Algebraicity criteria for compact complex manifolds. (English) Zbl 0606.32018
Hironaka exhibited a Moishezon 3-fold M which is not projective algebraic; namely, M contains 1-cycle cohomologous to zero. In this paper, an attempt to characterize such Moishezon 3-folds is considered. The author proves the following result: Theorem: Let M be a Moishezon 3- fold. Let us assume that M contains i) no effective curves cohomologous to zero and ii) no effective curves C and no positive closed currents T (arising from C) such that \(C+T\) is cohomologous to zero. Then M is projective algebraic.
This result also has been established by F. Campana [C. R. Acad. Sci., Paris, Ser. I 302, 477-480 (1986; Zbl 0595.32037)] by slightly different techniques. Also, it would be interesting to find out whether the condition in the previous result does indeed occur.
This result also has been established by F. Campana [C. R. Acad. Sci., Paris, Ser. I 302, 477-480 (1986; Zbl 0595.32037)] by slightly different techniques. Also, it would be interesting to find out whether the condition in the previous result does indeed occur.
Reviewer: Vo Van Tan
Citations:
Zbl 0595.32037References:
| [1] | Demailly, J. P.: Courants positifs extrémaux et conjecture de Hodge. Invent. Math.69, 347-374 (1982) · Zbl 0488.58001 · doi:10.1007/BF01389359 |
| [2] | Demailly, J. P.: Construction d’hypersurfaces irreductibles avec lieu donné dans 672-1. Ann. Inst. Fourier30, 219-236 (1980) · Zbl 0414.32004 |
| [3] | Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math.11, 263-292 (1970) · Zbl 0202.07602 · doi:10.1007/BF01403182 |
| [4] | Griffiths, Ph. A., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001 |
| [5] | Hartshorne, R.: Ample subvarieties of algebraic varieties. Lect. Notes Math. 156. Berlin, Heidelberg, New York: 1970 · Zbl 0208.48901 |
| [6] | Harvey, R.: Holomorphic chains and their boundaries. Proc. Pure Math.30, 309-382 (1975) |
| [7] | Harvey, R., Lawson, H. B.: An intrinsic characterization of Kähler manifolds. Invent. Math.74, 169-198 (1983) · Zbl 0553.32008 · doi:10.1007/BF01394312 |
| [8] | Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann.261, 43-46 (1982) · doi:10.1007/BF01456407 |
| [9] | Lelong, P.: Fonctions plurisousharmoniques et formes differentielles positives. New York: Gordon and Breach 1968 · Zbl 0195.11603 |
| [10] | Moi?ezon, B. G.: Onn-dimensional compact varieties withn algebraically independent meromorphic functions. Am. Math. Soc. Transl.63, 51-177 (1967) |
| [11] | Peternell, T.: Fast-positive Geradenbündel auf kompakten komplexen Mannigfaltigkeiten. Reine Angew. Math.356, 119-139 (1985) · Zbl 0549.32021 · doi:10.1515/crll.1985.356.119 |
| [12] | Peternell, T.: Hermitian metrics with singularities. In preparation |
| [13] | Schaefer, H. H.: Topological vector spaces. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0212.14001 |
| [14] | Siu, Y. T.: Analyticity of sets associated to Lelong numbers and the extension of positive closed currents. Invent. Math.27, 53-156 (1974) · Zbl 0289.32003 · doi:10.1007/BF01389965 |
| [15] | Skoda, H.: Prolongement des courants positifs fermé de masse finie. Invent. Math.66, 361-376 (1982) · Zbl 0488.58002 · doi:10.1007/BF01389217 |
| [16] | Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lect. Notes Math. 439. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0299.14007 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
