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Behnke-Stein theorem on complex spaces with singularities

Published online by Cambridge University Press:  22 January 2016

M. Colţoiu
Affiliation:
Institute of Mathematics of the Romanian Academy, P. 0. Box 1-764 RO-70700 Bucharest, Romania
A. Silva
Affiliation:
Dipartimento di Matematica, “G. Castelnuovo” Università di Roma “La Sapienza” 1-00185 Roma, Italy
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Let X be a non-compact Riemann surface and DX an open subset. By a classical result due to Behnke and Stein [2] D is Runge in X (i.e. the restriction map has dense image) iff X\D has no compact connected components. In other words the obstruction to holomorphic approximation is purely topological. This result has been generalized to 1-dimensional Stein spaces by Mihalache in [11].

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1995

References

[ 1 ] Andreotti, A. and Grauert, H., Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193259.CrossRefGoogle Scholar
[ 2 ] Behnke, H. and Stein, K., Entwicklungen analytischer Funktionen auf Riemannsche Flachen. Math. Ann., 120 (1948), 430461.Google Scholar
[ 3 ] Coltoiu, M., w-concavity of w-dimensional complex spaces, Math. Z., 210 (1992), 203206.Google Scholar
[ 4 ] Demailly, J. P., Cohomology of q-convex spaces in top degrees, Math. Z., 204 (1990), 283295.CrossRefGoogle Scholar
[ 5 ] Diederich, K. and Fornaess, J. E., Smoothing q-convex functions in the singular case, Math. Ann., 273 (1986), 665671.Google Scholar
[ 6 ] Grauert, H. and Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Raurne, Invent. Math., 11 (1970), 263292.CrossRefGoogle Scholar
[ 7 ] Greene, R. E. and Embedding, H. Wu. of open riemannian manifolds by harmonic functions, Ann. Inst. Fourier, 25 (1975), 215235.CrossRefGoogle Scholar
[ 8 ] Hironaka, H., Desingularization of complex-analytic varieties, Actes Congrès Int. Math., 2 (1970), 627631.Google Scholar
[ 9 ] Eojasiewicz, S., Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa., 18 (1964), 449474.Google Scholar
[10] Lupacciolu, G., Topological properties of q-convex sets, Trans, Amer. Math. Soc, 337 (1993), 427435.Google Scholar
[11] Mihalache, N., The Runge theorem on 1-dimensional Stein spaces, Rev. Roumaine Math. Pures et Appl, 33 (1988), 7, 601611.Google Scholar
[12] Narasimhan, R., Complex analysis in one variable, Birkhäuser, 1985.CrossRefGoogle Scholar
[13] Ohsawa, T., Completeness of non-compact analytic spaces, Publ. Res. Inst. Math. Sci., 20 (1984), 683692.CrossRefGoogle Scholar
[14] Ohsawa, T., A vanishing theorem for proper direct images, Publ. Res. Inst. Math. Sci., 23 (1987), 243250.CrossRefGoogle Scholar
[15] Peternell, M., Algebraische Varietäten und Q-vollständige komplexe Räurne, Math. Z., 200 (1989), 547581.CrossRefGoogle Scholar
[16] Silva, A., Behnke-Stein theorem for analytic spaces, Trans. Amer. Math. Soc, 199 (1974), 317326.CrossRefGoogle Scholar
[17] Sorani, G., Homologie des q-paires de Runge, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 319332.Google Scholar
[18] Spanier, E. H., Algebraic topology, Me Graw Hill, 1966.Google Scholar
[19] Takegoshi, K., Relative vanishing theorems in analytic spaces, Duke Math. J., 52 (1985), 273279.CrossRefGoogle Scholar
[20] Vâjâitu, V., Approximation theorems and homology of q-Runge domains in complex spaces, to appear in J. reine angew. Math., 1994.Google Scholar
[21] Weinstock, B. M., An approximation theorem for ∂-closed forms of type (n, n =1), Proc. Amer. Math. Soc, 26 (1970), 625628.Google Scholar