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

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.