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Kawaguchi–Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

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Abstract

We prove Kawaguchi–Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.

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References

  1. Ambro, F.: The moduli b-divisor of an lc-trivial fibration. Compos. Math. 141(2), 385–403 (2005)

    Article  MathSciNet  Google Scholar 

  2. 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 

  3. Bombieri, E., Gubler, W.: Heights in Diophantine Geometry. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  4. Broustet, A., Gongyo, Y.: Remarks on log Calabi-Yau structure of varieties admitting polarized endomorphisms. Taiwan. J. Math. 21(3), 569–582 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Cascini, P., Meng, S., Zhang, D.-Q.: Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. Math. Ann. 378(1–2), 637–665 (2020)

    Article  MathSciNet  Google Scholar 

  7. Dang, N.-B.: Degrees of iterates of rational maps on normal projective varieties. Proc. Lond. Math. Soc. 121(5), 1268–1310 (2020)

    Article  MathSciNet  Google Scholar 

  8. Debarre, O.: Higher-dimensional Algebraic Geometry, Universitext. Springer, New York (2001)

    Book  Google Scholar 

  9. Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Am. J. Math. 123(6), 1135–1169 (2001)

    Article  MathSciNet  Google Scholar 

  10. Dinh, T.-C., Nguyên, V.-A.: Comparison of dynamical degrees for semi-conjugate meromorphic maps. Comment. Math. Helv. 86(4), 817–840 (2011)

    Article  MathSciNet  Google Scholar 

  11. Dinh, T.-C., Sibony, N.: Equidistribution problems in complex dynamics of higher dimension. Int. J. Math. 28, 1750057 (2007)

    Article  MathSciNet  Google Scholar 

  12. Gongyo, Y.: Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs. J. Algebraic Geom. 22(3), 549–564 (2013)

    Article  MathSciNet  Google Scholar 

  13. Hindry, M., Silverman, J.H.: Diophantine Geometry. An introduction, Graduate Text in Mathematics, no. 20. Springer, New York (2000)

    MATH  Google Scholar 

  14. Jonsson, M., Wulcan, E.: Canonical heights for plane polynomial maps of small topological degree. Math. Res. Lett. 19(6), 1207–1217 (2012)

    Article  MathSciNet  Google Scholar 

  15. Kawaguchi, S., Silverman, J.H.: Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties. Trans. Am. Math. Soc. 368, 5009–5035 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kawaguchi, S., Silverman, J.H.: On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. 713, 21–48 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  18. Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983)

    Book  Google Scholar 

  19. Lesieutre, J., Satriano, M.: Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. Int. Math. Res. Not. IMRN 2021(10), 7677–7714 (2021)

    Article  Google Scholar 

  20. Lin, J.-L.: On the arithmetic dynamics of monomial maps. Ergodic Theory Dynam. Systems 39(12), 3388–3406 (2019)

    Article  MathSciNet  Google Scholar 

  21. Matsuzawa, Y.: On upper bounds of arithmetic degrees. Am. J. Math. 142(6), 1797–1820 (2020). https://doi.org/10.1353/ajm.2020.0045

    Article  MathSciNet  MATH  Google Scholar 

  22. Matsuzawa, Y.: Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties. Adv. Math. 366, 107086 (2020)

    Article  MathSciNet  Google Scholar 

  23. Matsuzawa, Y., Sano, K.: Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties. Ergodic Theory Dynam. Systems 40(6), 1655–1672 (2020)

    Article  MathSciNet  Google Scholar 

  24. Matsuzawa, Y., Sano, K., Shibata, T.: Arithmetic degrees and dynamical degrees of endomorphisms on surfaces. Algebra Number Theory 12(7), 1635–1657 (2018)

    Article  MathSciNet  Google Scholar 

  25. Matsuzawa, Y., Yoshikawa, S.: Int-amplified endomorphisms on normal projective surfaces. Taiwanese J. Math. 25(4), 681–697 (2021)

    Article  MathSciNet  Google Scholar 

  26. Meng, S.: Building blocks of amplified endomorphisms of normal projective varieties. Math. Z. (2019). https://doi.org/10.1007/s00209-019-02316-7

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Meng, S., Zhang, D.-Q.: Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. Math. Res. Lett. 27(2), 523–549 (2020)

    Article  MathSciNet  Google Scholar 

  29. Meng, S. and Zhang, D.-Q.: Kawaguchi-Silverman conjecture for surjective endomorphisms, arXiv:1908.01605

  30. Silverman, J.H.: Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergodic Theory Dyn. Syst. 34(2), 647–678 (2014)

    Article  MathSciNet  Google Scholar 

  31. Silverman, J.H.: Arithmetic and dynamical degrees on abelian varieties. J. Théor. Nombres Bprdeaux 29(1), 151–167 (2017)

    Article  MathSciNet  Google Scholar 

  32. Truong, T.T.: Relative dynamical degrees of correspondences over a field of arbitrary characteristic. J. Reine Angew. Math. 758, 139–182 (2020)

    Article  MathSciNet  Google Scholar 

  33. Yoshikawa, S.: Global F-splitting of surfaces admitting an int-amplified endomorphism, to appear in Manuscripta Math

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Acknowledgements

The authors would like to thank the organizers of “Younger generations in Algebraic and Complex geometry VI” where this collaboration started. The first author would like to thank Sheng Meng and De-Qi Zhang for stimulating discussions. The first author is supported by JSPS Research Fellowship for Young Scientists and KAKENHI Grant Number 18J11260. The second author is supported by the Program for Leading Graduate Schools, MEXT, Japan.

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Authors and Affiliations

  1. Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Yohsuke Matsuzawa

  2. Graduate school of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Shou Yoshikawa

Authors
  1. Yohsuke Matsuzawa
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  2. Shou Yoshikawa
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Correspondence to Yohsuke Matsuzawa.

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Communicated by Vasudevan Srinivas.

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Matsuzawa, Y., Yoshikawa, S. Kawaguchi–Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism. Math. Ann. 382, 1681–1704 (2022). https://doi.org/10.1007/s00208-021-02305-4

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  • DOI: https://doi.org/10.1007/s00208-021-02305-4

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