Skip to main content

Arnol’d’s type theorem on a neighborhood of a cycle of rational curves

Abstract

Arnol’d showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies a Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.

This is a preview of subscription content, access via your institution.

References

  1. V. I. Arnol’d, Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves, Funkcional Anal. i Prilozhen. 10–4 (1976), 1–12; English translation: Funct. Anal. Appl. 10–4 (1977), 249–257.

    Google Scholar 

  2. C. Camacho and H. Movasati, Neighborhoods of analytic varieties in complex manifolds, arXiv:math/0208058 [math.CV].

  3. M. Brunella, On Kähler surfaces with semipositive Ricci curvature, Riv. Mat. Univ. Parma(7) 1 (2010), 441–450.

    MATH  Google Scholar 

  4. H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.

    MathSciNet  Article  Google Scholar 

  5. K. Kodaira and D. C. Spencer, A theorem of completeness o f characteristic systems of complete continuous systems, Amer. J. Math. 81 (1959), 477–500.

    MathSciNet  Article  Google Scholar 

  6. T. Koike, On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, Ann. Inst. Fourier (Grenoble) 65 (2015), 1953–1967.

    MathSciNet  Article  Google Scholar 

  7. T. Koike, Ueda theory for compact curves with nodes, Indiana U. Math. J. 66 (2017), 845–876.

    MathSciNet  Article  Google Scholar 

  8. T. Koike and T. Uehara, A gluing construction of K3 surfaces, arXiv:1903.01444 [math.CV].

  9. H. B. Laufer, Normal Two-Dimensional Singularities, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1971.

    MATH  Google Scholar 

  10. T. Ohsawa, Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci. 20 (1984), 21–38.

    MathSciNet  Article  Google Scholar 

  11. C. L. Siegel, Iterations of analytic functions, Ann. of Math. (2) 43 (1942), 607–612.

    MathSciNet  Article  Google Scholar 

  12. Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976), 89–100.

    MathSciNet  Article  Google Scholar 

  13. H. Tsuji, Complex structures on S3× S3, Tohoku Math. J. (2) 36, 3 (1984), 351–376.

    MathSciNet  Google Scholar 

  14. T. Ueda, On the neighborhood of a compact complex curve with topologically trivial normal bundle, Math. Kyoto Univ. 22 (1983), 583–607.

    MathSciNet  MATH  Google Scholar 

  15. T. Ueda, Neighborhood of a rational curve with a node, Publ. RIMS, Kyoto Univ. 27 (1991), 681–693.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgment

The authorwould like to give heartful thanks to Prof. Tetsuo Ueda whose enormous support and insightful comments were invaluable. He thanks Dr. TakahiroMatsushita and Dr. Yuta Nozaki who gave him many valuable comments on the topological aspects of Levi-flat hypersurfaces which is treated in §5. He is supported by Leading Initiative for Excellent Young Researchers (No. J171000201).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takayuki Koike.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Koike, T. Arnol’d’s type theorem on a neighborhood of a cycle of rational curves. JAMA (2022). https://doi.org/10.1007/s11854-022-0208-5

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s11854-022-0208-5