Properties of compact complex manifolds carrying closed positive currents. (English) Zbl 0784.32009
The authors show that a compact complex manifold is Moishezon if and only if it carries a strictly positive integral (1,1)-current. Also the problem of intersecting closed positive currents is treated. Smoothing and approximation techniques are an essential part of the paper. Also several results of J. P. Demailly are used who was able to show in the meantime an analytic characterization, by singular Hermitian metrics, of big line bundles.
Reviewer: S.Kosarew (Saint-Martin-d’Hères)
MSC:
| 32C30 | Integration on analytic sets and spaces, currents |
| 32J99 | Compact analytic spaces |
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
References:
| [1] | Chow, W.-L.; Kodaira, K., On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sci. U.S.A., 38, 319-325 (1952) · Zbl 0046.30903 · doi:10.1073/pnas.38.4.319 |
| [2] | Demailly, J.-P., Champs magnétiques et inégalités de Morse pour la d″-cohomologie, Ann. Inst. Fourier, Grenoble, 35, 189-229 (1985) · Zbl 0565.58017 |
| [3] | Demailly, J.-P., Singular Hermitian metrics on positive line bundles, Complex Algebraic Varieties, Lecture Notes in Mathematics 1507, 87-104 (1992), New York: Springer-Verlag, New York · Zbl 0784.32024 · doi:10.1007/BFb0094512 |
| [4] | Demailly, J.-P. Regularization of closed positive currents and intersection theory. Preprint, 1991. |
| [5] | Gunning, R. C.; Rossi, H., Analytic Functions of Several Complex Variables (1965), Englewood Cliffs, NJ: Prentice-Hall, Englewood Cliffs, NJ · Zbl 0141.08601 |
| [6] | Harvey, R.; Knapp, A. W., Positive (p,p)-forms, Wirtinger’s inequality and currents, Value-Distribution Theory, 43-62 (1974), New York: Marcel Dekker, New York · Zbl 0287.53046 |
| [7] | Ji, S., Smoothing of currents and Moišezon manifolds, Several Complex Variables and Complex Geometry, A. M. S. Proceedings of Symposia in Pure Mathematics, 52, 273-282 (1991) · Zbl 0745.32013 |
| [8] | Ji, S. Currents, metrics and Moishezon manifolds. Pacific J. Math., to appear. · Zbl 0785.32011 |
| [9] | Miyaoka, Y., Extension theorems for Kähler metrics, Proc. Japan Acad., 50, 407-410 (1974) · Zbl 0354.32010 · doi:10.3792/pja/1195518893 |
| [10] | Moishezon, B., Onn -dimensional compact varieties withn algebraically independent meromorphic functions, Amer. Math. Soc. Translations, 63, 51-177 (1967) · Zbl 0186.26204 |
| [11] | Shiffman, B.; Sommese, A. J., Vanishing Theorems on Complex Manifolds (1985), Boston: Birkhäuser, Boston · Zbl 0578.32055 |
| [12] | Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27, 53-156 (1974) · Zbl 0289.32003 · doi:10.1007/BF01389965 |
| [13] | Siu, Y.-T., Every Stein subvariety admits a Stein neighborhood, Invent. Math., 38, 89-100 (1976) · Zbl 0343.32014 · doi:10.1007/BF01390170 |
| [14] | Siu, Y.-T., A vanishing theorem for semipositive line bundle over non-Kähler manifolds, J. Diff. Geom., 19, 431-452 (1984) · Zbl 0577.32031 |
| [15] | Siu, Y.-T., Some recent results in complex manifold theory related to vanishing theorems for the semipositive case, Arbeitstagung Bonn 1984, Lecture Notes in Mathematics 1111, 169-192 (1985), New York: Springer-Verlag, New York · Zbl 0577.32032 · doi:10.1007/BFb0084590 |
| [16] | Ueno, K., Classification Theory of Algebraic Varieties and Compact Complex Spaces (1975), New York: Springer-Verlag, New York · Zbl 0299.14007 |
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