Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

Abstract

We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories, and gives as a byproduct a simple proof of a theorem of Debarre and Kuznetsov about the fibers of the period map for Gushel-Mukai varieties. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.

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

“…Thus Φ induces a Hodge isometry between the numerical Grothendieck groups. Using this property, combined with the result in [BP22], there is a more general version of Theorem 9.2. More precisely, using [BP22, Theorem 5.12] and the same argument in [PS22, Remark 6.16], one can prove that any two GM fourfolds X and X ′ are period partners or duals if and only if there exists an equivalence Φ : Ku(X) ≃ Ku(X ′ ) such that the induced Hodge isometry [Φ] : N (Ku(X)) → N (Ku(X ′ )) maps Λ 1 , Λ 2 to Λ ′ 1 , Λ ′ 2 .…”

Categorical Torelli theorems: results and open problems

“…One could ask the intermediate question whether Φ induces a Hodge isometry H 4 van (X 1 , ℤ) ≅ H 4 van (X 2 , ℤ) between the degree-four vanishing cohomologies of X 1 and X 2 (see [46,Section 3.3] for the definition). This is answered positively by [15,Proposition 1.12].…”

“…Proof of Theorem 7. 15 Of course, if X 1 ≅ X2 , then there is a Hodge isometry H 4 prim (X 1 , ℤ) ≅ H 4 prim (X 2 , ℤ) . For the other implication, we start with a Hodge isometry…”

Derived categories of hearts on Kuznetsov components

“…In the case of the quartic double solid it was observed in [BT16, Theorem 7.2] that Theorem 1.3 implies the failure of original Fano threefolds Kuznetsov's Conjecture [Kuz09, Conjecture 3.7]. Note that Fano threefolds Conjecture has been disproved in [Zha21] and [BP22], independently, in a stronger sense, namely that the Kuznetsov component of a quartic double solid is never equivalent to that of a Gushel-Mukai threefold.…”