L-equivalence and Fourier--Mukai partners of cubic fourfolds

Abstract

We study L-equivalence in the Grothendieck ring of varieties and its interaction with categorical invariants of cubic fourfolds. Assuming a Derived Torelli-type criterion for Kuznetsov components and a mild condition on the discriminant of the transcendental lattice, we prove a counting formula for Fourier--Mukai partners of such cubic fourfolds. As an application, we exhibit cubic fourfolds with a fixed algebraic lattice admitting a unique non-trivial Fourier--Mukai partner, which is trivially L-equivalent to the original. Finally, we show that L-equivalence classes of cubic fourfolds are finite.