  We consider modifications of the classical dbar-Neumann conditions that
define Fredholm problems for the Spin_C Dirac operator. In part II, 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.
