϕ∗1), R−ind(0,ϕ∗1)=Ind(ðeX)−Ind(ðeX0,e0)+Ind(ðeX1,e1). For the special case that X0 and X1 are strictly pseudoconvex complex manifolds and 0 and 1 are the classical Szegő projectors defined by the complex structures this formula implies that R−ind(0,ϕ∗1)=Ind(ðeX)−χ′(X0)+χ′(X1), 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].