On the Inhomogeneous Yang-Mills Equation d_D^*R ^D=f

Abstract

We build a canonical family {D_s } of Hermitian connections in a Hermitian CR-holomorphic vector bundle (E,h ) over a nondegenerate CR manifold M, parametrized by S ∈ Γ ^∞(End (E )), S skewsymmetric. Consequently, we prove an existence and uniqueness result for the solution to the inhomogeneous Yang-Mills equation d_D^*R ^D =f on M. As an application we solve for D ∈D (E,h ) when E is either the trivial line bundle, or a locally trivial CR-holomorphic vector bundle over a nondegenerate real hypersurface in a complex manifold, or a canonical bundle over a pseudo-Einstein CR manifold.