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.

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 [13]. 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 [3, 6, 10, 11, 19, 33, 46, 55] (see [54] for a survey on this topic).…”

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

“…(2) σ is the unique Serre invariant stability condition modulo the GL + 2 (R)-action. Namely, by a combination of various works [PY22, FP23, JLLZ21a, PR21] (see also [PS22, §5.5]), this holds for Fano threefolds of Picard number 1, index 2, and degree d ≥ 2, as well as for Fano threefolds of Picard number 1, index 1, and and degree d ≥ 10. 1 Note that stability conditions which differ by the GL + 2 (R)-action have the same set of (semi)stable objects, so they are the same from the perspective of their moduli spaces; therefore, when σ as above exists, its moduli spaces are canonical to Ku(Y ).…”

“…There is a rich emerging theory of moduli spaces of stable objects in Kuznetsov components of Fano threefolds, with applications including the realization of classical moduli spaces in these terms [LMS15, APR19, Zha21, LZ21, FP23] and the proofs of (categorical) Torelli theorems [BMMS12, BBF + 20, JLLZ21b]; see [PS22] for a survey. What is still lacking, however, are general structure theorems for these moduli spaces, parallel to the situation for K3 categories discussed in §1.1.…”

Categorical Torelli theorems for Gushel–Mukai threefolds

“…There is a very nice survey article [50] on recent results and remaining open questions on this topic. In [8] and [51], the authors prove categorical Torelli theorems for cubic threefolds.…”