A Bogomolov unobstructedness theorem for log-symplectic manifolds in general position

TL;DR

This paper proves unobstructed deformation properties for certain log-symplectic structures on compact Kähler manifolds, linking deformation spaces to mixed Hodge structures and showing local triviality of degeneracy loci under deformations.

Contribution

It establishes a Bogomolov-type unobstructedness theorem for log-symplectic manifolds in general position, clarifies the deformation theory, and relates it to Hodge structures and degeneracy locus behavior.

Findings

Deformations are unobstructed and their tangent space relates to Hodge structures.

The deformation space coincides with a specific Hodge component when h^{2,0}_X=0.

Degeneracy loci deform trivially under deformations of the structure.

Abstract

We consider compact K\"ahlerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\Pi$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\Pi)$. We prove that $(X, \Pi)$ has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^2$ of the open symplectic manifold $X\setminus D(\Pi)$, and in fact coincides with this $H^2$ provided the Hodge number $h^{2,0}_X=0$, and finally that the degeneracy locus $D(\Pi)$ deforms locally trivially under deformations of $(X, \Pi)$. It has been pointed out that the general position hypothesis in the original paper is not strong enough and this is corrected in an appended erratum/corrigendum to the revised version.