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We settle the last open case of Kuznetsov's conjecture on the derivedcategories of Fano threefolds. Contrary to the original conjecture, we provethe Kuznetsov components of quartic double solids and Gushel-Mukai threefoldsare never equivalent, as recently shown independently by Zhang. On the otherhand, we prove the modified conjecture asserting their deformation equivalence.Our proof of nonequivalence combines a categorical Enriques-K3 correspondencewith the Hodge theory of categories. Along the way, we obtain a categoricaldescription of the periods of Gushel-Mukai varieties, which we use to resolve aconjecture of Kuznetsov and the second author on the birational categoricalTorelli problem, as well as to give a simple proof of a theorem of Debarre andKuznetsov on the fibers of the period map. Our proof of deformation equivalencerelies on results of independent interest about obstructions to enhancing groupactions on categories.

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