  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.
