×

zbMATH — the first resource for mathematics

The Hartogs-type extension theorem for meromophic mappings into compact Kähler manifolds. (English) Zbl 0738.32008
The author proves the following conjecture of P. Griffiths: Theorem. Let \(X\) be a compact Kähler manifold. Then any meromorphic map from a domain in a Stein manifold into \(X\) extends to a meromorphic map from the envelope of holomorphy of this domain into \(X\). The proof uses results about limits of sequences of analytic disks of bounded area and existence of Stein neighborhoods of such limits. An application to the covering manifolds of compact Kähler manifolds is given.
Reviewer: S.Ivashkovich

MSC:
32D15 Continuation of analytic objects in several complex variables
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32E10 Stein spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [D-G] Docquier, F., Grauert, H.: Levisches Problem und Rungescher Satz f?r Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann.140, 94-123 (1960) · Zbl 0095.28004
[2] [Gr] Griffiths, P.: Two theorems on extensions of holomorphic mappings. Invent. math.14, 27-62 (1971) · Zbl 0223.32016
[3] [Gm] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. math.82, 307-347 (1985) · Zbl 0592.53025
[4] [C-H] Carlson, J., Harvey, R.: A remark on the universal cover of a Moishezon space. Duke Math. J.43, 497-500 (1976) · Zbl 0333.32012
[5] [H] Hirschowitz, A.: Les deux types de m?romorphie different. Journ. reine und angew. Math.313, 157-160 (1980) · Zbl 0412.32006
[6] [Iv] Ivashkovich, S.: The Hartogs phenomenon for holomorphically convex K?hler manifolds. Math. USSR Izvestiya29, No. 1, 225-232 (1987) · Zbl 0618.32011
[7] [M-W] Mok N., Wong, B.: Characterization of bounded domains covering Zariski dense subsets of compact complex spaces. Amer. J. Math.105, 1481-1487 (1983) · Zbl 0553.32012
[8] [R] Remmert, R.: Holomorphe und meromorph? Abbildungen komplexer R?ume. Math. Ann.133, 328-370 (1957) · Zbl 0079.10201
[9] [Rh] de Rham, G.: Vari?t?s differentiables. Hermann, Paris, 1960
[10] [S-U] Sacks, J., Uhlenbeck, K.: The existence of minimal 2-spheres. Annals of Math.113, 1-24 (1981) · Zbl 0462.58014
[11] [Sh] Shiffman, B.: Extensions of holomorphic maps into Hermitian manifolds. Math. Ann.194, 249-258 (1971) · Zbl 0219.32007
[12] [Sb] Sibony, N.: Quelques problems de prolongement de courants en analyse complexe. Duke Math. J.52, 157-197 (1985) · Zbl 0578.32023
[13] [Si1] Siu, Y.-T.: Extension of meromorphic maps into K?hler manifolds. Annals of Math.102, 421-462 (1975) · Zbl 0318.32007
[14] [Si2] Siu, Y.-T.: Every Stein subvariety admits a Stein neighbourhood. Invent. Math.38, 89-100 (1976) · Zbl 0343.32014
[15] [Sz] Stolzenberg, G.: Volumes, Limits, and Extension of Analytic Varieties, Lecture Notes in Math.19 (1966) · Zbl 0142.33801
[16] [St] Stoll, W.: ?ber meromorphe Modifikation, II. Math. Z.61, 467-488 (1955)
[17] [Sg] Siegel, C.: Analytic functions of several complex variables. Institute for Advanced study, Princeton, 1949
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.