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. 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.

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch

References and Relationships

• Bibliographic citation: Kuznetsov’s Fano threefold conjecture via K3 categories and enhanced group actions ; volume:2023 ; number:800 ; year:2023 ; pages:107-153 ; extent:47 ; Journal für die reine und angewandte Mathematik ; 2023, Heft 800 (2023), 107-153 (gesamt 47)

Contributors, Places and Time

• Creator: Bayer, Arend ; Perry, Alexander

Further information

• DOI: 10.1515/crelle-2023-0021
• URN: urn:nbn:de:101:1-2023070514105366530360
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 14.08.2025, 10:44 AM CEST