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Endomorphisms of a variety of Ueno type and Kawaguchi–Silverman conjecture

Mathematical Sciences, The University of Tokyo, Meguro Komaba 3-8-1 Tokyo, Japan

National Center for Theoretical Sciences, Mathematics Division, National Taiwan University, Taipei, Taiwan

https://doi.org/10.1142/S0129167X24500551Cited by:0 (Source: Crossref)

Abstract

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over $\bar{ℚ}$ .

Communicated by Jungkai A. Chen

Keywords:

AMSC: 14J20, 14J40, 14J50

References

1. E. Amerik , Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Math. Res. Lett. 18 (2011) 251–256. Crossref, Web of Science, Google Scholar
2. A. Beauville , Some remarks on Kähler manifolds with $c_{1} = 0$ , in Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progress in Mathematics, Vol. 39 (Birkhäuser Boston, MA, 1983), pp. 1–26. Google Scholar
3. P. Cascini, S. Meng and D.-Q. Zhang , Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020) 637–665. Crossref, Web of Science, Google Scholar
4. F. Catanese, K. Oguiso and A. Verra , On the unirationality of higher dimensional Ueno-type manifolds, Rev. Roumaine Math. Pures Appl. 60 (2015) 337–353. Google Scholar
5. J.-A. Chen, H.-Y. Lin and K. Oguiso, On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties, preprint (2022), arXiv:2204.09845. Google Scholar
6. J.-L. Colliot-Thélène , Rationalité d’un fibré en coniques, Manuscr. Math. 147 (2015) 305–310. Google Scholar
7. T.-C. Dinh, H.-Y. Lin, K. Oguiso and D.-Q. Zhang , Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations, Geom. Funct. Anal. 32(3) (2022) 568–594. Crossref, Web of Science, Google Scholar
8. T.-C. Dinh and V.-A. Nguyên , Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86(4) (2011) 817–840. Crossref, Web of Science, Google Scholar
9. T.-C. Dinh and N. Sibony , Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. Math. 161 (2005) 1637–1644. Crossref, Web of Science, Google Scholar
10. Y. Fujimoto , Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension, Publ. RIMS, Kyoto Univ. 38 (2002) 33–92. Crossref, Web of Science, Google Scholar
11. M. Hindry and J. H. Silverman , Diophantine Geometry. An Introduction, Graduate Texts in Mathematics, Vol. 201 (Springer-Verlag, New York, 2000). Crossref, Google Scholar
12. S. Kawaguchi , Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint, Amer. J. Math. 130 (2008) 159–186. Crossref, Web of Science, Google Scholar
13. S. Kawaguchi and J. H. Silverman , Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and NEF canonical heights on abelian varieties, Trans. Amer. Math. Soc. 368 (2016) 5009–5035. Crossref, Web of Science, Google Scholar
14. S. Kawaguchi and J. H. Silverman , On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016) 21–48. Crossref, Web of Science, Google Scholar
15. J.-H. Keum, K. Oguiso and D.-Q. Zhang , Conjecture of Tits type for complex varieties and theorem of Lie–Kolchin type for a cone, Math. Res. Lett. 16 (2009) 133–148. Crossref, Web of Science, Google Scholar
16. J. Kollár , Rational Curves on Algebraic Varieties, A Series of Modern Surveys in Mathematics, Vol. 32 (Springer-Verlag, Berlin, 1996). Crossref, Google Scholar
17. J. Kollár and M. Larsen , Quotients of Calabi–Yau varieties, in Algebra, Arithmetic, and Geometry in honor of Y. I. Manin, Vol. 2, Progress in Mathematics, Vol. 270 (Birkhäuser Boston, Boston, MA, 2009), pp. 179–211. Crossref, Google Scholar
18. J. Lesieutre and M. Satriano . Canonical heights on hyper-Kahler varieties and the Kawaguchi–Silverman conjecture, Int. Math. Res. Not. 2021(10) (2021) 7677–7714. Crossref, Web of Science, Google Scholar
19. N.-H. Lee and K. Oguiso , Connecting certain rigid birational non-homeomorphic Calabi–Yau threefolds via Hilbert scheme, Comm. Anal. Geom. 17 (2009) 283–303. Crossref, Web of Science, Google Scholar
20. S. Li, Kawaguchi–Silverman conjecture on automorphisms of projective threefolds, Int. J. Math. preprint (2024), doi:10.1142/S0129167X24500022, arXiv:2209.06815. Google Scholar
