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 \cite[Conjecture 0.3]{H} 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 \cite{IMOU}). This disproves the original version of a conjecture of Kuznetsov and Schinder \cite[Conjecture 1.6]{KS}. 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 $\mathrm{End}(X)=\mathbb{Z}$ implies that $X$ and $Y$ are isomorphic.