no. 6
On holomorphic maps into compact non-Kähler manifolds
Kato, Masahide1; Okada, Noboru
1 Sophia University, Department of Mathematics, 7-1 Kioicho, Chiyoda-ku, Tokyo, 102-8554 (Japan)
Annales de l'Institut Fourier, Volume 54 (2004) no. 6, pp. 1827-1854.

We study the extension problem of holomorphic maps σ:H→X of a Hartogs domain H with values in a complex manifold X. For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain Ω σ of extension for σ over Δ is contained in a subdomain of Δ. For such manifolds, we define, in this paper, an invariant Hex n (X) using the Hausdorff dimensions of the singular sets of σ’s and study its properties to deduce informations on the complex structure of X.

On étudie le prolongement des applications holomorphes σ:H→X définies sur un ouvert de Hartogs H et à valeurs dans une variété holomorphe X. Pour les variétés kähleriennes compactes ainsi que pour certaines variétés compactes non kähleriennes le domaine maximal Ω σ de prolongement de σ au dessus du polydisque Δ est un domaine contenu dans Δ. Pour de telles variétés compactes, on définit, dans cet article, un invariant Hex n (X) qui utilise la dimension de Hausdorff de l’ensemble singulier de σ et on étudie ses propriétés afin d’en déduire des informations sur la structure complexe de X.

DOI: 10.5802/aif.2068
Classification: 32D10, 32D15, 32H02, 32J17, 32J18
Keywords: extension of holomorphic map, envelope of holomorphy, non-Kähler manifold
Mot clés : prolongement des applications holomorphes, enveloppe d'holomorphie, variété non-kählérienne
Kato, Masahide 1; Okada, Noboru 

1 Sophia University, Department of Mathematics, 7-1 Kioicho, Chiyoda-ku, Tokyo, 102-8554 (Japan) 
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Kato, Masahide; Okada, Noboru. On holomorphic maps into compact non-Kähler manifolds. Annales de l'Institut Fourier, Volume 54 (2004) no. 6, pp. 1827-1854. doi : 10.5802/aif.2068. https://aif.centre-mersenne.org/articles/10.5802/aif.2068/

[D] G. Dloussky Enveloppes d'holomorphie et prolongements d'hypersurfaces, Séminaire Pierre Lelong 1975-76 (Lec. Notes in Math), Volume 578 (1977), pp. 215-235 | Zbl

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

[I1] S. M. Ivashkovich The Hartogs-type extension theorem for the meromorphic maps into compact Kähler manifolds, Invent. math., Volume 109 (1992), pp. 47-54 | MR | Zbl

[I2] S. M. Ivashkovich Extension properties of meromorphic mappings with values in non-Kähler complex manifolds (2003) (preprint) | Zbl

[Kr] M. Krachni Prolongement d'applications holomorphes, Bull. Soc. math. France, Volume 118 (1990), pp. 229-240 | Numdam | MR | Zbl

[Kt1] Ma. Kato Factorization of compact complex 3-folds which admit certain projective structures, Tohoku Math. J., Volume 41 (1989), pp. 359-397 | MR | Zbl

[Kt2] Ma. Kato Examples on an Extension Problem of Holomorphic Maps and a Holomorphic 1-Dimensional Foliation, Tokyo J. Math, Volume 13 (1990), pp. 139-146 | MR | Zbl

[M] B. Malgrange Lectures on the theory of functions of several complex variables (1958) | Zbl

[O] N. Okada An example of holomorphic maps which cannot be extended meromorphically across a closed fractal subset, Mini-Conference on Algebraic Geometry (Saitama University, Urawa) (2000), pp. 42-53

[Sh] B. Shiffman On the removal of singularities of analytic sets, Michigan Math. J, Volume 15 (1968), pp. 111-120 | MR | Zbl

[Si] Y. T. Siu Techniques of extension of analytic objects, Dekker, New York, 1974 | MR | Zbl

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