Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler manifolds

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. As a consequence, we provide examples of hyper-Kähler manifolds which are L-equivalent but not birational. We also provide further examples in support of the fact that L-equivalent hyper-Kähler manifolds should be D-equivalent, as conjectured by Meinsma.