Courants kählériens et surfaces compactes

Lamari, Ahcène

Annales de l'Institut Fourier, Tome 49 (1999) no. 1, pp. 263-285.

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.

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

[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