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On holomorphic maps into compact non-Kähler manifolds. (English) Zbl 1077.32003
Author’s abstract: We study the extension problem of holomorphic maps \(\sigma : H \to X\) 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 \(\Omega_ \sigma\) of extension for \(\sigma\) over \(\Delta\) is contained in a subdomain of \(\Delta\). For such manifolds, we define, in this paper, an invariant Hex\(_n(X)\) using the Hausdorff dimensions of the singular sets of \(\sigma\)’s and study its properties to deduce informations on the complex structure of \(X\).

32D15 Continuation of analytic objects in several complex variables
32D10 Envelopes of holomorphy
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32J17 Compact complex \(3\)-folds
32J18 Compact complex \(n\)-folds
Full Text: DOI Numdam EuDML
[1] Enveloppes d’holomorphie et prolongements d’hypersurfaces, Séminaire Pierre Lelong 1975-76, 578, 215-235, (1977), Springer · Zbl 0372.32008
[2] Levisches problem und rungescher satz für teilgebiete steinscher mannigfaltigkeiten, Math. Ann, 140, 94-123, (1960) · Zbl 0095.28004
[3] The Hartogs-type extension theorem for the meromorphic maps into compact Kähler manifolds, Invent. math., 109, 47-54, (1992) · Zbl 0738.32008
[4] Extension properties of meromorphic mappings with values in non-Kähler complex manifolds, (2003) · Zbl 1081.32010
[5] Prolongement d’applications holomorphes, Bull. Soc. math. France, 118, 229-240, (1990) · Zbl 0718.32013
[6] Factorization of compact complex 3-folds which admit certain projective structures, Tohoku Math. J., 41, 359-397, (1989) · Zbl 0686.32016
[7] Examples on an extension problem of holomorphic maps and a holomorphic 1-dimensional foliation, Tokyo J. Math, 13, 139-146, (1990) · Zbl 0718.32014
[8] Lectures on the theory of functions of several complex variables, (1958), Tata Inst. Fund. Research, Bombay · Zbl 0184.10903
[9] An example of holomorphic maps which cannot be extended meromorphically across a closed fractal subset, Mini-Conference on Algebraic Geometry (Saitama University, Urawa), 42-53, (2000)
[10] On the removal of singularities of analytic sets, Michigan Math. J, 15, 111-120, (1968) · Zbl 0165.40503
[11] Techniques of extension of analytic objects, (1974), Dekker, New York · Zbl 0294.32007
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