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Normal Projective Varieties Admitting Polarized or Int-amplified Endomorphisms

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Abstract

Let X be a normal projective variety admitting a polarized or int-amplified endomorphism f. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor, and Kodaira dimension. Then, we run the equivariant minimal model program with respect to not just the single f but also the monoid SEnd(X) of all surjective endomorphisms of X, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.

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References

  1. Amerik, E., Rovinsky, M., Van de Ven, A.: A boundedness theorem for morphisms between threefolds. Ann. Inst. Fourier (Grenoble) 49, 405–415 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  Google Scholar 

  4. Birkar, C., Waldron, J.: Existence of Mori fibre spaces for 3-folds in char p. Adv. Math. 313, 62–101 (2017)

    Article  MathSciNet  Google Scholar 

  5. Boucksom, S., de Fernex, T., Favre, C.: The volume of an isolated singularity. Duke Math. J. 161(8), 1455–1520 (2012)

    Article  MathSciNet  Google Scholar 

  6. Briend, J.-Y., Duval, J.: Erratum: Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \(p^{k}(\mathbb {C})\). Publ. Math. Inst. Hautes Études Sci. 109, 295–296 (2009)

    Article  Google Scholar 

  7. Brion, M., Zhang, D.-Q.: Log Kodaira dimension of homogeneous Varieties. Algebraic varieties and automorphism groups, 1-6, Adv. Stud. Pure Math, 75, Math. Soc. Japan, Tokyo (2017)

  8. Broustet, A., Höring, A.: Singularities of varieties admitting an endomorphism. Math. Ann. 360(1–2), 439–456 (2014)

    Article  MathSciNet  Google Scholar 

  9. Brown, M., Mckernan, J., Svaldi, R., Zong, H.: A geometric characterisation of toric varieties. Duke Math. J. 167(5), 923–968 (2018)

    Article  MathSciNet  Google Scholar 

  10. Cascini, P., Meng, S., Zhang, D.-Q.: Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. arXiv:1710.01903

  11. Cerveau, D., Lins Neto, A.: Hypersurfaces exceptionnelles des endomorphismes de CP(n). Bol. Soc. Brasil. Mat. (N.S.) 31(2), 155–161 (2000)

    Article  MathSciNet  Google Scholar 

  12. Dinh, T.-C.: Analytic multiplicative cocycles over holomorphic dynamical systems. Complex Var. Elliptic Equ. 54(3–4), 243–251 (2009)

    Article  MathSciNet  Google Scholar 

  13. Dinh, T.-C., Sibony, N.: Dynamique des applications d’allure polynomiale. J. Math. Pures Appl. 82, 367–423 (2003)

    Article  MathSciNet  Google Scholar 

  14. Dinh, T.-C., Sibony, N.: Equidistribution speed for endomorphisms of projective spaces. Math. Ann. 347(3), 613–626 (2010)

    Article  MathSciNet  Google Scholar 

  15. Fakhruddin, N.: Questions on self-maps of algebraic varieties. J. Ramanujan Math. Soc. 18(2), 109–122 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Fujino, O.: Some remarks on the minimal model program for log canonical pairs. J. Math. Sci. Univ. Tokyo 22(1), 149–192 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Greb, D., Kebekus, S., Peternell, T.: Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties. Duke Math. J. 165 (10), 1965–2004 (2016)

    Article  MathSciNet  Google Scholar 

  18. Hacon, C., Xu, C.: On the three dimensional minimal model program in positive characteristic. J. Am. Math. Soc. 28(3), 711–744 (2015)

    Article  MathSciNet  Google Scholar 

  19. Hu, F.: A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic. Math. Ann., 1–30 (2018). arXiv:1801.06555

  20. Hwang, J.-M., Mok, N.: Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles. J. Alg. Geom. 12, 627–651 (2003)

    Article  MathSciNet  Google Scholar 

  21. Hwang, J.M., Nakayama, N.: On endomorphisms of Fano manifolds of Picard number one. Pure Appl. Math. Q. 7(4), 1407–1426 (2011)

