In this talk I will discuss recent joint work with Brent Pym concerning Hodge theory for holomorphic Poisson manifolds, with a view towards nonperturbative deformation quantization. I will show how to every logarithmic Poisson manifold one can associate a canonical mixed Hodge structure and describe how, in the case of torus invariant Poisson structures on projective space, the associated period map recovers the q-parameters defining noncommutative tori.