On the deformation theory of Fourier-Mukai transforms between Calabi-Yau varieties

TL;DR

This paper investigates how Fourier-Mukai transforms between Calabi-Yau varieties deform, revealing that obstructions are governed by Hodge theory and providing a new formula for the obstruction class related to the Chern character and Kodaira-Spencer class.

Contribution

It generalizes previous work by establishing that obstructions to deforming Fourier-Mukai transforms are controlled by Hodge theory in Calabi-Yau cases and introduces a new formula for the obstruction class.

Findings

Obstructions to deformation are governed by Hodge theory.

A new formula for the obstruction class involving the Chern character and Kodaira-Spencer class.

Extension of deformation results to mixed characteristic settings.

Abstract

We study the deformation theory of fully faithful Fourier-Mukai transforms in both characteristic zero and mixed characteristic. Our main result shows that obstructions to deforming such transforms can be completely controlled by Hodge theory when the source variety has trivial canonical bundle, generalizing work of Addington-Thomas and Lieblich-Olsson. The main technical contribution is a formula for the obstruction class measuring the failure of a Chern character to remain within the Hodge filtration as a cup product with a (derived) Kodaira-Spencer class.