Recursive formula for covariant derivatives and geometric classification of quotient modules

Abstract

Let H be a vector-valued holomorphic reproducing kernel function space and H0n be the subspace of functions vanishing to order n on an analytic hyper-surface. We determine unitary equivalence of the quotient space H⊖H0n by covariant derivatives of the curvature on the vector bundle associated to the reproducing kernel of H. The proof is a combination of some geometric and combinatorial results. In particular, we show that on a holomorphic Hermitian vector bundle, matrix representations for covariant derivatives of its curvature satisfy a recursive formula, which is of independent interest.