  In this paper we show that there is a well defined modified dbar-Neumann
problem for a spin_c manifold with a strictly pseudoconvex boundary (in the
contact geometry sense). We show that the index of the associated boundary
value problem can be computed as the relative index between the Calderon
projector on the boundary and the projector defining the boundary condition. To
make sense of this statement we need to develop a considerable extension of the
classical notion of a Fredholm pair of subspaces of a Hilbert space. We call
this the theory of "tame Fredholm pairs;" it is presented in the appendix to
the paper. Using these tools, we give a proof of a conjecture of Atiyah and
Weinstein for the index of a Fourier integral operator defined by a contact
transformation between the boundaries of two strictly pseudoconvex manifolds.
In fact, we prove a much more general result expressing the relative index
between two generalized Szego projectors as the index of a Dirac operator on a
compact "glued" space with correction terms coming from boundary value
problems. The Atiyah-Weinstein conjecture is a simple special case of this
general formula. Special cases of this result were earlier proved by Epstein
and Melrose, and Leichtnam, Nest and Tsygan. Using our general formula for
relative indices, we also obtain a formula for the relative index between two
Szego projectors defined by embeddable CR-structures on a contact three
manifold. Using this formula, we reduce a conjecture about the boundedness of
these relative indices to a conjecture of A. Stipsicz on the boundedness of the
Euler characteristics and signatures of Stein manifolds with a given contact
boundary. This latter conjecture has been proved in many cases of interest.
