Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T08:44:49.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Free abelian group actions on normal projective varieties: submaximal dynamical rank case

Part of: Complex spaces with a group of automorphisms Automorphic functions Topological dynamics Holomorphic mappings and correspondences Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  14 May 2020

Fei Hu  and
Sichen Li
Fei Hu
Affiliation:
University of British Columbia, Vancouver, BCV6T 1Z2, Canada and Pacific Institute for the Mathematical Sciences, Vancouver, BCV6T 1Z4, Canada Current address: University of Waterloo, Waterloo, ONN2L 3G1, Canada e-mail: [email protected] URL: https://sites.google.com/view/feihu90s/
Sichen Li
Affiliation:
School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai200241, China and Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore119076, Republic of Singapore Current address: School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, China e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

Keywords

Automorphismdynamical degreedynamical ranktopological entropyKodaira dimensionweak Calabi–Yau varietyspecial MRC fibration

MSC classification

Primary: 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces 32H50: Iteration problems 37B40: Topological entropy
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1057 - 1073
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beauville, A., Some remarks on Kähler manifolds with $\;{c}_1=0$ . Classification of algebraic and analytic manifolds (Katata, 1982). Progr. Math. 39(1983), 1–26. MR 728605.Google Scholar
Birkar, C., Cascini, P., Hacon, C. D., and McKernan, J., Existence of minimal models for varieties of log general type . J. Am. Math. Soc. 23(2010), no. 2, 405–468. MR 2601039. http://doi.org/10.1090/S0894-0347-09-00649-3 CrossRefGoogle Scholar
Boucksom, S., Demailly, J.-P., Păun, M., and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension . J. Algebraic Geom. 22(2013), no. 2, 201–248. MR 3019449. http://doi.org/10.1090/S1056-3911-2012-00574-8 CrossRefGoogle Scholar
Campana, F., Connexité rationnelle des variétés de Fano . Ann. Sci. École Norm. Sup. 25(1992), no. 5, 539–545. MR 1191735. http://doi.org/10.24033/asens.1658 CrossRefGoogle Scholar
Campana, F., Wang, F., and Zhang, D.-Q., Automorphism groups of positive entropy on projective threefolds . Trans. Am. Math. Soc. 366(2014), no. 3, 1621–1638. MR 3145744. http://doi.org/10.1090/S0002-9947-2013-05838-2 CrossRefGoogle Scholar
Cantat, S., Dynamique des automorphismes des surfaces projectives complexes . C. R. Acad. Sci. Paris Sér. I Math. 328(1999), no. 10, 901–906. MR 1689873. http://doi.org/10.1016/S0764-4442(99)80294-8 CrossRefGoogle Scholar
Cantat, S. and Zeghib, A., Holomorphic actions, Kummer examples, and Zimmer program . Ann. Sci. Éc. Norm. Supér. 45(2012), no. 3, 447–489. MR 3014483. http://doi.org/10.24033/asens.2170 CrossRefGoogle Scholar
Dinh, T.-C., Hu, F., and Zhang, D.-Q., Compact Kähler manifolds admitting large solvable groups of automorphisms . Adv. Math. 281(2015), 333–352. MR 3366842. http://doi.org/10.1016/j.aim.2015.05.002 CrossRefGoogle Scholar
Dinh, T.-C. and Nguyên, V.-A., Comparison of dynamical degrees for semi-conjugate meromorphic maps . Comment. Math. Helv. 86(2011), no. 4, 817–840. MR 2851870. http://doi.org/10.4171/CMH/241 CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N., Groupes commutatifs d’automorphismes d’une variété kählérienne compacte . Duke Math. J. 123(2004), no. 2, 311–328. MR 2066940. http://doi.org/10.1215/S0012-7094-04-12323-1 CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N., Equidistribution problems in complex dynamics of higher dimension . Int. J. Math. 28(2017), no. 7, 1750057, 31 pp. MR 3667901. http://doi.org/10.1142/S0129167X17500574 CrossRefGoogle Scholar
Druel, S., A decomposition theorem for singular spaces with trivial canonical class of dimension at most five . Invent. Math. 211(2018), no. 1, 245–296. MR 3742759. http://doi.org/10.1007/s00222-017-0748-y CrossRefGoogle Scholar
Graber, T., Harris, J., and Starr, J., Families of rationally connected varieties . J. Am. Math. Soc. 16(2003), no. 1, 57–67. MR 1937199. http://doi.org/10.1090/S0894-0347-02-00402-2 CrossRefGoogle Scholar
