Courants kählériens et surfaces compactes
Lamari, Ahcène
Annales de l'Institut Fourier, Tome 49 (1999), p. 263-285 / Harvested from Numdam
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Le théorème de régularisation de Demailly ramène l’existence d’une métrique kählérienne sur une surface compacte à celle d’un (1-1)-courant strictement positif d-fermé (“courant kählérien”). Après avoir démontré un critère d’existence d’un tel courant, nous utilisons la symétrie de Hodge pour donner une démonstration unifiée du caractère kählérien des surfaces compactes à premier nombre de Betti pair.

A compact complex surface is shown to be Kähler if and only if it carries a strictly positive d-closed current (in other words, a Kähler current), thanks to Demailly’s regularization theorem. We prove a Harvey-Lawson type characterization of compact manifolds carrying such a current. Using Hodge symmetry, we then give a unified proof of kählerianity for surfaces with even first Betti number.

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     author = {Lamari, Ahc\`ene},
     title = {Courants k\"ahl\'eriens et surfaces compactes},
     journal = {Annales de l'Institut Fourier},
     volume = {49},
     year = {1999},
     pages = {263-285},
     doi = {10.5802/aif.1673},
     mrnumber = {2000d:32034},
     zbl = {0926.32026},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_1_263_0}
}

Lamari, Ahcène. Courants kählériens et surfaces compactes. Annales de l'Institut Fourier, Tome 49 (1999) pp. 263-285. doi : 10.5802/aif.1673. http://gdmltest.u-ga.fr/item/AIF_1999__49_1_263_0/

[BPV84] W. Barth, C. Peters et A. Van De Ven, Compact complex surfaces, Springer, Berlin, 1984. | MR 86c:32026 | Zbl 0718.14023

[Be85] A. Beauville, Toutes les surfaces K3 sont kählériennes, Astérisque 126, Paris (1985). | Zbl 0574.32040

[CK52] W.-L. Chow et K. Kodaira, On analytic surfaces with two independent meromorphic functions, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319-325. | MR 14,37e | Zbl 0046.30903

[De92] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom., 1 (1992), 361-409. | MR 93e:32015 | Zbl 0777.32016

[De93] J.-P. Demailly, 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. | MR 94k:32009 | Zbl 0792.32006

[F83] A. Fujiki, On compact complex manifolds in C without holomorphic 2-forms Publ. Res. Inst. Math. Sci. Kyoto, 19 (1983), 193-202. | MR 84m:32037 | Zbl 0522.32024

[Gau77] P. Gauduchon, Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris, série A, 285 (1977), 387-390. | MR 57 #10664 | Zbl 0362.53024

[Gau85] P. Gauduchon, Les métriques standard sur une surface à b1 pair, Astérisque 126, Paris (1985).

[H74] R. Harvey, Removable singularities for positive currents, Amer. J. Math., 96 (1974), 67-78. | MR 50 #13602 | Zbl 0293.32015

[H77] R. Harvey, Holomorphic chains and their boundaries Proc. Symp. Pure Math., 30, Part I, AMS, Providence, R.I. (1977), 309-382. | MR 56 #5929 | Zbl 0374.32002

[Hi75] H. Hironaka, Flattening theorem in complex analytic geometry, Amer. J. Math., 97 (1975), 503-547. | MR 52 #14365 | Zbl 0307.32011

[HL83] R. Harvey et H.B. Jr Lawson, An intrinsic characterization of Kähler manifolds, Invent. Math., 74 (1983), 261-295. | Zbl 0553.32008

[JS93] S. Ji et B. Shiffman, Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal., 3 (1993), 37-62. | MR 93m:32014 | Zbl 0784.32009

[Ji93] S. Ji, Currents, metrics and Moishezon manifolds, Pacific Journal of Math., 158 (1993), 335-351. | MR 94m:32044 | Zbl 0785.32011

[K64] K. Kodaira, On the structure of compact complex analytic surfaces (I), Amer. J. Math., 86 (1964), 751-798. | MR 32 #4708 | Zbl 0137.17501

[KM71] K. Kodaira et J. Morrow, Complex manifolds, New York: Holt, Rinehart and Winston, 1971. | MR 46 #2080 | Zbl 0325.32001

[Le68] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Dunod, Paris, 1968. | MR 39 #4436 | Zbl 0195.11603

[M83] M.-L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math., 143 (1983), 261-295. | Zbl 0531.53053

[Miy74a] Y. Miyaoka, Extension theorems for Kähler metrics, Proc. Japan Acad., 50 (1974), 407-410. | MR 57 #3432 | Zbl 0354.32010

[Miy74b] Y. Miyaoka, Kähler metrics on elliptic surfaces, Proc. Japan Acad., 50 (1974), 533-536. | MR 57 #723 | Zbl 0354.32011

[Sh] B. Shiffman, Extension of positive line bundles and meromorphic maps, Invent. Math., 15 (1972), 332-347. | MR 51 #10689 | Zbl 0223.32017

[Siu74] Y.-T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27 (1974), 53-156. | MR 50 #5003 | Zbl 0289.32003

[Siu83] Y.-T. Siu, Every K3 surface is Kähler, Invent. Math., 73 (1983), 130-150. | MR 84j:32036 | Zbl 0557.32004

[Su76] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. | MR 55 #6440 | Zbl 0335.57015

[V] J. Varouchas, 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