Holomorphic Koszul-Brylinski Homology

TL;DR

This paper investigates the Koszul-Brylinski homology of holomorphic Poisson manifolds, establishing its isomorphism with Lie algebroid cohomology, proving the nondegeneracy of a key pairing, and computing specific examples.

Contribution

It introduces a new isomorphism between Koszul-Brylinski homology and Lie algebroid cohomology for holomorphic Poisson manifolds, and demonstrates the nondegeneracy of the Evens-Lu-Weinstein pairing.

Findings

Homology is isomorphic to Lie algebroid cohomology.

Evens-Lu-Weinstein pairing is nondegenerate.

Explicit computation for Poisson structures on P^1 P^1.

Abstract

In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homology is nondegenerate. Finally we compute the Koszul-Brylinski homology for Poisson structures on $\CP^1\times\CP^1$.