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Degeneracy of Holomorphic Curves into Algebraic Varieties II

Abstract

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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Acknowledgements

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). https://doi.org/10.1007/s10013-013-0051-1

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  • DOI: https://doi.org/10.1007/s10013-013-0051-1

Keywords

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

Mathematics Subject Classification (2000)

  • 32H30
  • 14E99
  • 32H25