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

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 projective hyper-Kähler manifolds should be D-equivalent, as conjectured by Meinsma [Counterexamples to the Kuznetsov–Shinder L-equivalence conjecture. Preprint, arXiv: 2503.21511 , 2025].