Hostname: page-component-76c49bb84f-229nc Total loading time: 0 Render date: 2025-07-12T11:26:43.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Polarized endomorphisms of uniruled varieties. With an appendix by Y. Fujimoto and N. Nakayama

Part of: Birational geometry Surfaces and higher-dimensional varieties Holomorphic mappings and correspondences

Published online by Cambridge University Press:  21 December 2009

De-Qi Zhang
De-Qi Zhang*
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore (email: matzdq@nus.edu.sg)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on ℚ-Fano threefolds, Gorenstein log del Pezzo surfaces and ℙ1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree greater than one is rational.

Keywords

polarized endomorphismuniruled varietyrationality of variety

MSC classification

Secondary: 14E20: Coverings 14J45: Fano varieties 14E08: Rationality questions 32H50: Iteration problems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 145 - 168
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Amerik, E., Rovinsky, M. and Van de Ven, A., A boundedness theorem for morphisms between threefolds, Ann. Inst. Fourier (Grenoble) 49 (1999), 405–415.Google Scholar
[2]Birkhoff, G., Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274–276.Google Scholar
[3]Chen, J. A., Chen, M. and Zhang, D.-Q., A non-vanishing theorem for -divisors on surfaces, J. Algebra 293 (2005), 363–384.CrossRefGoogle Scholar
[4]Cutkosky, S., Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), 521–525.Google Scholar
[5]Dinh, T.-C. and Sibony, N., Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 307–336.CrossRefGoogle Scholar
[6]Fakhruddin, N., Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109–122.Google Scholar
[7]Favre, C., Holomorphic self-maps of singular rational surfaces, arXiv:0809.1724.Google Scholar
[8]Fujimoto, Y. and Nakayama, N., Complex projective manifolds which admit non-isomorphic surjective endomorphisms, RIMS Kôkyûroku Bessatsu B9 (2008), 51–80.Google Scholar
[9]Hwang, J.-M. and Mok, N., Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), 627–651.CrossRefGoogle Scholar
[10]Hwang, J.-M. and Nakayama, N., On endomorphisms of Fano manifolds of Picard number one, Preprint RIMS-1628, Kyoto University (2008).Google Scholar
[11]Iskovskikh, V. A., On the rationality problem for algebraic threefolds, Proc. Steklov Inst. Math. 218 (1997), 186–227.Google Scholar
[12]Kawamata, Y., The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), 606–633.Google Scholar
[13]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 ed. Oda, T. (Kinokuniya, Tokyo, and North-Holland, 1987), 283–360.Google Scholar
[14]Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).Google Scholar
[15]Kollár, J. and Xu, C., Fano varieties with large degree endomorphisms, arXiv:0901.1692v1.Google Scholar
[16]Miyanishi, M., Algebraic methods in the theory of algebraic threefolds—surrounding the works of Iskovskikh, Mori and Sarkisov, Algebraic varieties and analytic varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1 (North-Holland, Amsterdam, 1983), 69–99.Google Scholar
[17]Mori, S. and Mukai, S., Classification of Fano 3-folds with B2⩾2, Manuscripta Math. 36 (1981), 147–162; Manuscripta Math. 110 (2003), 407 (Erratum).Google Scholar
[18]Mori, S. and Prokhorov, Y., On ℚ-conic bundles, Publ. Res. Inst. Math. Sci. 44 (2008), 315–369.Google Scholar
[19]Nakayama, N., Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56 (2002), 433–446.Google Scholar
[20]Nakayama, N., Zariski-decomposition and abundance, MSJ Memoirs, vol. 14 (Mathematical Society of Japan, Tokyo, 2004).Google Scholar
[21]Nakayama, N., Intersection sheaves over normal schemes, Preprint RIMS-1614, Kyoto University (2007), Proc. London Math. Soc. (3), doi: la i112/plms/pdp015.Google Scholar
[22]Nakayama, N., On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms, Preprint (2008), Math. Ann., to appear.Google Scholar
[23]Nakayama, N. and Zhang, D.-Q., Building blocks of étale endomorphisms of complex projective manifolds, Preprint RIMS-1577, Kyoto University (2007), Proc. London Math. Soc. (3), doi:10.1112/plms/pdp015.Google Scholar
[24]Nakayama, N. and Zhang, D.-Q., Polarized endomorphisms of complex normal varieties, Preprint RIMS-1613, Kyoto University (2007), Math. Ann., to appear.Google Scholar
[25]Sakai, F., Classification of normal surfaces, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46, Part 1 (American Mathematical Society, Providence, RI, 1987), 451–465.Google Scholar
[26]Serre, J.-P., Analogues kählériens de certaines conjectures de Weil, Ann. of Math. (2) 71 (1960), 392–394.CrossRefGoogle Scholar
[27]Zhang, D.-Q., On endomorphisms of algebraic surfaces, in Topology and geometry: commemorating SISTAG, Contemporary Mathematics, vol. 314 (American Mathematical Society, Providence, RI, 2002), 249–263.Google Scholar
[28]Zhang, D.-Q., Dynamics of automorphisms of compact complex manifolds, in Proceedings of the Fourth International Congress of Chinese Mathematicians (ICCM2007), vol. II, 678–689 (Higher Education Press, Beijing, China), arXiv:0801.0843.Google Scholar
[29]Zhang, D.-Q., Rationality of rationally connected varieties, Preprint (2008).Google Scholar
[30]Zhang, D.-Q., Polarized endomorphisms of uniruled varieties, Preprint (2008).Google Scholar
[31]Zhang, S.-W., Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. 10 (International Press, Somerville, MA, 2006), 381–430.Google Scholar