Advertisement

Selecta Mathematica

, Volume 24, Issue 4, pp 3753–3762 | Cite as

Some remarks on L-equivalence of algebraic varieties

  • Alexander I. Efimov
Article
  • 53 Downloads

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 (Int J Math 16(1):13–36, 2005, Conjecture 0.3) 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 Ito et al. Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. arXiv:1612.08497). This disproves the original version of a conjecture of Kuznetsov and Shinder (Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Sel Math New Ser. arXiv:1612.07193, Conjecture 1.6). 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 \({\text {End}}(X)=\mathbb {Z}\) implies that X and Y are isomorphic.

Keywords

Grothendieck rings of varieties Hodge structures Krull–Schmidt categories 

Mathematics Subject Classification

18F30 14C30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.: On the Krull-Schmidt theorem with application to sheaves. Bull. Soc. Math. Fr. 84, 307–317 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander, M.: Representation dimension of Artin algebras. In: Queen Mary College Mathematics Notes, London (1971)Google Scholar
  3. 3.
    Auslander, M.: Representation theory of Artin algebras. I, II. Commun. Algebra 1, 177–268 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auslander, M.: Representation theory of Artin algebras. I, II. Commun. Algebra 1, 269–310 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bridgeland, T., Maciocia, A.: Complex surfaces with equivalent derived categories. Math. Z. 236, 677–697 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Ph.D. thesis Cornell (2000)Google Scholar
  8. 8.
    Gabriel, P., Roiter, A.V.: Representations of Finite-Dimensional Algebras, Translated from the Russian, reprint of the 1992 English translation. Springer, Berlin (1997)Google Scholar
  9. 9.
    Hosono, S., Lian, B.H., Oguiso, K., Yau, S.-T.: Kummer structures on a K3 surface—an old question of T. Shioda. Duke Math. J. 12, 635–647 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huybrechts, D.: Motives of isogenous K3 surfaces. Commentarii Mathematici Helvetici. arXiv:1705.04063 (preprint)
  11. 11.
    Huybrechts, D.: Generalized Calabi-Yau structures, K3 surfaces, and \(B\)-fields. Int. J. Math. 16(1), 13–36 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huybrechts, D., Stellari, P.: Equivalences of twisted K3 surfaces. Math. Ann. 332(4), 901–936 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ito, A., Miura, M., Okawa, S., Ueda, K.: Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. arXiv:1612.08497 (preprint)
  14. 14.
    Krause, H.: Krull-Schmidt categories and projective covers. Expositiones Math. 33(4), 535–549 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuznetsov, A., Shinder, E.: Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. E. Sel. Math. New Ser. (2017).  https://doi.org/10.1007/s00029-017-0344-4
  16. 16.
    Looijenga, E.: Motivic measures. Séminaire Bourbaki 42, 267–297 (1999–2000)Google Scholar
  17. 17.
    Manin, Y.I.: Correspondences, motifs and monoidal transformations. Mat. Sb. (N.S.), 77(119):4, 475–507 (1968). English translation: Math. USSR-Sb., 6:4, 439–470 (1968)Google Scholar
  18. 18.
    Orlov, D.: Equivalences of derived categories and K3 surfaces. Algebraic geometry, 7. J. Math. Sci. (New York) 84(5), 1361–1381 (1997)Google Scholar
  19. 19.
    Reiner, I.: Maximal Orders. Academic Press, London (1975)zbMATHGoogle Scholar
  20. 20.
    Stellari, P.: Derived categories and Kummer varieties. Math. Z. 256(2), 425–441 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zarhin, Y.: Hodge groups of K3 surfaces. J. Reine Angew. Math. 341, 193–220 (1983)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of RASMoscowRussia

Personalised recommendations