Greb, D., Guenancia, H., and Kebekus, S., Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups . Geom. Topol. 23(2019), no. 4, 2051–2124. MR 3988092. http://doi.org/10.2140/gt.2019.23.2051 CrossRefGoogle Scholar
Greb, D., Kebekus, S., and Peternell, T., Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties . Duke Math. J. 165(2016), no. 10, 1965–2004. MR 3522654. http://doi.org/10.1215/00127094-3450859 CrossRefGoogle Scholar
Greb, D., Kebekus, S., and Peternell, T., Singular spaces with trivial canonical class. Minimal models and extremal rays (Kyoto, 2011). Adv. Stud. Pure Math. 70(2016), 67–113. MR 3617779. http://doi.org/10.2969/aspm/07010067 CrossRefGoogle Scholar
Gromov, M., On the entropy of holomorphic maps . Enseign. Math. 49(2003), no. 3–4, 217–235, preprint SUNY (1977). MR 2026895. http://doi.org/10.5169/seals-66687 Google Scholar
Höring, A. and Peternell, T., Algebraic integrability of foliations with numerically trivial canonical bundle . Invent. Math. 216(2019), no. 2, 395–419. MR 3953506. http://doi.org/10.1007/s00222-018-00853-2 CrossRefGoogle Scholar
Hu, F., A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic (with an appendix by Tomohide Terasoma) . Math. Ann. 377(2020), no. 3–4, 1573–1602. MR 4126902. http://doi.org/10.1007/s00208-019-01812-9 CrossRefGoogle Scholar
Kawamata, Y., Minimal models and the Kodaira dimension of algebraic fiber spaces . J. Reine Angew. Math. 363(1985), 1–46. MR 814013. http://doi.org/10.1515/crll.1985.363.1 Google Scholar
Keum, J. H., Oguiso, K., and Zhang, D.-Q., Conjecture of Tits type for complex varieties and theorem of Lie-Kolchin type for a cone . Math. Res. Lett. 16(2009), no. 1, 133–148. MR 2480567. http://doi.org/10.4310/MRL.2009.v16.n1.a13 CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, UK, 1998. MR 1658959. http://doi.org/10.1017/CBO9780511662560 CrossRefGoogle Scholar
Kollár, J., Miyaoka, Y., and Mori, S., Rationally connected varieties . J. Algebraic Geom. 1(1992), no. 3, 429–448. MR 1158625.Google Scholar
Kollár, J., Rational curves on algebraic varieties . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge. A Series of Modern Surveys in Mathematics, 32, Springer-Verlag, Berlin, 1996. MR 1440180. http://doi.org/10.1007/978-3-662-03276-3 Google Scholar
Lesieutre, J., Some constraints of positive entropy automorphisms of smooth threefolds . Ann. Sci. Éc. Norm. Supér. 51(2018), no. 6, 1507–1547. MR 3940903. http://doi.org/10.24033/asens.2380 CrossRefGoogle Scholar
Nakayama, N., Zariski-decomposition and abundance. MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208. http://doi.org/10.2969/msjmemoirs/014010000 Google Scholar
Nakayama, N., Intersection sheaves over normal schemes. J. Math. Soc. Japan. 62(2010), no. 2, 487–595. MR 2662853. http://doi.org/10.2969/jmsj/06220487CrossRefGoogle Scholar
Nakayama, N. and Zhang, D.-Q., Polarized endomorphisms of complex normal varieties. Math. Ann. 346(2010), no. 4, 991–1018. MR 2587100. http://doi.org/10.1007/s00208-009-0420-y CrossRefGoogle Scholar
Oguiso, K., Automorphisms of hyperkähler manifolds in the view of topological entropy. In: Algebraic geometry contemporary mathematics, Amer. Math. Soc. 422, Providence, RI, 2007, pp. 173–185. MR 2296437. http://doi.org/10.1090/conm/422/08060 CrossRefGoogle Scholar
Oguiso, K. and Sakurai, J., Calabi-Yau threefolds of quotient type. Asian J. Math. 5(2001), no. 1, 43–77. MR 1868164. http://doi.org/10.4310/AJM.2001.v5.n1.a5 CrossRefGoogle Scholar
Oguiso, K. and Truong, T. T., 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. MR 3329200.Google Scholar
Ueno, K., Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Mathematics, 439, Springer-Verlag, Berlin, 1975. MR 0506253. http://doi.org/10.1007/BFb0070570CrossRefGoogle Scholar
Yomdin, Y., Volume growth and entropy. Israel J. Math. 57(1987), no. 3, 285–300. MR 889979. http://doi.org/10.1007/BF02766215 CrossRefGoogle Scholar
Zhang, D.-Q., A theorem of Tits type for compact Kähler manifolds. Invent. Math. 176(2009), no. 3, 449–459. MR 2501294. http://doi.org/10.1007/s00222-008-0166-2 CrossRefGoogle Scholar
Zhang, D.-Q., Algebraic varieties with automorphism groups of maximal rank. Math. Ann. 355(2013), no. 1, 131–146. MR 3004578. http://doi.org/10.1007/s00208-012-0783-3 CrossRefGoogle Scholar
Zhang, D.-Q., $n$ -Dimensional projective varieties with the action of an abelian group of rank $n-1$ . Trans. Am. Math. Soc. 368(2016), no. 12, 8849–8872. MR 3551591. http://doi.org/10.1090/tran/6629 CrossRefGoogle Scholar