Complex-Foliated Structures. I. Cohomology of the Dolbeault-Kostant Complexes

Abstract

Abstract.We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable subbundles F of the complex tangent bundle of a manifold M. When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a "polarization" then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M, for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.

On quasi-classical limits of DQ-algebroids

“…In this section we briefly review basic facts regarding differential calculus in the presence of an integrable complex distribution. We refer the reader to [Kos70, Raw77] and [FW79] for details and proofs.…”