Holomorphic Koszul-Brylinski homology via Dolbeault cohomology

TL;DR

This paper explores the holomorphic Koszul-Brylinski homology on Poisson manifolds using Dolbeault cohomology, establishing key theorems and relations, especially around blow-up transformations, extending previous work.

Contribution

It introduces new theorems like Leray-Hirsch, Mayer-Vietoris, and K"unneth for holomorphic Koszul-Brylinski homology, linking it with Dolbeault cohomology and blow-up transformations.

Findings

Established Leray-Hirsch theorem for Hochschild homology

Derived Mayer-Vietoris sequence for holomorphic Koszul-Brylinski homology

Proved K"unneth theorem in this context

Abstract

We use the Dolbeault cohomology to investigate the Koszul-Brylinski homology on holomorphic Poisson manifolds. We obtain the Leray-Hirsch theorem for Hochschild homology and the Mayer-Vietoris sequence, K\"{u}nneth theorem for holomorphic Koszul-Brylinski homology. In particular, we show some relations of holomorphic Koszul-Brylinski homologies around a blow-up transformation for the general case (\emph{not necessarily compact}) by our previous works on the Dolbeault cohomology.