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On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties

  • Jungkai Alfred Chen , Hsueh-Yung Lin and Keiji Oguiso EMAIL logo
Published/Copyright: February 10, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

We study the main open parts of the Kawaguchi–Silverman conjecture, asserting that for a birational self-map f of a smooth projective variety X defined over ℚ ¯ , the arithmetic degree α f ⁢ ( x ) exists and coincides with the first dynamical degree δ f for any ℚ ¯ -point x of X with a Zariski dense orbit. Among other results, we show that this holds when X has Kodaira dimension zero and irregularity q ⁢ ( X ) ≥ dim ⁡ X - 1 or X is an irregular threefold (modulo one possible exception). We also study the existence of Zariski dense orbits, with explicit examples.

Keywords: Kawaguchi–Silverman conjecture; irregular varieties
MSC 2020: 14J50; 14E07; 37P55

Dedicated to the memory of the late Professor Nessim Sibony

Communicated by Shigeharu Takayama

Funding statement: Jungkai-Alfred Chen is partially supported by NSTC 110-2123-M-002-005 and by NCTS. Hsueh-Yung Lin is partially supported by Taiwan Ministry of Education Yushan Young Scholar Fellowship (NTU-110VV006), and NSTC (110-2628-M-002-006). Keiji Oguiso is partially supported by JSPS Grant-in-Aid (A) 15H05738, JSPS Grant-in-Aid (B) 15H03611, and by NCTS Scholar Program.

Acknowledgements

We would like to express our thanks to the NCTS and the staff members there for financial support and hospitality. This joint work was initiated during the third named author’s stay at the NCTS in October 7–December 16 2021. We thank Professors Yohsuke Matsuzawa and Junyi Xie for helpful comments, as well as Professors Shu Kawaguchi, De-Qi Zhang and Doctor Long Wang for their interest in this work and discussions. We also would like to express our thanks for the referee for her/his careful reading and valuable comments.

References

[1] E. Amerik, Existence of non-preperiodic algebraic points for a rational self-map of infinite order, Math. Res. Lett. 18 (2011), no. 2, 251–256. 10.4310/MRL.2011.v18.n2.a5Search in Google Scholar

[2] J. P. Bell, J. Diller and M. Jonsson, A transcendental dynamical degree, Acta Math. 225 (2020), no. 2, 193–225. 10.4310/ACTA.2020.v225.n2.a1Search in Google Scholar

[3] J. P. Bell, J. Diller, M. Jonsson and H. Krieger, Birational maps with transcendental dynamical degree, Proc. Lond. Math. Soc. (3) 128 (2024), no. 1, Article ID e12573. 10.1112/plms.12573Search in Google Scholar

[4] M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers, Birkhäuser, Basel, 1992. 10.1007/978-3-0348-8632-1Search in Google Scholar

[5] G. S. Call and J. H. Silverman, Canonical heights on varieties with morphisms, Compos. Math. 89 (1993), no. 2, 163–205. Search in Google Scholar

[6] T.-C. Dinh, H.-Y. Lin, K. Oguiso and D.-Q. Zhang, Derived length and nilpotency class of zero entropy group acting on compact Kähler manifolds, preprint (2020), https://arxiv.org/abs/1810.04827v2; to appear in PAMQ. Search in Google Scholar

[7] T.-C. Dinh, V.-A. Nguyên and T. T. Truong, On the dynamical degrees of meromorphic maps preserving a fibration, Commun. Contemp. Math. 14 (2012), no. 6, Article ID 1250042. 10.1142/S0219199712500423Search in Google Scholar

[8] T.-C. Dinh and N. Sibony, Regularization of currents and entropy, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), no. 6, 959–971. 10.1016/j.ansens.2004.09.002Search in Google Scholar

[9] T.-C. Dinh and N. Sibony, Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2) 161 (2005), no. 3, 1637–1644. 10.4007/annals.2005.161.1637Search in Google Scholar

[10] D. Ghioca and T. Scanlon, Density of orbits of endomorphisms of abelian varieties, Trans. Amer. Math. Soc. 369 (2017), no. 1, 447–466. 10.1090/tran6648Search in Google Scholar

[11] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar

[12] M. Hindry and J. H. Silverman, Diophantine Geometry. An Introduction, Grad. Texts in Math. 201, Springer, New York, 2000. 10.1007/978-1-4612-1210-2Search in Google Scholar

[13] V. A. Iskovskikh and I. R. Shafarevich, Algebraic surfaces, Algebraic Geometry. II, Encyclopaedia Math. Sci. 35, Springer, Berlin (1996), 127–262. 10.1007/978-3-642-60925-1_2Search in Google Scholar

[14] J. Jia, T. Shibata, J. Xie and D.-Q. Zhang, Endomorphisms of quasi-projective varieties: towards Zariski dense orbit and Kawaguchi–Silverman conjectures, Math. Res. Lett. 31 (2024), no. 3, 701–746. 10.4310/MRL.241113041354Search in Google Scholar

