L-equivalence and Fourier--Mukai partners of cubic fourfolds
Reinder Meinsma, Riccardo Moschetti
TL;DR
The paper investigates L-equivalence in the Grothendieck ring and its interaction with FM-partners for cubic fourfolds. By assuming a Derived Torelli principle for Kuznetsov components and a discriminant condition on the transcendental lattice, it derives a precise lattice-theoretic formula for counting Fourier–Mukai partners and constructs concrete pairs of $\mathbb{L}$-equivalent cubic fourfolds, including cases with a unique non-trivial FM partner. It further proves that $\mathbb{L}$-equivalence classes of cubic fourfolds are finite and that $\mathbb{L}$-equivalence passes to FM-equivalence and related invariants under the stated assumptions. The results illuminate the link between motivic and derived invariants and provide explicit families illustrating trivially $\mathbb{L}$-equivalent examples and the finiteness phenomenon in this context.
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.