A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

We consider compact Kahlerian 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 unobstructed 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 \ D(Pi), and in fact coincides with this H-2 provided the Hodge number h(X)(2,0) = 0, and finally that the degeneracy locus D(Pi) deforms locally trivially under deformations of (X, Pi).