Differential complexes and Hodge theory on log-symplectic manifolds

TL;DR

This paper explores differential complexes on log-symplectic manifolds, relating their properties to geometric features like degeneracy divisors and Hodge numbers, and investigates implications for deformation theory.

Contribution

It introduces new complexes of differential forms on log-symplectic manifolds and establishes their connections to geometric and cohomological properties, including local cohomology and deformation obstructions.

Findings

Computed local cohomology of complexes in holomorphic cases

Established relations between degeneracy divisor multiplicities and Hodge numbers

Linked vanishing of Hodge numbers to unobstructed deformations

Abstract

We study certain complexes of differential forms, including reverse de Rham complexes, on (real or complex) Poisson manifolds, especially holomorphic log-symplectic ones. We relate these to the degeneracy divisor and rank loci of the Poisson bivector. In some good holomorphic cases we compute the local cohomology of these complexes. In the Kahlerian case, we deduce a relation between the multiplicity loci of the degeneracy divisor and the Hodge numbers of the manifold. We also show that vanishing of one of these Hodge numbers is related tounobstructed deformations of the normalized degeneracy divisor with its induced Poisson structure.