Categorical Torelli theorems: results and open problems

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.Contents 24 6. Cubic fourfolds 40 References 47

Derived categories of hearts on Kuznetsov components

“…In 1997, Bondal and Orlov proved that smooth projective varieties with ample (anti)canonical bundle and equivalent bounded derived categories are isomorphic [BO01]. Similar reconstruction statements, called Categorical Torelli theorems, have been obtained for admissible subcategories of the bounded derived category, arising as residual components of exceptional collections in semiorthogonal decompositions, of certain Fano threefolds and fourfolds [BMMS12, BBF + 20, PY20, BT16, APR19, HR19, BLMS17, LPZ18] (see [PS22] for a survey on this topic).…”

Serre functors of residual categories via hybrid models

“…The subcategory R 𝑋 is called the residual component (or Kuznetsov component) of D b (𝑋). It controls much of the interesting behaviour of D b (𝑋): the deformation theory, moduli spaces of objects, stability conditions, and so on [3,7,15]. It also has a Serre functor 𝕊, and one can ask about the behaviour of 𝕊.…”

Brill--Noether theory for Kuznetsov components and refined categorical Torelli theorems for index one Fano threefolds

“…The question of whether Ku(X) determines X up to isomorphism, known as the Categorical Torelli question, has been studied for certain cases in the setting of Fano threefolds. There is a nice survey [PS22] on recent results. In the case of index two prime Fano threefolds, in [BMMS12], the authors show that the Kuznetsov component completely determines cubic threefolds up to isomorphism.…”