Fourier–Mukai transform in the quantized setting

François Petit

doi:10.1016/j.aim.2014.01.019

Abstract

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.

Highlights

Fourier-Mukai transform has been extensively studied in algebraic geometry and is still an active area of research
Several works have extended to the framework of deformation quantization of complex varieties some important aspects of the theory of integral transforms
The main result of this paper is Theorem 3.16 which states that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves

Summary

Introduction

Fourier-Mukai transform has been extensively studied in algebraic geometry and is still an active area of research (see [1] and [4]). In [6], Kashiwara and Schapira have developed the necessary formalism to study integral transforms in the framework of DQ-modules and some classical results have been extended to the quantized setting. The main result of this paper is Theorem 3.16 which states that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves. Both of these functors preserve compact generators. Aknowledgement: I would like to thank Oren Ben-Bassat, Andrei Caldararu, Carlo Rossi, Pierre Schapira, Nicolò Sibilla, Geordie Williamson for many useful discussions and Damien Calaque and Michel Vaquié for their careful reading of early version of the manuscript and numerous suggestions which have allowed substantial improvements

Some recollections on DQ-modules

Findings

Fourier-Mukai functors in the quantized setting