Some remarks on L-equivalence of algebraic varieties
Alexander I. Efimov
TL;DR
The paper investigates L-equivalence of complex algebraic varieties through integral Hodge realization and provides counterexamples to the conjectures of Huybrechts and Kuznetsov–Schinder, including isogenous K3s that are not L-equivalent and derived-equivalent twisted K3s with non-L-equivalent underlying K3s. It develops general Grothendieck-group results for additive categories with finitely generated morphisms, proving that stable isomorphism classes contain finitely many isomorphism classes and that End$(X)\cong \mathbb{Z}$ forces equality of objects when $[X]=[Y]$. It then applies these tools to abelian varieties and K3 surfaces, showing that $[H^{1}(A)]=[H^{1}(A')]$ determines $A\cong A'$ under $\operatorname{End}(A)=\mathbb{Z}$, and that L-equivalence implies isomorphism in this setting, while D-equivalence does not imply L-equivalence. For K3 surfaces, finiteness of L-equivalence classes is established and counterexamples to the Huybrechts conjecture are constructed via limitations of Hodge structures on $H^{2}$ and transcendental lattices. The results have implications for distinguishing L-equivalence classes via integral Hodge data and for understanding the relationship between D-equivalence and L-equivalence in geometric contexts.
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.