Some remarks on L-equivalence of algebraic varieties
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866909480931819520 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1707_08997 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Some remarks on L-equivalence of algebraic varieties Efimov, Alexander I. Algebraic Geometry Category Theory K-Theory and Homology 18F30, 14C30 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. |
| title | Some remarks on L-equivalence of algebraic varieties |
| topic | Algebraic Geometry Category Theory K-Theory and Homology 18F30, 14C30 |
| url | https://arxiv.org/abs/1707.08997 |