Subelliptic Spin_ℂDirac operators, III. The Atiyah–Weinstein conjecture

Abstract

In this paper we extend the results obtained in [9], [10] to manifolds with Spin C -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, R eo + , which give subelliptic Fredholm problems for the Spin C -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 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, 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 φ : bX 1 → bX 0 , a contact diffeomorphism.Let S 0 , S 1 denote generalized Szegő projectors on bX 0 , bX 1 , respectively, and R eo 0 , R eo 1 , the subelliptic boundary conditions they define.If X 1 is the manifold X 1 with its orientation reversed, then the glued manifold X = X 0 φ X 1 has a canonical Spin C -structure and Dirac operator, ð eo X .Applying these results and those of our previous papers we obtain a formula for the relative index, R-Ind(S 0 , φ * S 1 ), R-Ind(S 0 , φ * S 1 ) = Ind(ð e X ) -Ind(ð e X0 , R e 0 ) + Ind(ð e X1 , R e 1 ).( 1) For the special case that X 0 and X 1 are strictly pseudoconvex complex manifolds and S 0 and S 1 are the classical Szegő projectors defined by the complex structures this formula implies that R-Ind(S 0 , φ * S 1 ) = Ind(ð e X )χ O (X 0 ) + χ O (X 1 ), (2)

Locations

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