21. Y. Matsuzawa , On upper bounds of arithmetic degrees, Amer. J. Math. 142 (2020) 1797–1820. Crossref, Web of Science, Google Scholar
22. Y. Matsuzawa, Recent advances on Kawaguchi–Silverman conjecture, preprint (2023), arXiv:2311.15489, to appear in Proc. Simons Symposia on Algebraic, Complex, and Arithmetic Dynamics. Google Scholar
23. Y. Matsuzawa, K. Sano and T. Shibata , Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, Algebra Number Theory 12 (2018) 1635–1657. Crossref, Web of Science, Google Scholar
24. Y. Matsuzawa and S. Yoshikawa , Kawaguchi–Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism, Math. Ann. 382 (2022) 1681–1704. Crossref, Web of Science, Google Scholar
25. S. Meng and D.-Q. Zhang , Kawaguchi–Silverman conjecture for certain surjective endomorphisms, Doc. Math. 27 (2022) 1605–1642. Crossref, Web of Science, Google Scholar
26. S. Meng and D.-Q. Zhang, Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics, preprint (2023), arXiv:2311.16369, to appear in Proc. Simons Symposia on Algebraic, Complex, and Arithmetic Dynamics. Google Scholar
27. Y. Miyaoka , The Chern classes and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai, 1985, Advances Studies in Pure Mathematics, Vol. 10 (North-Holland, Amsterdam, 1987), pp. 449–476. Crossref, Google Scholar
28. D. Mumford , Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5 (Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970). Google Scholar
29. N. Nakayama and D.-Q. Zhang , Building blocks of étale endomorphisms of complex projective manifolds, Proc. London Math. Soc. 99 (2009) 725–756. Crossref, Web of Science, Google Scholar
30. K. Oguiso , On the complete classification of Calabi–Yau threefolds of type $I I I_{0}$ , in Higher-Dimensional Complex Varieties (Trento, 1994) (de Gruyter, 1996), pp. 329–339. Crossref, Google Scholar
31. K. Oguiso , Some aspects of explicit birational geometry inspired by complex dynamics, in Proc. Int. Congress of Mathematicians, Seoul, 2014. Vol. 2 (Kyung Moon Sa, Seoul, 2014), pp. 695–721. Google Scholar
32. K. Oguiso , Pisot units, Salem numbers, and higher dimensional projective manifolds with primitive automorphisms of positive entropy, Int. Math. Res. Not. 2019 (2016) 1373–1400. Crossref, Google Scholar
33. K. Oguiso, Fibered Calabi–Yau threefolds with relative automorphisms of positive entropy and $c_{2}$ -contractions, preprint (2024), arXiv:2401.04298. Google Scholar
34. K. Oguiso and J. Sakurai , Calabi–Yau threefolds of quotient type, Asian J. Math. 5 (2001) 43–77. Crossref, Google Scholar
35. K. Oguiso and T.-T. Truong , Explicit examples of rational and Calabi–Yau threefolds with primitive automorphisms of positive entropy, J. Math. Sci. Univ. Tokyo 22 (2015) 361–385. Google Scholar
36. J. H. Silverman , Arithmetic and dynamical degrees on abelian varieties, J. Theor. Nombres Bordeaux 29 (2017) 151–167. Crossref, Web of Science, Google Scholar
37. T.-T. Truong, (Relative) dynamical degrees of rational maps over an algebraic closed field, preprint (2015), arXiv:1501.01523. Google Scholar
38. T.-T. Truong , Relative dynamical degrees of correspondences over a field of arbitrary characteristic, J. Reine Angew. Math. 758 (2020) 139–182. Crossref, Web of Science, Google Scholar
39. K. Ueno , Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Mathematics, Vol. 439 (Springer-Verlag, 1975). Crossref, Google Scholar
40. E. Viehweg and K. Zuo , On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001) 781–799. Web of Science, Google Scholar
41. P. M. H. Wilson , The role of $c_{2}$ in Calabi–Yau classification — A preliminary survey, in Mirror Symmetry, II, AMS/IP Studies in Advanced Mathematics, Vol. 1 (American Mathematical Society, Providence, RI, 1997), pp. 381–392. Google Scholar
42. S. Yoshikawa , Structure of Fano fibrations of varieties admitting an int-amplified endomorphism, Adv. Math. 391 (2021) 107964. Crossref, Web of Science, Google Scholar