[15] S. Kawaguchi and J. H. Silverman, Examples of dynamical degree equals arithmetic degree, Michigan Math. J. 63 (2014), no. 1, 41–63. 10.1307/mmj/1395234358Search in Google Scholar

[16] 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), no. 7, 5009–5035. 10.1090/tran/6596Search in Google Scholar

[17] 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. 10.1515/crelle-2014-0020Search in Google Scholar

[18] Y. Kawamata, Characterization of abelian varieties, Compos. Math. 43 (1981), no. 2, 253–276. 10.1007/BF01457819Search in Google Scholar

[19] Y. Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57–71. 10.1007/BF01404756Search in Google Scholar

[20] Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. 10.1515/crll.1985.363.1Search in Google Scholar

[21] Y. Kawamata, Abundance theorem for minimal threefolds, Invent. Math. 108 (1992), no. 2, 229–246. 10.1007/BF02100604Search in Google Scholar

[22] Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423. 10.2977/prims/1210167332Search in Google Scholar

[23] S. Kobayashi and T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7–16. 10.1007/BF01389863Search in Google Scholar

[24] J. Lesieutre, Some constraints of positive entropy automorphisms of smooth threefolds, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 6, 1507–1547. 10.24033/asens.2380Search in Google Scholar

[25] J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi–Silverman conjecture, Int. Math. Res. Not. IMRN 2021 (2021), no. 10, 7677–7714. 10.1093/imrn/rnz067Search in Google Scholar

[26] M. Levine, Deformations of irregular threefolds, Algebraic Threefolds (Varenna 1981), Lecture Notes in Math. 947, Springer, Berlin (1982), 269–286. 10.1007/BFb0093595Search in Google Scholar

[27] S. Li and Y. Matsuzawa, A note on Kawaguchi–Silverman conjecture, Internat. J. Math. 32 (2021), no. 11, Article ID 2150085. 10.1142/S0129167X21500853Search in Google Scholar

[28] Y. Matsuzawa, On upper bounds of arithmetic degrees, Amer. J. Math. 142 (2020), no. 6, 1797–1820. 10.1353/ajm.2020.0045Search in Google Scholar

[29] Y. Matsuzawa, K. Sano and T. Shibata, Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, Algebra Number Theory 12 (2018), no. 7, 1635–1657. 10.2140/ant.2018.12.1635Search in Google Scholar

[30] N. Nakayama and D.-Q. Zhang, Building blocks of étale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. (3) 99 (2009), no. 3, 725–756. 10.1112/plms/pdp015Search in Google Scholar

[31] K. Oguiso, Pisot units, Salem numbers, and higher dimensional projective manifolds with primitive automorphisms of positive entropy, Int. Math. Res. Not. IMRN 2019 (2019), no. 5, 1373–1400. 10.1093/imrn/rnx142Search in Google Scholar

[32] 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), no. 1, 361–385. Search in Google Scholar

[33] K. Sano, Dynamical degree and arithmetic degree of endomorphisms on product varieties, Tohoku Math. J. (2) 72 (2020), no. 1, 1–13. 10.2748/tmj/1585101618Search in Google Scholar

[34] J. H. Silverman, Rational points on K ⁢ 3 surfaces: a new canonical height, Invent. Math. 105 (1991), no. 2, 347–373. 10.1007/BF01232270Search in Google Scholar

[35] J. H. Silverman, Arithmetic and dynamical degrees on abelian varieties, J. Théor. Nombres Bordeaux 29 (2017), no. 1, 151–167. 10.5802/jtnb.973Search in Google Scholar

[36] T. T. Truong, (Relative) dynamical degrees of rational maps over an algebraic closed field, preprint (2015), https://arxiv.org/abs/1501.01523. Search in Google Scholar

[37] T. T. Truong, Relative dynamical degrees of correspondences over a field of arbitrary characteristic, J. Reine Angew. Math. 758 (2020), 139–182. 10.1515/crelle-2017-0052Search in Google Scholar

[38] K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Math. 439, Springer, Berlin, 1975. 10.1007/BFb0070570Search in Google Scholar

[39] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic Varieties and Analytic Varieties (Tokyo 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam (1983), 329–353. 10.2969/aspm/00110329Search in Google Scholar

[40] E. Viehweg and K. Zuo, On the isotriviality of families of projective manifolds over curves, J. Algebraic Geom. 10 (2001), no. 4, 781–799. Search in Google Scholar

[41] J. Xie, Periodic points of birational transformations on projective surfaces, Duke Math. J. 164 (2015), no. 5, 903–932. 10.1215/00127094-2877402Search in Google Scholar

[42] J. Xie, The existence of Zariski dense orbits for endomorphisms of projective surfaces, J. Amer. Math. Soc. 38 (2025), no. 1, 1–62. 10.1090/jams/1004Search in Google Scholar

[43] J. Xie, Algebraic dynamics and recursive inequalities, preprint (2024), https://arxiv.org/abs/2403.09181. Search in Google Scholar
Received: 2023-10-04
Revised: 2024-10-16
Published Online: 2025-02-10
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2023-0350
Keywords for this article
Kawaguchi–Silverman conjecture; irregular varieties

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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