Some remarks on L-equivalence of algebraic varieties

Abstract

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts (Int J Math 16(1):13–36, 2005, Conjecture 0.3) stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying K3 surfaces are not L-equivalent. We also give examples showing that D-equivalent abelian varieties can be non-L-equivalent (the same examples were obtained independently in Ito et al. Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. arXiv:1612.08497). This disproves the original version of a conjecture of Kuznetsov and Shinder (Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Sel Math New Ser. arXiv:1612.07193, Conjecture 1.6). We deduce the statements on (non-)L-equivalence from the very general results on the Grothendieck group of an additive category, whose morphisms are finitely generated abelian groups. In particular, we show that in such a category each stable isomorphism class of objects contains only finitely many isomorphism classes. We also show that a stable isomorphism between two objects X and Y with ${\text {End}}(X)=\mathbb {Z}$ implies that X and Y are isomorphic.