Rationality of rationally connected threefolds admitting non-isomorphic endomorphisms
HTML articles powered by AMS MathViewer
- by De-Qi Zhang PDF
- Trans. Amer. Math. Soc. 364 (2012), 6315-6333 Request permission
Abstract:
We prove a structure theorem for non-isomorphic endomorphisms of weak $\mathbb {Q}$-Fano threefolds or, more generally, for threefolds with a big anti-canonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be rational. As a consequence, we show (without using the classification) that every smooth Fano threefold having a non-isomorphic surjective endomorphism is rational.References
- E. Amerik, M. Rovinsky, and A. Van de Ven, A boundedness theorem for morphisms between threefolds, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 405–415 (English, with English and French summaries). MR 1697369
- Garrett Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274–276. MR 214605, DOI 10.2307/2316020
- Jungkai Alfred Chen, Meng Chen, and De-Qi Zhang, A non-vanishing theorem for ${\Bbb Q}$-divisors on surfaces, J. Algebra 293 (2005), no. 2, 363–384. MR 2172344, DOI 10.1016/j.jalgebra.2005.05.005
- Najmuddin Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109–122. MR 1995861
- Charles Favre, Holomorphic self-maps of singular rational surfaces, Publ. Mat. 54 (2010), no. 2, 389–432. MR 2675930, DOI 10.5565/PUBLMAT_{5}4210_{0}6
- Jun-Muk Hwang and Ngaiming Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), no. 4, 627–651. MR 1993759, DOI 10.1090/S1056-3911-03-00319-9
- V. A. Iskovskikh, On the rationality problem for conic bundles, Duke Math. J. 54 (1987), no. 2, 271–294. MR 899398, DOI 10.1215/S0012-7094-87-05416-0
- V. A. Iskovskikh, On the rationality problem for algebraic threefolds, Tr. Mat. Inst. Steklova 218 (1997), no. Anal. Teor. Chisel i Prilozh., 190–232 (Russian); English transl., Proc. Steklov Inst. Math. 3(218) (1997), 186–227. MR 1642381
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- 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
- J. Kollár and C. Xu, Fano Varieties with Large Degree Endomorphisms, arXiv:0901.1692.
- Curtis T. McMullen, Dynamics on $K3$ surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002), 201–233. MR 1896103, DOI 10.1515/crll.2002.036
- Masayoshi Miyanishi, Algebraic methods in the theory of algebraic threefolds—surrounding the works of Iskovskikh, Mori and Sarkisov, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 69–99. MR 715647, DOI 10.2969/aspm/00110069
- Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82), no. 2, 147–162. MR 641971, DOI 10.1007/BF01170131
- Shigefumi Mori and Yuri Prokhorov, On $\Bbb Q$-conic bundles, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 315–369. MR 2426350, DOI 10.2977/prims/1210167329
- Noboru Nakayama, Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56 (2002), no. 2, 433–446. MR 1934136, DOI 10.2206/kyushujm.56.433
- N. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms, Preprint 2 September 2008.
- 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
- Yu. G. Prokhorov and V. V. Shokurov, Towards the second main theorem on complements, J. Algebraic Geom. 18 (2009), no. 1, 151–199. MR 2448282, DOI 10.1090/S1056-3911-08-00498-0
- D.-Q. Zhang, On endomorphisms of algebraic surfaces, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 249–263. MR 1941637, DOI 10.1090/conm/314/05437
- 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
- D. -Q. Zhang, Dynamics of automorphisms of compact complex manifolds, Proceedings of The Fourth International Congress of Chinese Mathematicians (ICCM2007), Vol. II, pp. 678-689; also: arXiv:0801.0843.
- De-Qi Zhang, Polarized endomorphisms of uniruled varieties, Compos. Math. 146 (2010), no. 1, 145–168. With an appendix by Y. Fujimoto and N. Nakayama. MR 2581245, DOI 10.1112/S0010437X09004278
- Qi Zhang, Rational connectedness of log $\textbf {Q}$-Fano varieties, J. Reine Angew. Math. 590 (2006), 131–142. MR 2208131, DOI 10.1515/CRELLE.2006.006
Additional Information
- De-Qi Zhang
- Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
- MR Author ID: 187025
- ORCID: 0000-0003-0139-645X
- Email: matzdq@nus.edu.sg
- Received by editor(s): September 18, 2009
- Received by editor(s) in revised form: September 25, 2010
- Published electronically: June 29, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6315-6333
- MSC (2010): Primary 14E20, 14J45, 14E08, 32H50
- DOI: https://doi.org/10.1090/S0002-9947-2012-05500-0
- MathSciNet review: 2958937