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Regular version of the site

Article

Some remarks on L-equivalence of algebraic varieties

Selecta Mathematica, New Series. 2018. Vol. 24. No. 4. P. 3753-3762.

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 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. This disproves the original version of a conjecture of Kuznetsov and Shinder.

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 such that the endomorphism ring of X is Z, implies that X and Y are isomorphic.