Results in Mathematics

, Volume 33, Issue 1–2, pp 161–168 | Cite as

Cohomological q-convexity versus local q-completeness with corners

  • Viorel Vâjâitu


We show that if (D, π) is an unramified Riemann domain over a distinguished complex manifold X such that D is cohomologically q-convex, then π is locally q-complete with corners. We call X distinguished if for every point x of X there is a holomorphic line bundle \(\cal L\) on X (which may depend on x) so that the global sections \(\Gamma (X \cal L)\) of \(\cal L\) generate its 1-jets at x. Examples of distinguished complex manifolds include all complex submanifolds of Cm × Pn; in particular all Stein or projectively algebraic manifolds.

Key words

Unramified Riemann domains q-convexity functions q-convex with corners 

Mathematics Subject Classification (1991)

32 F 10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Andreotti, A.; Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes, Bull Soc. Math. France, 90 (1962), 193–259.MathSciNetMATHGoogle Scholar
  2. [2]
    Colţoiu, M.: A counterexample to the q-Levi problem in Pn, J. Math. Kyoto Univ., vol. 36, No. 2 (1996), 385–387.MathSciNetMATHGoogle Scholar
  3. [3]
    Docquier, F.; Grauert, H.: Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., 140 (1960), 94–123.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Eastwood, M. G.; Vigna Suria, G.: Cohomologically complete and pseudoconvex domains, Comment. Math. Helv., 55 (1980), 413–426.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Fujita, O.: Domaines pseudoconvexes d’ordre général et fonctions pseudoconvexes d’ordre général, J. Math. Kyoto Univ., 30 (1990), 637–649.MathSciNetMATHGoogle Scholar
  6. [6]
    Lelong, P.: Domaines convexes par raport aux fonctions plurisousharmoniques, J. An. Math., 2 (1952), 178–208.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Matsumoto, K.: Pseudoconvex Riemann domains of general order over Stein manifolds, Mem. Fac. Sci. Kyushu Univ., Ser. A, 44 (1990), 95–109.MATHGoogle Scholar
  8. [8]
    Peterneil, M.: Continuous q-convex exhaustion functions, Invent, math., 85(1986), 249–262.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Vâjâitu, V.: Some convexity properties of morphisms of complex spaces, Math. Z., 217 (1994), 215–245.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Vâjâitu, V.: On \(\cal P\)-complete morphisms of complex spaces, Proc. 3rd Int. Res. Inst., Hayama, 1995, Noguchi et. al eds., World Scientific Publ, Singapore (1996), 653–665.Google Scholar
  11. [11]
    Vâjâitu, V.: Cohomological q-convexity in projective manifolds, Kobe J. Math., 13(1996), 117–122.MathSciNetMATHGoogle Scholar
  12. [12]
    Vâjâitu, V.: Locally q-complete open sets in Stein spaces with isolated singularities, Kyushu J. Math., Vol. 51, No.2 (1997), 355–368.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1998

Authors and Affiliations

  1. 1.Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.FB 7 MathematikBergische Universität WuppertalWuppertalGermany

Personalised recommendations