On certain non-Kahlerian strongly pseudoconvex manifolds

  • Vo Van Tan


It has been conjectured that strongly pseudoconvex manifoldsX such that its exceptional setS is an irreducible curve can be embedded biholomorphically into some ℂ N ×P m . In this paper we show that this is true, with one exception, namely when dim X = 3 and its first Chern classc 1 (K X ¦S) = 0 whereSP 1 andK X is the canonical bundle ofX. On the other hand, we explicitly exhibit such a 3-foldX that is not Kahlerian; also we construct non-Kahlerian strongly pseudoconvex 3-foldX whose exceptional setS is a ruled surface; those concrete examples naturally raise the possibility of classifying non-Kahlerian strongly pseudoconvex 3-folds.

Math Subject Classification

32F10 32C17 

Key Words and Phrases

Embeddable 1 -convex manifolds canonical bundle rational curves 


  1. [C]
    Coltoiu, M. On the embedding of 1-convex manifolds with 1-dimensional exceptional set.Comment. Math. Helve. 60, 458–465 (1985).MATHCrossRefMathSciNetGoogle Scholar
  2. [GR]
    Grauert, H. Uber Modifikationenund exzeptionelle analytische Mengen.Math. Ann. 146, 331–368 (1962).MATHCrossRefMathSciNetGoogle Scholar
  3. [HR]
    Hironaka, H., and Rossi, H. On the equivalence of imbeddings of exceptional complex spaces.Math. Ann. 156, 313–333(1964).MATHCrossRefMathSciNetGoogle Scholar
  4. [K]
    Kollar, J. Flips, Flops, Minimal models, etc.Surv. in Diff. Geom. 1, 113–199 (1991).MathSciNetGoogle Scholar
  5. [L]
    Laufer, H. OnC P 1 as exceptional set.Ann. Math. Studies, 261–275 (1981).Google Scholar
  6. [Mo]
    Moishezon, B. G. Onn-dimensional compact varieties withn algebraically independent meromorphic functionsI, II, III.AMS Translations Ser. 2 63, 51–177 (1967).MATHGoogle Scholar
  7. [Mr1]
    Mori, S. Projective manifolds with ample tangent bundles.Ann. Math. 110, 593–606 (1979).CrossRefGoogle Scholar
  8. [Mr2]
    Mori, S. Threefolds whose canonical bundles are not numerically effective.Ann. Math. 116, 133–176 (1982).CrossRefGoogle Scholar
  9. [N]
    Nakano, S. On the inverse of monoidal transformation.Publ. RIMS Kyoto University 6, 483–502 (1971).Ibid. 7, 637–644(1972).CrossRefGoogle Scholar
  10. [Ny]
    Nakayama, N. The lower semi-continuity of the plurigenera of complex varieties. Algebraic geometry, Sendai 1985.Adv. Studies in Pure Math. 10, 551–590 (1987).MathSciNetGoogle Scholar
  11. [R]
    Reid, M. Minimal models of canonical threefolds. Algebraic Varieties & Analytic Varieties.Adv. Stud. Pure Math. 1, 131–180 (1981).Google Scholar
  12. [V1]
    Tan, Vo Van. On the embedding problem for 1-convex spaces.Trans. A.M.S. 256, 185–197 (1979).MATHCrossRefGoogle Scholar
  13. [V2]
    Tan, Vo Van. Vanishing theorems and Kahlerity for strongly pseudoconvex manifolds.Trans. A.M.S. 261, 297–302 (1980).Ibid. 291 379–380 (1985).MATHCrossRefGoogle Scholar
  14. [V3]
    Tan, Vo Van. Embedding theorems & Kahlerity for 1-convex spaces.Comment. Math. Helve. 57, 196–201 (1982).MATHCrossRefGoogle Scholar
  15. [V4]
    Tan, Vo Van. On compactifiable strongly pseudoconvex threefolds.Manus. Mathe. 69, 333–338 (1990).MATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1994

Authors and Affiliations

  • Vo Van Tan
    • 1
  1. 1.Department of MathematicsSuffolk UniversityBoston

Personalised recommendations