In this paper we extend the results obtained in ,  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.
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.
We assume that the manifold with boundary, X, has a SpinC -structure with spinor bundle S Along the boundary, this structure agrees with the /. structure deﬁned by an inﬁnite order, integrable, almost complex structure and the metric is K¨hler. In this case the SpinC -Dirac operator . agrees with a ¯ ¯ ∂ + ∂ ∗ along the boundary. The induced CR-structure on bX is integrable and either strictly pseudoconvex or strictly pseudoconcave.