Deformations and Fourier-Mukai transforms

TL;DR

This paper constructs explicit infinitesimal deformations of the category of coherent sheaves on a smooth projective variety and shows that Fourier-Mukai transforms extend to these deformed categories, preserving equivalences.

Contribution

It provides an explicit method for deforming the category of coherent sheaves and demonstrates the extension of Fourier-Mukai transforms to these deformations.

Findings

Explicit construction of infinitesimal deformations of Coh(X)

Extension of Fourier-Mukai transforms to deformed categories

Preservation of derived equivalences under deformation

Abstract

The aim of this paper is twofold: First we give an explicit construction of the infinitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly we show that any Fourier-Mukai transform $/Phi/colon D^b(X)/to D^b(Y)$ extends to an equivalence between the derived categories of the deformed Abelian categories.