Some remarks on L-equivalence of algebraic varieties

TL;DR

This paper investigates L-equivalence of algebraic varieties, disproves some conjectures relating to K3 surfaces and abelian varieties, and explores the structure of the Grothendieck group in additive categories.

Contribution

It provides counterexamples to conjectures on L-equivalence for K3 surfaces and abelian varieties, and analyzes the finiteness and isomorphism conditions within the Grothendieck group framework.

Findings

Disproved that isogenous K3 surfaces are L-equivalent.

Constructed examples of D-equivalent twisted K3 surfaces that are not L-equivalent.

Showed that stable isomorphism classes contain finitely many isomorphism classes.

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.