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. We assume that E → X is a complex vector bundle, which has an inﬁnite order, integrable, complex structure along bX, compatible with that deﬁned.