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Degeneracy of Holomorphic Curves into Algebraic Varieties II


In our former paper (Noguchi et al. in J. Math. Pures Appl. 88:293–306, 2007) we proved an algebraic degeneracy of entire holomorphic curves into a variety X which carries a finite morphism to a semi-abelian variety, but which is not isomorphic to a semi-abelian variety by itself. The finiteness condition of the morphism is necessary in general by example. In this paper we improve that finiteness condition under an assumption such that some open subset of non-singular points of X is of log-general type, and simplify the proof in (Noguchi et al. in J. Math. Pures Appl. 88:293–306, 2007), which was rather involved. As a corollary it implies that every entire holomorphic curve \(f:\mathbb{C} \to V\) into an algebraic variety V with \(\bar{q}(V)\geq\dim V=\bar {\kappa}(V)\) is algebraically degenerate, which is due to Winkelmann (dimV=2) (Winkelmann in Ann. Inst. Fourier 61:1517–1537, 2011) and Lu–Winkelmann (Lu and Winkelmann in Forum Math. 24:399–418, 2012).

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  1. Abhyankar, S.S.: Local Analytic Geometry. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  2. Lu, S.S.Y., Winkelmann, J.: Quasiprojective varieties admitting Zariski dense entire holomorphic curves. Forum Math. 24, 399–418 (2012)

    MATH  MathSciNet  Google Scholar 

  3. Noguchi, J.: Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties. Nagoya Math. J. 83, 213–233 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Noguchi, J., Ochiai, T.: Geometric Function Theory in Several Complex Variables. Transl. Math. Mon., vol. 80. Am. Math. Soc., Providence (1990)

    MATH  Google Scholar 

  5. Noguchi, J., Winkelmann, J., Yamanoi, K.: Degeneracy of holomorphic curves into algebraic varieties. J. Math. Pures Appl. 88, 293–306 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Noguchi, J., Winkelmann, J., Yamanoi, K.: The second main theorem for holomorphic curves into semi-abelian varieties II. Forum Math. 20, 469–503 (2008)

    MATH  MathSciNet  Google Scholar 

  7. Szabó, E.: Divisorial log terminal singularities. J. Math. Sci. Univ. Tokyo 1, 631–639 (1994)

    MATH  MathSciNet  Google Scholar 

  8. Winkelmann, J.: Degeneracy of entire curves in log surfaces with \(\bar{q}=2\). Ann. Inst. Fourier 61, 1517–1537 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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The authors thank Professor Osamu Fujino very much for discussions about singularities of algebraic varieties. Also we thank Professors Kenji Matsuki and Chikara Nakayama for informing us on literature about resolution of singularities.

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Correspondence to Junjiro Noguchi.

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Research of J.N. is supported in part by Grant-in-Aid for Scientific Research (B) 23340029.

Research of J.W. is supported in part by DFG-SFB/TR 12 (“Symmetries and Universality in mesoscopic systems”).

Research of K.Y. is supported in part by Grant-in-Aid for Scientific Research (C) 24540069.

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Noguchi, J., Winkelmann, J. & Yamanoi, K. Degeneracy of Holomorphic Curves into Algebraic Varieties II. Viet J Math 41, 519–525 (2013).

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  • Holomorphic curve
  • Entire curve
  • log-Bloch–Ochiai Theorem
  • Green–Griffiths Conjecture

Mathematics Subject Classification (2000)

  • 32H30
  • 14E99
  • 32H25