    Article  MathSciNet  Google Scholar 

  22. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Math, vol. 134. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  23. Krieger, H., Reschke, P.: Cohomological conditions on endomorphisms of projective varieties. Bull. Soc. Math. France 145(3), 449–468 (2017)

    Article  MathSciNet  Google Scholar 

  24. Meng, S.: Building blocks of amplified endomorphisms of normal projective varieties. arXiv:1712.08995

  25. Meng, S., Zhang, D.-Q.: Building blocks of polarized endomorphisms of normal projective varieties. Adv. Math. 325, 243–273 (2018)

    Article  MathSciNet  Google Scholar 

  26. Meng, S., Zhang, D.-Q.: Characterizations of toric varieties via polarized endomorphisms. Math. Z. (to appear). arXiv:1702.07883

  27. Meng, S., Zhang, D.-Q.: Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. Math. Res. Lett. (to appear). arXiv:1806.05828 (2019)

  28. Nakayama, N., Zhang, D.-Q.: Building blocks of étale endomorphisms of complex projective manifolds. Proc. Lond. Math. Soc. 99(3), 725–756 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Okawa, S.: Extensions of two Chow stability criteria to positive characteristics. Michigan Math. J. 60(3), 687–703 (2011)

    Article  MathSciNet  Google Scholar 

  31. Paranjape, K.H., Srinivas, V.: Self maps of homogeneous spaces. Invent. Math. 98, 425–444 (1989)

    Article  MathSciNet  Google Scholar 

  32. Reschke, P.: Distinguished line bundles for complex surface automorphisms. Transform. Groups 19(1), 225–246 (2014)

    Article  MathSciNet  Google Scholar 

  33. Shokurov, V.V.: Complements on surfaces. J. Math. Sci. (New York) 102(2), 3876–3932 (2000). Algebraic Geometry 10

    Article  MathSciNet  Google Scholar 

  34. Waldron, J.: The LMMP for log canonical 3-folds in characteristic p > 5. Nagoya Math. J. 230, 48–71 (2018). arXiv:1603.02967

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Yuan, X., Zhang, S.: The arithmetic Hodge index theorem for adelic line bundles. Math. Ann. 367(3–4), 1123–1171 (2017)

    Article  MathSciNet  Google Scholar 

  37. Zhang, D.-Q.: Polarized endomorphisms of uniruled varieties. Compos. Math. 146(1), 145–168 (2010)

    Article  MathSciNet  Google Scholar 

  38. Zhang, D.-Q.: N-dimensional projective varieties with the action of an abelian group of rank n − 1. Trans. Am. Math. Soc. 368(12), 8849–8872 (2016)

    Article  MathSciNet  Google Scholar 

  39. Zhang, S.W.: Distributions in algebraic dynamics, vol. 10, pp. 381–430. International Press, Somerville (2006). Survey in Differential Geometry

    Google Scholar 

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Acknowledgements

The second named author would like to thank the organizing committee for the kind invitation and warm hospitality during the International Conference: Nevanlinna Theory and Complex Geometry in Honor of Le Van Thiem’s Centenary, February–March, 2018, Hanoi, Vietnam. Both authors would like to thank the referee for the very careful reading and the suggestions to improve and clarify the paper.

Funding

The first named author is supported by a Research Assistantship of NUS. The second named author is supported by an Academic Research Fund of NUS.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Republic of Singapore

    Sheng Meng & De-Qi Zhang

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  1. Sheng Meng
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  2. De-Qi Zhang
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Correspondence to Sheng Meng.

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Meng, S., Zhang, DQ. Normal Projective Varieties Admitting Polarized or Int-amplified Endomorphisms. Acta Math Vietnam 45, 11–26 (2020). https://doi.org/10.1007/s40306-019-00333-6

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  • DOI: https://doi.org/10.1007/s40306-019-00333-6

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