Rungescher Satz and a condition for Steiness for the limit of an increasing sequence of Stein spaces
Annales de l'Institut Fourier, Volume 28 (1978) no. 2, p. 187-200

A necessary and sufficient condition, which is a weak converse of a classical theorem of Behnke-Stein, in order that a limit of Stein spaces be again a Stein space is proved.

Une condition nécessaire et suffisante pour qu’une limite d’espaces de Stein soit un espace de Stein est prouvée. Cette condition donne une réciproque faible d’un théorème classique de Behnke-Stein.

@article{AIF_1978__28_2_187_0,
     author = {Silva, Alessandro},
     title = {Rungescher Satz and a condition for Steiness for the limit of an increasing sequence of Stein spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {28},
     number = {2},
     year = {1978},
     pages = {187-200},
     doi = {10.5802/aif.695},
     zbl = {0365.32008},
     mrnumber = {58 \#22656},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1978__28_2_187_0}
}
Rungescher Satz and a condition for Steiness for the limit of an increasing sequence of Stein spaces. Annales de l'Institut Fourier, Volume 28 (1978) no. 2, pp. 187-200. doi : 10.5802/aif.695. http://www.numdam.org/item/AIF_1978__28_2_187_0/

[1] A. Andreotti et H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-260. | Numdam | MR 27 #343 | Zbl 0106.05501

[2] A. Andreotti et E. Vesentini, Les Théorèmes fondamentaux de la théorie des espaces holomorphiquement complets, in Sem. Ehresmann, Paris, (1962). | Numdam

[3] H. Behnke und K. Stein, Konvergente Folgen von Regularitatsbereichen und die meromorphe Konvexitat, Math. Annalen, 116 (1939), 204-216. | JFM 64.0322.03 | Zbl 0020.37803

[4] A. Cassa, Coomologia separata sulle varieta analitiche complesse, Annali SNS Pisa, 25 (1971), 291-323. | Numdam

[5] J. E. Fornaess, An increasing sequence of Stein manifolds whose limit is not Stein, Math. Annalen, 223 (1976), 275-277. | MR 54 #5498 | Zbl 0334.32017

[6] A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. Journal, II, 9 (1957), 119-183. | MR 21 #1328 | Zbl 0118.26104

[7] A. Hirschowitz, Pseudoconvexité au-dessus d'espaces plus ou moins homogènes, Inventiones Math., 26 (1974), 303-322. | MR 50 #10323 | Zbl 0275.32009

[8] Y. T. Siu, Non countable dimension of cohomology groups of analytic sheaves and domains of holomorphy, Math. Zeit., 102 (1967), 17-29. | MR 36 #5394 | Zbl 0167.06802

[9] F. Treves, Locally convex spaces and linear partial differential equations, Springer, Berlin (1967). | MR 36 #6986 | Zbl 0152.32104

[10] V. Villani, Un teorema di passaggio al limite per la coomologia degli spazi complessi, Rend. Sc. fis. mat. e nat. Accad. Lincei, 43 (1967), 168-170. | MR 37 #6487 | Zbl 0157.40501

[11] J. Wermer, An example concerning polynomial convexity, Math. Annalen, 139 (1959), 147-150. | MR 22 #12238 | Zbl 0094.28302

[12] A. Markoe, Runge families and inductive limits of Stein spaces, Ann. Inst. Fourier, 27 (1977), 117-128. | Numdam | MR 58 #28665 | Zbl 0323.32014

[13] J.-P. Ramis, G. Ruget et J. L. Verdier, Dualité Relative en Géométrie Analytique Complexe, Inv. Math., 13 (1971), 261-283. | MR 46 #7553 | Zbl 0218.14010

[14] A. Ogus, Local cohomological dimension, Ann. of Math., 98 (1973), 327-365. | MR 58 #22059 | Zbl 0308.14003