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Science in China Series A: Mathematics

, Volume 48, Supplement 1, pp 123–145 | Cite as

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

  • Ngaiming MokEmail author
Article

Abstract

We study holomorphic immersions f:XM from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.

Keywords

second fundamental form harmonic form tangent sequence totally geodesic immersion holomorphic splitting 

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References

  1. 1.
    Gunning, R., On Uniformizations of Complex Manifolds, Mathematical Notes 22, Princeton: Princeton Univ. Press, 1978.Google Scholar
  2. 2.
    Van de Ven, A., A property of algebraic varieties in complex projective spaces, Colloque de géom. diff. gl., Bruxelles, 1958.Google Scholar
  3. 3.
    Mutsaţ, M., Popa, M., A new proof of a theorem of A. Van de Ven, Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser., 1997, 40: 49–55.Google Scholar
  4. 4.
    Jahnke, P., Submanifolds with splitting tangent sequence, arXiv:math.AG/0304223 v3, 2004.Google Scholar
  5. 5.
    Griffiths, P., Hermitian differential geometry, Chern classes and positive vector bundles, in Global Analysis, Tokyo and Princeton, 1969.Google Scholar
  6. 6.
    Molzon, R., Mortensen, K., The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. AMS, 1996, 348: 3015–3036.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Mok, N., Yeung, S. -K., Geometric realizations of uniformization of conjugates of Hermitian locally symmetric manifolds, Complex Analysis and Geometry, New York: Plenum Press, 1993, 253–270.Google Scholar
  8. 8.
    Feder, S., Immersions and embeddings in complex projective spaces, Topology, 1965, 4: 143–158.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cao, H., Mok, N., Holomorphic immersions between compact hyperbolic space forms, Invent. Math., 1990, 100: 49–61.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mok, N., Factorization of semisimple discrete representations of Kähler groups, Invent. Math., 1992, 110: 557–614.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Okonek, C., Schneider, M., Spindler, H., Vector Bundles on Complex Projective Spaces, Progress in Mathematics, Boston-Basel-Stuttgart: Birkhäuser, 1980.Google Scholar
  12. 12.
    Horrocks, G., Mumford, D., A rank 2 vector bundle on P4 with 15,000 symmetries, Topology, 1973, 12: 63–81.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong UniversityHong Kong,China

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