Semiclassical Hodge theory for log Poisson manifolds

We construct a mixed Hodge structure on the topological K-theory of smooth Poisson varieties, depending weakly on a choice of compactification. We establish a package of tools for calculations with these structures, such as functoriality statements, projective bundle formulae, Gysin sequences and Torelli properties. We show that for varieties with trivial A-hat class, the corresponding period maps for families can be written as exponential maps for bundles of tori, which we call the "quantum parameters". As justification for the terminology, we show that in many interesting examples, the quantum parameters of a Poisson variety coincide with the parameters appearing in its known deformation quantizations. In particular, we give a detailed implementation of an argument of Kontsevich, to prove that his canonical quantization formula, when applied to Poisson tori, yields noncommutative tori with parameter "$q = e^\hbar$".