Compact Kähler manifolds with automorphism groups of maximal rank
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Abstract:
For an automorphism group $G$ on an $n$-dimensional ($n \ge 3$) normal projective variety or a compact Kähler manifold $X$ so that $G$ modulo its subgroup $N(G)$ of null entropy elements is an abelian group of maximal rank $n-1$, we show that $N(G)$ is virtually contained in $\mathrm {Aut}_0(X)$, the $X$ is a quotient of a complex torus $T$ and $G$ is mostly descended from the symmetries on the torus $T$, provided that both $X$ and the pair $(X, G)$ are minimal.References
- Arnaud Beauville, Some remarks on Kähler manifolds with $c_{1}=0$, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 1–26. MR 728605
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- Garrett Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274–276. MR 214605, DOI 10.2307/2316020
- Frederic Campana, Fei Wang, and De-Qi Zhang, Automorphism groups of positive entropy on projective threefolds, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1621–1638. MR 3145744, DOI 10.1090/S0002-9947-2013-05838-2
- S. Cantat and A. Zeghib, Holomorphic actions of higher rank lattices in dimension three, Preprint 2009.
- Tien-Cuong Dinh and Nessim Sibony, Groupes commutatifs d’automorphismes d’une variété kählérienne compacte, Duke Math. J. 123 (2004), no. 2, 311–328 (French, with English and French summaries). MR 2066940, DOI 10.1215/S0012-7094-04-12323-1
- Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. MR 481142, DOI 10.1007/BF01403162
- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI 10.1515/crll.1985.363.1
- JongHae Keum, Keiji Oguiso, and De-Qi Zhang, 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, DOI 10.4310/MRL.2009.v16.n1.a13
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- David I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 140–186. MR 521918
- Noboru Nakayama and De-Qi Zhang, Building blocks of étale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. (3) 99 (2009), no. 3, 725–756. MR 2551469, DOI 10.1112/plms/pdp015
- Noboru Nakayama and De-Qi Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991–1018. MR 2587100, DOI 10.1007/s00208-009-0420-y
- Keiji Oguiso, Automorphisms of hyperkähler manifolds in the view of topological entropy, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 173–185. MR 2296437, DOI 10.1090/conm/422/08060
- Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953
- N. I. Shepherd-Barron and P. M. H. Wilson, Singular threefolds with numerically trivial first and second Chern classes, J. Algebraic Geom. 3 (1994), no. 2, 265–281. MR 1257323
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
- De-Qi Zhang, Automorphism groups and anti-pluricanonical curves, Math. Res. Lett. 15 (2008), no. 1, 163–183. MR 2367182, DOI 10.4310/MRL.2008.v15.n1.a14
- De-Qi Zhang, A theorem of Tits type for compact Kähler manifolds, Invent. Math. 176 (2009), no. 3, 449–459. MR 2501294, DOI 10.1007/s00222-008-0166-2
- De-Qi Zhang, Dynamics of automorphisms on projective complex manifolds, J. Differential Geom. 82 (2009), no. 3, 691–722. MR 2534992
- De-Qi Zhang, The $g$-periodic subvarieties for an automorphism $g$ of positive entropy on a compact Kähler manifold, Adv. Math. 223 (2010), no. 2, 405–415. MR 2565534, DOI 10.1016/j.aim.2009.08.010
- De-Qi Zhang, Automorphism groups of positive entropy on minimal projective varieties, Adv. Math. 225 (2010), no. 5, 2332–2340. MR 2680167, DOI 10.1016/j.aim.2010.04.022
- De-Qi Zhang, Algebraic varieties with automorphism groups of maximal rank, Math. Ann. 355 (2013), no. 1, 131–146. MR 3004578, DOI 10.1007/s00208-012-0783-3
Additional Information
- De-Qi Zhang
- Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
- MR Author ID: 187025
- ORCID: 0000-0003-0139-645X
- Email: matzdq@nus.edu.sg
- Received by editor(s): August 23, 2012
- Published electronically: March 5, 2014
- Additional Notes: The author was supported by an ARF of NUS
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 3675-3692
- MSC (2010): Primary 32H50, 37C85, 32M05, 14J50, 32Q15
- DOI: https://doi.org/10.1090/S0002-9947-2014-06227-2
- MathSciNet review: 3192612