Holomorphic Koszul-Brylinski homologies of Poisson blow-ups

TL;DR

This paper establishes a blow-up formula for holomorphic Koszul-Brylinski homologies on compact holomorphic Poisson manifolds, explores spectral sequence invariance, and computes homologies for specific complex surfaces and nilmanifolds.

Contribution

It introduces a new blow-up formula for holomorphic Koszul-Brylinski homologies and applies it to analyze spectral sequence invariance and compute homologies for particular manifolds.

Findings

Derived a blow-up formula for holomorphic Koszul-Brylinski homologies.

Proved invariance of spectral sequence degeneracy under Poisson blow-ups.

Computed homologies for del Pezzo surfaces and certain nilmanifolds.

Abstract

We derive a blow-up formula for holomorphic Koszul-Brylinski homologies of compact holomorphic Poisson manifolds. As applications, we investigate the invariance of the $E_{1}$-degeneracy of the Dolbeault-Koszul-Brylinski spectral sequence under Poisson blow-ups, and compute the holomorphic Koszul-Brylinski homology for del Pezzo surfaces and two complex nilmanifolds with holomorphic Poisson structures.