Holomorphic line bundles and divisors on a domain of a Stein manifold

Abe, Makoto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 2, pp. 323-330.

Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^k(D,𝒪)=0$ for $2\le k\le n-1$. We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$.

Classification : 32E10, 32L10, 32Q28

[1] M. Abe, Holomorphic line bundles on a domain of a two-dimensional Stein manifold, Ann. Polon. Math. 83 (2004), 269-272. | MR | Zbl

[2] A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259. | Numdam | MR | Zbl

[3] E. Ballico, Finitezza e annullamento di gruppi di coomologia su uno spazio complesso, Boll. Un. Mat. Ital. B (6) 1 (1982), 131-142. | MR | Zbl

[4] E. Ballico, Cousin I condition and Stein spaces, Complex Var. Theory Appl. 50 (2005), 23-25. | MR | Zbl

[5] F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94-123. | MR | Zbl

[6] H. Grauert and R. Remmert, “Analytische Stellenalgebren”, Grundl. Math. Wiss., Vol. 176, Springer, Heidelberg, 1971. | MR | Zbl

[7] H. Grauert and R. Remmert, “Theory of Stein Spaces”, Grundl. Math. Wiss., Vol. 236, Springer, Berlin-Heidelberg-New York, 1979, Translated by A. Huckleberry. | MR | Zbl

[8] H. Grauert and R. Remmert, “Coherent Analytic Sheaves”, Grundl. Math. Wiss., Vol. 265, Springer, Berlin-Heidelberg-New York-Tokyo, 1984. | MR | Zbl

[9] R. C. Gunning, “Introduction to Holomorphic Functions of Several Variables”, Vol. 3, Wadsworth, Belmont, 1990. | Zbl

[10] J. Kajiwara and H. Kazama, Two dimensional complex manifold with vanishing cohomology set, Math. Ann. 204 (1973), 1-12. | MR | Zbl

[11] H. B. Laufer, On sheaf cohomology and envelopes of holomorphy, Ann. of Math. 84 (1966), 102-118. | MR | Zbl

[12] M. Raimondo and A. Silva, The cohomology of an open subspace of a Stein space, J. Reine Angew. Math. 318 (1980), 32-35. | MR | Zbl

[13] H.-J. Reiffen, Riemannsche Hebbarkeitssätze für Cohomologieklassen mit kompaktem Träger, Math. Ann. 164 (1966), 272-279. | MR | Zbl

[14] J.-P. Serre, 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] J.-P. Serre, “Algèbre Locale. Multiplicités”, 3rd ed., Lecture Notes in Math., Vol. 11, Springer, Berlin-Heidelberg-New York, 1975. | Zbl

[16] Y.-T. Siu, Non-countable dimensions of cohomology groups of analytic sheaves and domains of holomorphy, Math. Z. 102 (1967), 17-29. | MR | Zbl

[17] Y.-T. Siu, Analytic sheaf cohomology groups of dimension $n$ of $n$-dimensional complex spaces, Trans. Amer. Math. Soc. 143 (1969), 77-94. | MR | Zbl