On complete intersections

Forstnerič, Franc

Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 497-512.

Nous construisons des sous-variétés complexes fermées de ${ℂ}^n$, qui sont différentiablement des intersections complètes, mais qui ne le sont pas holomorphiquement. Nous établissons aussi un principe d’homotopie pour l’élimination des intersections avec certaines sous-variétés holomorphes de ${ℂ}^n$.

We construct closed complex submanifolds of ${ℂ}^n$ which are differential but not holomorphic complete intersections. We also prove a homotopy principle concerning the removal of intersections with certain complex subvarieties of ${ℂ}^n$.

[FRo] F. Forstnerič; J.-P. Rosay Erratum: ``Approximation of biholomorphic mappings by automorphisms of ${ℂ}^n$, Invent. Math., Volume 118 (1994), pp. 573-574 | DOI | MR | Zbl

[Bis] E. Bishop Differentiable manifolds in complex Euclidean space, Duke Math. J., Volume 32 (1965), pp. 1-21 | DOI | MR | Zbl

[BK] J. Bochnak; W. Kucharz Complete intersections in differential topology and analytic geometry, Boll. Un. Mat. Ital.B (7), Volume 10 (1996) no. 4, pp. 1019-1041 | MR | Zbl

[Car] H. Cartan Espaces fibrés analytiques, Symposium Internat. de topologia algebraica, Mexico (1958) (Oeuvres), Volume vol. 2 (1979), pp. 97-121 | MR | Zbl

[Chi] E.M. Chirka Complex analytic sets, Kluwer Academic Publishers Group, Dordrecht, 1989 | MR | Zbl

[DE] P.G. Dixon; J. Esterle Michael's problem and the Poincaré--Fatou--Bieberbach phenomenon, Bull. Amer. Math. Soc. (N.S.), Volume 15 (1986), pp. 127-187 | DOI | MR | Zbl

[EGr] Y. Eliashberg; M. Gromov Embeddings of Stein manifolds, Ann. Math., Volume 136 (1992), pp. 123-135 | MR | Zbl

[For1] F. Forstnerič Complex tangents of real surfaces in complex surfaces, Duke Math. J., Volume 67 (1992), pp. 353-376 | DOI | MR | Zbl

[For2] F. Forstnerič Interpolation by holomorphic automorphisms and embeddings in ${ℂ}^n$, J. Geom. Anal., Volume 9 (1999) no. 1, pp. 93-118 | DOI | MR | Zbl

[FP1] F. Forstnerič; J. Prezelj Oka's principle for holomorphic fiber bundles with sprays, Math. Ann., Volume 317 (2000), pp. 117-154 | DOI | MR | Zbl

[FP2] F. Forstnerič; J. Prezelj Oka's principle for holomorphic submersions with sprays (1999) (Preprint) | Zbl

[FRa] O. Forster; K.J. Ramspott Analytische Modulgarben und Endromisbündel, Invent. Math., Volume 2 (1966), pp. 145-170 | DOI | MR | Zbl

[FRo] F. Forstnerič; J.-P. Rosay Approximation of biholomorphic mappings by automorphisms of ${ℂ}^n$, Invent. Math., Volume 112 (1993), pp. 323-349 | DOI | MR | Zbl

[GR] C. Gunning; H. Rossi Analytic functions of several complex variables, Prentice--Hall, Englewood Cliffs, 1965 | MR | Zbl

[Gra] H. Grauert Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., Volume 135 (1958), pp. 263-273 | DOI | MR | Zbl

[GRe] H. Grauert; R. Remmert Theory of Stein Spaces, Grundl. Math. Wiss., 227, Springer, New York, 1977 | MR | Zbl

[Gro] M. Gromov Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., Volume 2 (1989), pp. 851-897 | MR | Zbl

[Ham] H.A. Hamm Zur Homotopietyp Steinscher Räume, J. Reine Angew. Math., Volume 338 (1983), pp. 121-135 | DOI | MR | Zbl

[Hus] D. Husemoller Fibre bundles, Graduate Texts in Mathematics, 20, Springer-Verlag, New York, 1994 | MR | Zbl

[RRu] J.-P. Rosay; W. Rudin Holomorphic maps from ${ℂ}^n$ to ${ℂ}^n$, Trans. Amer. Math. Soc., Volume 310 (1988), pp. 47-86 | MR | Zbl

[Sch] M. Schneider On the number of equations needed to describe a variety, Complex analysis of several variables (Madison, Wis., 1982) (Proc. Sympos. Pure Math.), Volume 41 (1984), pp. 163-180 | Zbl

[Schür] J. Schürmann Embeddings of Stein spaces into affine spaces of minimal dimension, Math. Ann., Volume 307 (1997), pp. 381-399 | DOI | MR | Zbl

[Whi] G.W. Whitehead Elements of homotopy theory, Graduate Texts in Mathematics, 61, Springer, New York--Berlin, 1978 | MR | Zbl