Deformation quantization modules I:Finiteness and duality

TL;DR

This paper develops a framework for understanding modules over deformation quantization algebroids on complex Poisson manifolds, establishing coherence and duality properties using cohomological completeness.

Contribution

It introduces cohomological completeness for sheaves of Z[ħ]-modules and proves new coherence and duality results for modules over deformation quantization algebroids.

Findings

Proves coherence of convolutions of kernels under properness conditions

Constructs dualizing complexes in the deformation quantization setting

Shows convolution commutes with duality in this framework

Abstract

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing complexes in this framework and show that the convolution of kernels commutes with duality.