Semiclassical Hodge theory for log Poisson manifolds

TL;DR

This paper develops a mixed Hodge structure framework for the topological K-theory of smooth Poisson varieties, linking it to deformation quantization parameters and providing computational tools for these structures.

Contribution

It introduces a new mixed Hodge structure on K-theory of Poisson varieties and connects quantum parameters with deformation quantizations.

Findings

Quantum parameters coincide with deformation quantization parameters.

Period maps can be expressed as exponential maps for tori.

Application of Kontsevich's quantization formula to Poisson tori.

Abstract

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 "$q = e^\hbar$".