Tohoku Mathematical Journal

Compact complex surfaces admitting non-trivial surjective endomorphisms

Yoshio Fujimoto and Noboru Nakayama

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Smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphism. The algebraic case was dealt with earlier by the authors. The following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, a successive blowups of an Inoue surface with curves whose centers are nodes of curves, and an Inoue surface without curves satisfying a rationality condition.

Article information

Tohoku Math. J. (2), Volume 57, Number 3 (2005), 395-426.

First available in Project Euclid: 7 October 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32J15: Compact surfaces
Secondary: 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35} 14J27: Elliptic surfaces

Endomorphism elliptic surface non-algebraic surface $\roma{7}_{0}$ surface Kodaira surface Inoue surface


Fujimoto, Yoshio; Nakayama, Noboru. Compact complex surfaces admitting non-trivial surjective endomorphisms. Tohoku Math. J. (2) 57 (2005), no. 3, 395--426. doi:10.2748/tmj/1128703004.

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  • W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, Berlin, 1984.
  • I. Enoki, Surfaces of class ( \roma7_0) with curves, Tohoku Math. J. (2) 33 (1981), 453--492.
  • Y. Fujimoto, Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension, Publ. Res. Inst. Math. Sci. 38 (2002), 33--92.
  • Y. Fujimoto and E. Sato, On smooth projective threefolds with non-trivial surjective endomorphisms, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 143--145.
  • M. Inoue, On surfaces of class ( \roma7_0), Invent. Math. 24 (1974), 269--310.
  • M. Inoue, New surfaces with no meromorphic functions, Proceedings of the International Congress of Mathematics (Vancouver, 1974), 423--426, Canad. Math. Congress, Montreal, 1975.
  • M. Inoue, New surfaces with no meromorphic functions II, Complex analysis and algebraic geometry, 91--106, Iwanami Shoten, Tokyo, 1977.
  • Ma. Kato, Compact complex manifolds containing global spherical shells, I, Proceeding of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1997), 45--84, Kinokuniya, Tokyo, 1978.
  • K. Kodaira, On compact complex analytic surfaces, I--III, Ann. of Math. 71 (1960), 111--152, 77 (1963), 563--626, 78 (1963), 1--40.
  • K. Kodaira, On the structure of compact complex analytic surfaces, I--IV, Amer. J. Math. 86 (1964), 751--798; 88 (1966), 682--721, 90 (1968), 55--83, 90 (1968), 170--192.
  • J. Li, S. T. Yau and F. Zheng, A simple proof of Bogomolov's theorem on class ( \roma7_0) surfaces with ( b_2 = 0), Illinois J. Math. 34 (1990), 217--220.
  • Y. Miyaoka, Kähler metrics on elliptic surfaces, Proc. Japan Acad. 50 (1974), 533--536.
  • I. Nakamura, On surfaces of class ( \roma7_0) with curves, Invent. Math. 78 (1984), 393--443.
  • I. Nakamura, On surfaces of class ( \roma7_0) with curves, II, Tohoku Math. J. (2) 42 (1990), 475--516.
  • N. Nakayama, On Weierstrass models, Algebraic Geometry and Commutative Algebra. vol. II, 405--431, Kinokuniya, Tokyo, 1988.
  • N. Nakayama, Projective algebraic varieties whose universal covering spaces are biholomorphic to (\BCC^n), J. Math. Soc. Japan 51 (1999), 643--654.
  • N. Nakayama, Local structure of an elliptic fibration, Higher dimensional birational geometry (Kyoto, 1997), 185--295, Adv. Stud. in Pure Math. 35, Math. Soc. Japan, Tokyo, 2002.
  • N. Nakayama, Global structure of an elliptic fibration, Publ. Res. Inst. Math. Sci. 38 (2002), 451--649.
  • N. Nakayama, Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56 (2002), 433--446.
  • T. Oda, Torus embeddings and applications, Based on joint work with K. Miyake, Tata Inst. Fund. Res. Lectures on Math. and Phys. 57, Tata Inst. Fund. Res., Bombay, by Springer-Verlag, Berlin-New York, 1978.
  • A. Teleman, Projectively flat surfaces and Bogomolov's theorem on class ( \roma7_0) surfaces, Internat. J. Math. 5 (1994), 253--264.