Let X be a compact K¨hler manifold with strictly pseudoconvex bounda ary, Y. In this setting, the SpinC Dirac operator is canonically identiﬁed with ¯ ¯ ∂ + ∂ ∗ : C ∞ (X; Λ0,e ) → C ∞ (X; Λ0,o ). We consider modiﬁcations of the classi¯ cal ∂-Neumann conditions that deﬁne Fredholm problems for the SpinC Dirac operator. In Part 2, , 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.