TAILIEUCHUNG - Đề tài " 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, + .eo . We introduce a generalization of Fredholm pairs to the “tame” category. | Annals of Mathematics Subelliptic SpinC Dirac operators III The Atiyah-Weinstein conjecture By Charles L. Epstein Annals of Mathematics 168 2008 299 365 Subelliptic Spinc Dirac operators III The Atiyah-Weinstein conjecture By Charles L. Epstein This paper is dedicated to my wife Jane for her enduring love and support. 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 Ỡ-Neumann boundary conditions defined by projection operators R which give subelliptic Fredholm problems for the Spinc-Dirac operator Se . We introduce a generalization of Fredholm pairs to the tame category. In this context we show that the index of the graph closure of 3e Re equals the relative index on the boundary between Re and the Calderon projector Pe . Using the relative index formalism and in particular the comparison operator T e 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 Ộ bX1 bX0 a contact diffeomorphism. Let 5o 51 denote generalized Szego projectors on bX0 bX1 respectively and R0 R1 the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed then the glued manifold X X0 Hộ X1 has a canonical SpinC-structure and Dirac operator 3e . Applying these results and those of our previous papers we obtain a formula for the relative index R-Ind 50 ộ s1 1 R-Ind ổ0 rS1 Ind 31 - Ind 310 R0 Ind 3Xi Re . For the special case that X0 and X1 are strictly pseudoconvex complex manifolds and 50 and 51 are the classical Szego projectors defined by the complex structures this formula implies that 2 R-Ind 50 r 1 Ind 3X - xO X0 xO X1 Research partially supported by NSF grant DMS02-03795 and the Francis J. .

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