Derived categories of hearts on Kuznetsov components

Abstract

We prove a general criterion which guarantees that an admissible subcategory K of the derived category of an abelian category is equivalent to the bounded derived category of the heart of a bounded t-structure. As a consequence, we show that K has a strongly unique dg enhancement, applying the recent results of Canonaco, Neeman and Stellari. We apply this criterion to the Kuznetsov component Ku(X) when X is a cubic fourfold, a Gushel-Mukai variety or a quartic double solid. In particular, we obtain that these Kuznetsov components have strongly unique dg enhancement and that exact equivalences of the form Ku(X) ∼ − → Ku(X ′ ) are of Fourier-Mukai type when X, X ′ belong to these classes of varieties, as predicted by a conjecture of Kuznetsov.

Categorical Torelli theorems: results and open problems

“…This can be proved exactly as in [57,Corollary 3.6]. Indeed, by [98] every equivalence as above is of Fourier-Mukai type. Then by [57, Corollary 3.5], it induces a Hodge isometry Φ H between the Mukai lattices of Ku(X 1 ) and Ku(X 2 ) .…”

“…Moreover, it is an indecomposable admissible subcategory due to the wellknown argument in [84,Section 2.6]. The fact that Question 2.7 has a positive answer is the content of [98] (based on [34,37]).…”

“…(2) By [67,Theorem 4.25] (see also [51,Remark 3.7]) there is a unique GL . Note that by [98] every equivalence is of Fourier-Mukai type in this case. (4) The Categorical Torelli theorem holds for Y 4 by [29], and for Y 5 since it is unique up to isomorphism by [66].…”

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

“…Related work. The nonemptiness result in Theorems 1.1 and 1.3 has been applied in [LPZ22] to show that the Kuznetsov components of quartic double solids and GM threefolds have a strongly unique enhancement.…”