On holomorphic maps into compact non-Kähler manifolds

Masahide Kato[1]; Noboru Okada

  • [1] Sophia University, Department of Mathematics, 7-1 Kioicho, Chiyoda-ku, Tokyo, 102-8554 (Japan)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 6, page 1827-1854
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Kato, Masahide, and Okada, Noboru. "On holomorphic maps into compact non-Kähler manifolds." Annales de l’institut Fourier 54.6 (2004): 1827-1854. <http://eudml.org/doc/116161>.

@article{Kato2004,
abstract = {We study the extension problem of holomorphic maps $\sigma : H \rightarrow X$ 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 $X$.},
affiliation = {Sophia University, Department of Mathematics, 7-1 Kioicho, Chiyoda-ku, Tokyo, 102-8554 (Japan)},
author = {Kato, Masahide, Okada, Noboru},
journal = {Annales de l’institut Fourier},
keywords = {extension of holomorphic map; envelope of holomorphy; non-Kähler manifold; Hartogs domain},
language = {eng},
number = {6},
pages = {1827-1854},
publisher = {Association des Annales de l'Institut Fourier},
title = {On holomorphic maps into compact non-Kähler manifolds},
url = {http://eudml.org/doc/116161},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Kato, Masahide
AU - Okada, Noboru
TI - On holomorphic maps into compact non-Kähler manifolds
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 6
SP - 1827
EP - 1854
AB - We study the extension problem of holomorphic maps $\sigma : H \rightarrow X$ 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 $X$.
LA - eng
KW - extension of holomorphic map; envelope of holomorphy; non-Kähler manifold; Hartogs domain
UR - http://eudml.org/doc/116161
ER -

References

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  3. S. M. Ivashkovich, The Hartogs-type extension theorem for the meromorphic maps into compact Kähler manifolds, Invent. math. 109 (1992), 47-54 Zbl0738.32008MR1168365
  4. S. M. Ivashkovich, Extension properties of meromorphic mappings with values in non-Kähler complex manifolds, (2003) Zbl1081.32010
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  6. Ma. Kato, Examples on an Extension Problem of Holomorphic Maps and a Holomorphic 1-Dimensional Foliation, Tokyo J. Math 13 (1990), 139-146 Zbl0718.32014MR1059019
  7. M. Krachni, Prolongement d'applications holomorphes, Bull. Soc. math. France 118 (1990), 229-240 Zbl0718.32013MR1087380
  8. B. Malgrange, Lectures on the theory of functions of several complex variables, (1958), Tata Inst. Fund. Research, Bombay Zbl0184.10903
  9. 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), 42-53 
  10. B. Shiffman, On the removal of singularities of analytic sets, Michigan Math. J 15 (1968), 111-120 Zbl0165.40503MR224865
  11. Y. T. Siu, Techniques of extension of analytic objects, (1974), Dekker, New York Zbl0294.32007MR361154

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