Some remarks on L-equivalence for cubic fourfolds and hyper-K\"ahler manifolds

TL;DR

The paper investigates L-equivalence among cubic fourfolds and hyper-K"ahler manifolds, establishing conditions under which L-equivalence implies isomorphism or Fourier-Mukai equivalence, and supporting the conjecture that L- and D-equivalence coincide.

Contribution

It proves that very general cubic fourfolds are isomorphic if L-equivalent, and explores L-equivalence implications for special cases and hyper-K"ahler manifolds.

Findings

L-equivalent very general cubic fourfolds are isomorphic

Existence of L-equivalent but non-isomorphic cubic fourfolds

L-equivalence implies Fourier-Mukai partners in certain cases

Abstract

We prove that if two very general cubic fourfolds are L-equivalent then they are isomorphic, and we observe that there exist special cubic fourfolds which are L-equivalent but not isomorphic. When the cubic fourfolds are very general in certain Hassett divisors, we prove that if they are L-equivalent then they are also Fourier-Mukai partners. We also provide further examples in support of the fact that L-equivalent hyper-K\"ahler manifolds should be D-equivalent, as conjectured by Meinsma.