Categorical Torelli theorems for Gushel--Mukai threefolds

Abstract

The Kuznetsov component Ku(X) of a Gushel-Mukai (GM) threefold has two numerical (−1)-classes with respect to the Euler form. We describe the Bridgeland moduli spaces for stability conditions on Kuznetsov components with respect to each of the (−1)-classes and prove refined and birational categorical Torelli theorems in terms of Ku(X). We also prove a categorical Torelli theorem for special GM threefolds. We study the smoothness and singularities on Bridgeland moduli spaces for all smooth GM threefolds and use this to prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use our moduli spaces to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds. We also prove the restatement of this conjecture infinitesimally using Hochschild (co)homology.

Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces

“…Birational categorical Torelli for GM varieties. In [JLLZ21], we show that the Kuznetsov component determines the birational isomorphic class for general GM threefolds while in the present article, we prove a similar statement for very general GM fourfolds. Since GM fivefolds and sixfolds are all rational, the analogous statements automatically hold in these cases.…”

“…In [JLLZ21], we show the conjecture is true for general GM threefolds. In this article, we prove the following theorem.…”

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

“…In Corollary 4.5 we show this criterion applies to the Kuznetsov component of a GM threefold. Note that this result was already known by [JLLZ21,Theorem 4.25]. In particular, all known stability conditions on KupXq for X a Fano threefold of Picard rank 1, index 2 or index 1 and even genus ě 6 are Serre-invariant.…”

“…In fact, the property of Serre-invariance is very helpful in the study of the properties of moduli spaces and the stability of objects, see for instance [JLLZ21,LZ21] for many recent applications. In [FP21] the notion of Serre-invariance is applied to show that the moduli space of stable Ulrich bundles of rank d ě 2 on a cubic threefold is irreducible.…”

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

“…Theorem 1.1 applies to cubic threefolds and to other Fano threefolds (see Remark 3.7). Note that in these cases the uniqueness result has been recently proved independently by [JLLZ21].…”