Derived categories and birationality

TL;DR

This paper investigates conditions under which derived equivalences between smooth projective varieties imply that the varieties are birational, exploring categorical analogs of classical Torelli theorems and introducing the concept of strongly filtered derived equivalence.

Contribution

It introduces and studies the notion of strongly filtered derived equivalence and its implications for birationality between varieties.

Findings

Strongly filtered derived equivalence can imply birationality in certain cases.

The paper proposes an open variant of the main question regarding derived equivalences and birationality.

Conditions under which derived equivalences lead to birationality are analyzed.

Abstract

We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.