Colmatage de surfaces holomorphes et classification des surfaces compactes
Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 713-741.

On considère le problème du colmatage en dimension 2, où l’on examine sous quelle condition une hypersurface strictement pseudoconvexe dans une surface holomorphe est le bord d’un espace de Stein. On montre que l’exemple de Rossi d’une hypersurface strictement pseudoconvexe Σ, qui est le bord de deux domaines non relativement compacts, n’est jamais le bord d’un espace de Stein bien que les fonctions holomorphes définies dans un voisinage de Σ donnent des cartes locales. On démontre que dans une surface holomorphe compacte sans fonctions méromorphes vérifiant b1(S)=1, un tel phéomène ne peut se produire.

We consider the problem of filling in holes in dimension 2 where we examine under which condition a strictly pseudoconvex hypersurface in an analytic surface is the boundary of a Stein space. We show that Rossi’s example of a strictly pseudoconvex hypersurface Σ, which bounds two nonrelatively compact domains, is not the boundary of a Stein space although holomorphic functions in a neighbourhood of Σ give local charts. We show that in a compact complex surface S without nonconstant meromorphic functions with b1(S)=1, such a phenomenon cannot exist

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     title = {Colmatage de surfaces holomorphes et classification des surfaces compactes},
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Dloussky, Georges. Colmatage de surfaces holomorphes et classification des surfaces compactes. Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 713-741. doi : 10.5802/aif.1352. https://www.numdam.org/articles/10.5802/aif.1352/

[1] A. Andreotti and Y-T Siu, Projective embedding of pseudo-concave spaces, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 231-278. | EuDML | Numdam | MR | Zbl

[2] G. Dloussky, Structure des surfaces de Kato, Mémoire de la S.M.F., n° 14 (1984). | EuDML | Numdam | MR | Zbl

[3] G. Dloussky, Une construction élémentaire des surfaces d'Inoue-Hirzebruch, Math. Ann., 280 (1988), 663-682. | EuDML | MR | Zbl

[4] I. Enoki, Surfaces of class VII0 with curves, Tôhoku Math. J., 33 (1981), 453-492. | MR | Zbl

[5] R.C. Gunning, H. Rossi, Analytic functions of several variables, Prentice-Hall, 1965. | MR | Zbl

[6] G.M. Henkin, J. Leiterer, Andreotti-Grauert Theory by Integral Formulas, Mathematical Research, Vol. 43, Akademie-Verlag-Berlin (1988). | MR | Zbl

[7] Masahide Kato, Compact complex surfaces containing global strongly pseudoconvex hypersurfaces, Tôhoku Math. J., vol. 31 (1979), 537-547. | MR | Zbl

[8] L. Kaup, B. Kaup, Holomorphic Functions of several variables, W. de Gruyter, 1983. | Zbl

[9] H. Kerner, Überlagerungen und Holomorphiehüllen, Math. Ann., 144 (1961), 126-134. | EuDML | MR | Zbl

[10] K. Kodaira, On the structure of compact complex analytic surface. I, Am. J. of Math., 86 (1964), 751-798. | MR | Zbl

[11] N. Kuhlmann, Über holomorphe Abbildungen komplexer Raüme, Arch. der Math., 15 (1964), 81-90. | MR | Zbl

[12] Iku Nakamura, On surfaces of class VII0 with curves, Invent. Math., 78 (1984), 393-443. | Zbl

[13] H. Rossi, Attaching analytic spaces to an analytic space along a pseudoconcave boundary. Proc. of the Conference on complex Analysis. Minneapolis 1964, Springer-Verlag (1964). | Zbl

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