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Mathematische Zeitschrift

, Volume 281, Issue 3–4, pp 967–991 | Cite as

Toward a higher codimensional Ueda theory

  • Takayuki KoikeEmail author
Article

Abstract

Ueda’s theory is a theory on a flatness criterion around a smooth hypersurface of a certain type of topologically trivial holomorphic line bundles. We propose a codimension two analogue of Ueda’s theory. As an application, we give a sufficient condition for the anti-canonical bundle of the blow-up of the three dimensional projective space at 8 points to be non semi-ample however admit a smooth Hermitian metric with semi-positive curvature.

Keywords

Flat line bundles Ueda’s theory The blow-up of the three dimensional projective space at eight points 

Mathematics Subject Classification

32J25 14C20 

Notes

Acknowledgments

The author would like to give heartful thanks to his supervisor Prof. Shigeharu Takayama whose comments and suggestions were of inestimable value for my study. He also thanks Prof. Tetsuo Ueda and Prof. Yoshinori Gongyo for helpful comments and warm encouragements. He is supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 25-2869) and the Grant-in-Aid for JSPS fellows. This work is supported by the Program for Leading Graduate Schools, MEXT, Japan.

References

  1. 1.
    Brunella, M.: On Kähler surfaces with semipositive Ricci curvature. Riv. Mat. Univ. Parma 1, 441–450 (2010)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Camacho, C., Movasati, H.: Neighborhoods of analytic varieties, Monografías del Instituto de Matemática y Ciencias Afines, 35. Instituto de Matemática y Ciencías Afines, IMCA, Lima; Pontificia Universidad Católica del Perú, Lima (2003)Google Scholar
  3. 3.
    Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kodaira, K., Spencer, D.C.: A theorem of completeness of characteristic systems of complete continuous systems. Am. J. Math. 81, 477–500 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Koike, T.: On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, to appear in Ann. Inst. Fourier (Grenoble). arXiv:1312.6402
  6. 6.
    Koike, T.: On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles, to appear in Kyoto J. Math. arXiv:1405.4698
  7. 7.
    Laufer, H.B.: Normal two-dimensional singularities. Ann. Math. Stud. 71. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1971)Google Scholar
  8. 8.
    Lesieutre, J., Ottem, J.C.: Curves disjoint from a nef divisor. arXiv:1410.4467
  9. 9.
    Neeman, A.: Ueda theory: theorems and problems. Am. Math. Soc. 81(415) (1989)Google Scholar
  10. 10.
    Rossi, H.: Strongly pseudoconvex manifolds. Lect. Mod. Anal. Appl. I Lect. Notes Math. 103, 10–29 (1969)Google Scholar
  11. 11.
    Siegel, C.L.: Iterations of analytic functions. Ann. Math. 43, 607–612 (1942)zbMATHCrossRefGoogle Scholar
  12. 12.
    Totaro, B.: Moving codimension-one subvarieties over finite fields. Am. J. Math. 131, 1815–1833 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Ueda, T.: On the neighborhood of a compact complex curve with topologically trivial normal bundle. Math. Kyoto Univ. 22, 583–607 (1983)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoMeguro-kuJapan

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