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

TL;DR

This paper resolves the last open case of Kuznetsov's conjecture on Fano threefolds by proving the deformation equivalence of certain derived categories, using categorical and Hodge-theoretic methods.

Contribution

It proves the deformation equivalence of Kuznetsov components for specific Fano threefolds and introduces new techniques for understanding group actions on categories.

Findings

Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are not equivalent.

Deformation equivalence of these components is established.

Categorical description of Gushel-Mukai periods and applications to Torelli problems.

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. Along the way, we obtain a categorical description of the periods of Gushel-Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.