Categorical Torelli theorems: results and open problems

TL;DR

This paper surveys recent advances in the Categorical Torelli problem, exploring how derived categories and their subcategories can uniquely determine certain algebraic varieties like Enriques surfaces, Fano threefolds, and cubic fourfolds.

Contribution

It provides a comprehensive overview of new results and open problems in reconstructing varieties from their derived categories, focusing on specific classes of algebraic surfaces and threefolds.

Findings

Reconstruction of Enriques surfaces from derived categories

Results on prime Fano threefolds and cubic fourfolds

Identification of open problems in categorical Torelli theory

Abstract

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.