Complexes, residues and obstructions for log-symplectic manifolds

TL;DR

This paper studies log-symplectic structures on compact Kähler manifolds, providing conditions under which such structures and their divisors deform unobstructedly, thus advancing understanding of their geometric stability.

Contribution

It introduces a linear inequality involving Poincaré residues that guarantees unobstructed deformations of log-symplectic structures on certain Kähler manifolds.

Findings

Unobstructed deformations of log-symplectic structures are ensured under specific residue conditions.

Divisors with simple normal crossings deform locally trivially under these conditions.

The results apply to even-dimensional compact Kähler manifolds with log-symplectic forms.

Abstract

We consider compact K\"ahlerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic structure $\Phi$, a generically nondegenerate closed 2-form with simple poles on a divisor $D$ with local normal crossings. A simple linear inequality involving the iterated Poincar\'e residues of $\Phi$ at components of the double locus of $D$ ensures that the pair $(X, \Phi)$ has unobstructed deformations and that $D$ deforms locally trivially.