Subelliptic SpinC Dirac operators, I

Let X be a compact Kahler manifold with strictly pseudoconvex boundary, Y. In this setting, the Spin(C) Dirac operator is canonically identified with partial derivative + partial derivative : C-infinity (X; Lambda(0,e)) -> C-infinity (X; Lambda(0,o)) . We consider modifications of the classical partial derivative-Neumann conditions that define Fredholm problems for the Spin(C) Dirac operator. In Part 2, [7], we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a Spin(C) Dirac operator with a subelliptic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface, then we obtain formulae for the holomorphic Euler characteristic of X as sums of indices of Spin(C) Dirac operators on the components. This is a subelliptic analogue of Bojarski's formula in the elliptic case.