Let be an open set of a Stein manifold of dimension such that for . We prove that is Stein if and only if every topologically trivial holomorphic line bundle on is associated to some Cartier divisor on .
@article{ASNSP_2007_5_6_2_323_0, author = {Abe, Makoto}, title = {Holomorphic line bundles and divisors on a domain of a {Stein} manifold}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {323--330}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {2}, year = {2007}, mrnumber = {2352521}, zbl = {1142.32007}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2007_5_6_2_323_0/} }
TY - JOUR AU - Abe, Makoto TI - Holomorphic line bundles and divisors on a domain of a Stein manifold JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 323 EP - 330 VL - 6 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2007_5_6_2_323_0/ LA - en ID - ASNSP_2007_5_6_2_323_0 ER -
%0 Journal Article %A Abe, Makoto %T Holomorphic line bundles and divisors on a domain of a Stein manifold %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 323-330 %V 6 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2007_5_6_2_323_0/ %G en %F ASNSP_2007_5_6_2_323_0
Abe, Makoto. Holomorphic line bundles and divisors on a domain of a Stein manifold. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, pp. 323-330. http://archive.numdam.org/item/ASNSP_2007_5_6_2_323_0/
[1] Holomorphic line bundles on a domain of a two-dimensional Stein manifold, Ann. Polon. Math. 83 (2004), 269-272. | MR | Zbl
,[2] Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. | Numdam | MR | Zbl
and ,[3] Finitezza e annullamento di gruppi di coomologia su uno spazio complesso, Boll. Un. Mat. Ital. B (6) 1 (1982), 131-142. | MR | Zbl
,[4] Cousin I condition and Stein spaces, Complex Var. Theory Appl. 50 (2005), 23-25. | MR | Zbl
,[5] Levisches Problem und Rungescher Satz fĂĽr Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94-123. | MR | Zbl
and ,[6] “Analytische Stellenalgebren”, Grundl. Math. Wiss., Vol. 176, Springer, Heidelberg, 1971. | MR | Zbl
and ,[7] “Theory of Stein Spaces”, Grundl. Math. Wiss., Vol. 236, Springer, Berlin-Heidelberg-New York, 1979, Translated by A. Huckleberry. | MR | Zbl
and ,[8] “Coherent Analytic Sheaves”, Grundl. Math. Wiss., Vol. 265, Springer, Berlin-Heidelberg-New York-Tokyo, 1984. | MR | Zbl
and ,[9] “Introduction to Holomorphic Functions of Several Variables”, Vol. 3, Wadsworth, Belmont, 1990. | Zbl
,[10] Two dimensional complex manifold with vanishing cohomology set, Math. Ann. 204 (1973), 1-12. | MR | Zbl
and ,[11] On sheaf cohomology and envelopes of holomorphy, Ann. of Math. 84 (1966), 102-118. | MR | Zbl
,[12] The cohomology of an open subspace of a Stein space, J. Reine Angew. Math. 318 (1980), 32-35. | MR | Zbl
and ,[13] Riemannsche Hebbarkeitssätze für Cohomologieklassen mit kompaktem Träger, Math. Ann. 164 (1966), 272-279. | MR | Zbl
,[14] Quelques problèmes globaux relatifs aux variétés de Stein, In: “Colloque sur les fonctions de plusieurs variables tenu à Bruxelles du 11 au 14 Mars 1953”, Centre belge de Recherches mathématiques, Librairie universitaire, Louvain, 1954, 57-68. | MR | Zbl
,[15] “Algèbre Locale. Multiplicités”, 3rd ed., Lecture Notes in Math., Vol. 11, Springer, Berlin-Heidelberg-New York, 1975. | Zbl
,[16] Non-countable dimensions of cohomology groups of analytic sheaves and domains of holomorphy, Math. Z. 102 (1967), 17-29. | MR | Zbl
,[17] Analytic sheaf cohomology groups of dimension of -dimensional complex spaces, Trans. Amer. Math. Soc. 143 (1969), 77-94. | MR | Zbl
,