Courants kählériens et surfaces compactes. (Kähler currents and compact surfaces). (French) Zbl 0926.32026
The unified proof of the existence of Kähler metric on a surface of even Betti first number is given. The problem was set up by Kodaira [K. Kodaira and J. Morrow, ‘Complex manifolds’, New York: Holt, Rinehart and Winston (1971)], and the solution was received by Y.-T. Siu [Invent. Math. 73, 139-150 (1983; Zbl 0557.32004)].
The author uses the Demailly regularization theorem and the Siu decomposition theorem to prove that the compact complex manifold \(X\) having a Kähler current is the Kähler manifold outside of some analytic subset of codimension at least two. Using the Hodge symmetry, which follows from the hypothesis \(b_1=2 h^{0,1},\) and the criterion similar to that of Harvey-Lawson, the author proves elementary the existence of a Kähler current on \(X\), that, due to above-stated result, is enough to ensure the existence of the Kähler metric on \(X.\)
The author uses the Demailly regularization theorem and the Siu decomposition theorem to prove that the compact complex manifold \(X\) having a Kähler current is the Kähler manifold outside of some analytic subset of codimension at least two. Using the Hodge symmetry, which follows from the hypothesis \(b_1=2 h^{0,1},\) and the criterion similar to that of Harvey-Lawson, the author proves elementary the existence of a Kähler current on \(X\), that, due to above-stated result, is enough to ensure the existence of the Kähler metric on \(X.\)
Reviewer: A.V.Chernecky (Odessa)
Keywords:
strictly positive \(d\)-closed currents; Kähler currents; regularization; Kähler surfaces; Kähler metrics; compact surfacesCitations:
Zbl 0557.32004References:
| [1] | [BPV84] , et , Compact complex surfaces, Springer, Berlin, 1984. · Zbl 0718.14023 |
| [2] | [Be85] , Toutes les surfaces K3 sont kählériennes, Astérisque 126, Paris (1985). · Zbl 0574.32040 |
| [3] | [CK52] et , On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319-325. · Zbl 0046.30903 |
| [4] | [De92] , Regularization of closed positive currents and intersection theory, J. Alg. Geom., 1 (1992), 361-409. · Zbl 0777.32016 |
| [5] | [De93] , Monge-Ampère operators, Lelong numbers and intersection theory, in Complex Analysis and Geometry, Univ. Series in Math., edited by V. Ancona and A. Silva, Plenum Press, New-York, 1993. · Zbl 0792.32006 |
| [6] | [F83] , On compact complex manifolds in C without holomorphic 2-forms Publ. Res. Inst. Math. Sci. Kyoto, 19 (1983), 193-202. · Zbl 0522.32024 |
| [7] | [Gau77] , Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris, série A, 285 (1977), 387-390. · Zbl 0362.53024 |
| [8] | [Gau85] , Les métriques standard sur une surface à b1 pair, Astérisque 126, Paris (1985). |
| [9] | [H74] , Removable singularities for positive currents, Amer. J. Math., 96 (1974), 67-78. · Zbl 0293.32015 |
| [10] | [H77] , Holomorphic chains and their boundaries Proc. Symp. Pure Math., 30, Part I, AMS, Providence, R.I. (1977), 309-382. · Zbl 0374.32002 |
| [11] | [Hi75] , Flattening theorem in complex analytic geometry, Amer. J. Math., 97 (1975), 503-547. · Zbl 0307.32011 |
| [12] | [HL83] et , An intrinsic characterization of Kähler manifolds, Invent. Math., 74 (1983), 261-295. · Zbl 0553.32008 |
| [13] | [JS93] et , Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal., 3 (1993), 37-62. · Zbl 0784.32009 |
| [14] | [Ji93] , Currents, metrics and Moishezon manifolds, Pacific Journal of Math., 158 (1993), 335-351. · Zbl 0785.32011 |
| [15] | [K64] , On the structure of compact complex analytic surfaces (I), Amer. J. Math., 86 (1964), 751-798. · Zbl 0137.17501 |
| [16] | [KM71] et , Complex manifolds, New York: Holt, Rinehart and Winston, 1971. · Zbl 0325.32001 |
| [17] | [Le68] , Fonctions plurisousharmoniques et formes différentielles positives, Dunod, Paris, 1968. · Zbl 0195.11603 |
| [18] | [M83] , On the existence of special metrics in complex geometry, Acta Math., 143 (1983), 261-295. · Zbl 0531.53053 |
| [19] | [Miy74a] , Extension theorems for Kähler metrics, Proc. Japan Acad., 50 (1974), 407-410. · Zbl 0354.32010 |
| [20] | [Miy74b] , Kähler metrics on elliptic surfaces, Proc. Japan Acad., 50 (1974), 533-536. · Zbl 0354.32011 |
| [21] | [Sh] , Extension of positive line bundles and meromorphic maps, Invent. Math., 15 (1972), 332-347. · Zbl 0223.32017 |
| [22] | [Siu74] , Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27 (1974), 53-156. · Zbl 0289.32003 |
| [23] | [Siu83] , Every K3 surface is Kähler, Invent. Math., 73 (1983), 130-150. · Zbl 0557.32004 |
| [24] | [Su76] , Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. · Zbl 0335.57015 |
| [25] | [V] , Propriétés cohomologiques d’une classe de variétés analytiques complexes compactes, Sem. d’Analyse Lelong-Dolbeault-Skoda 1983-1984, Lecture Notes in Math., Vol. 1198, Springer, Berlin, 1985, 245-259. · Zbl 0591.32032 |
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