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Singularities of varieties admitting an endomorphism

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Abstract

Let \(X\) be a normal variety such that \(K_X\) is \(\mathbb {Q}\)-Cartier, and let \(f:X \rightarrow X\) be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism \(f\). As a consequence we prove that if \(X\) is projective and \(f\) polarised, then \(X\) has at most log-canonical singularities.

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Notes

  1. The existence of log-canonical models would be a consequence of the MMP, including the abundance conjecture.

  2. Note that we do not assume that the boundary divisor \(\Delta \) is effective, so some authors would say that such a pair is sub-lc (resp. sub-klt). We follow the notation of [14].

  3. \(X\) is demi-normal in the sense of Kollár.

References

  1. Aprodu, M., Kebekus, S., Peternell, T.: Galois coverings and endomorphisms of projective varieties. Math. Z. 260(2), 431–449 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amerik, E.: On endomorphisms of projective bundles. Manuscripta Math. 111(1), 17–28 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boucksom, S., de Fernex, T., Favre, C.: The volume of an isolated singularity. Duke Math. J. 161, 1455–1520 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beauville, A.: Endomorphisms of hypersurfaces and other manifolds. Int. Math. Res. Notices (1), 53–58 (2001)

  5. Favre, C.: Holomorphic self-maps of singular rational surfaces. Publ. Mat. 54(2), 389–432 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fujimoto, Y., Nakayama, N.: Endomorphisms of smooth projective 3-folds with nonnegative Kodaira dimension. II. J. Math. Kyoto Univ. 47(1), 79–114 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Fujimoto, Y.: Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension. Publ. Res. Inst. Math. Sci. 38(1), 33–92 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fujino, O.: Fundamental theorems for the log minimal model program. Publ. Res. Inst. Math. Sci. 47(3), 727–789 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fulger, M.: Local volumes on normal algebraic varieties. arXiv preprint, 1105.2981 (2011)

  10. Graf, P.: Bogomolov-Sommese vanishing on log canonical pairs. arXiv preprint, 1210.0421 (2012)

  11. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. (28), 255 (1966)

  12. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, no. 52. Springer, New York (1977)

  13. Hacon, C.D., Kovács, S.J.: Classification of Higher Dimensional Algebraic Varieties, Volume 41 of Oberwolfach Seminars. Birkhäuser Verlag, Basel (2010)

    Book  Google Scholar 

  14. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). (With the collaboration of C. H. Clemens and A. Corti)

    Book  Google Scholar 

  15. Nakayama, N.: On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. preprint (2008)

  16. Nakayama, N., Zhang, D.-Q.: Polarized endomorphisms of complex normal varieties. Math. Ann. 346(4), 991–1018 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Odaka, Y., Xu, C.: Log-canonical models of singular pairs and its applications. Math. Res. Lett. 19(2), 325–334 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wahl, J.: A characteristic number for links of surface singularities. J. Am. Math. Soc. 3(3), 625–637 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, D.-Q.: Polarized endomorphisms of uniruled varieties. Compos. Math. 146(1), 145–168 (2010). (With an appendix by Y. Fujimoto and N. Nakayama)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, D.-Q.: Invariant hypersurfaces of endomorphisms of projective varieties. arXiv preprint, 1310.5944 (2013)

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Acknowledgments

We thank S. Druel and Y. Gongyo for pointing out some crucial references. The authors are partially supported by the ANR project CLASS (ANR-10-JCJC-0111). A.B is partially supported by the ANR project MACK (ANR-10-BLAN-0104) and the Labex CEMPI (ANR-11-LABX-0007-01). This work was done while A.H. was a member of the Institut de Mathématiques de Jussieu (UPMC).

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Authors and Affiliations

  1. Laboratoire Painlevé, UMR CNRS 8524, UFR de mathématiques, Université Lille 1, 59 655 , Villeneuve d’Ascq CEDEX, France

    Amaël Broustet

  2. Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, 06108 , Nice Cedex 02, France

    Andreas Höring

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  1. Amaël Broustet
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  2. Andreas Höring
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Correspondence to Amaël Broustet.

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Broustet, A., Höring, A. Singularities of varieties admitting an endomorphism. Math. Ann. 360, 439–456 (2014). https://doi.org/10.1007/s00208-014-1015-9

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  • DOI: https://doi.org/10.1007/s00208-014-1015-9

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