On compact Kähler surfaces
Buchdahl, Nicholas
Annales de l'Institut Fourier, Tome 49 (1999), p. 287-302 / Harvested from Numdam
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Sans utiliser la classification des surfaces compactes complexes, on démontre qu’une telle surface dont le premier nombre de Betti est pair possède une métrique kählérienne, et qu’une version réelle du critère classique de Nakai-Moishezon est valable sur la surface.

Without relying on the classification of compact complex surfaces, it is proved that every such surface with even first Betti number admits a Kähler metric and that a real form of the classical Nakai-Moishezon criterion holds on the surface.

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     author = {Buchdahl, Nicholas},
     title = {On compact K\"ahler surfaces},
     journal = {Annales de l'Institut Fourier},
     volume = {49},
     year = {1999},
     pages = {287-302},
     doi = {10.5802/aif.1674},
     mrnumber = {2000f:32029},
     zbl = {0926.32025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_1_287_0}
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Buchdahl, Nicholas. On compact Kähler surfaces. Annales de l'Institut Fourier, Tome 49 (1999) pp. 287-302. doi : 10.5802/aif.1674. http://gdmltest.u-ga.fr/item/AIF_1999__49_1_287_0/

[BPV] W. Barth, C. Peters and A. Van De Ven, Compact Complex Surfaces, Berlin-Heidelberg-New York, Springer, 1984. | MR 86c:32026 | Zbl 0718.14023

[CP] F. Campana and T. Peternell, Algebraicity of the ample cone of projective varieties, J. reine angew. Math., 407 (1990), 160-166. | MR 91c:14012 | Zbl 0728.14004

[D1] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom., 1 (1992), 361-409. | MR 93e:32015 | Zbl 0777.32016

[D2] J.-P. Demailly, Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, in: Contributions to complex analysis and analytic geometry: dedicated to Pierre Dolbeault, ed. H. Skoda and J. M. Trépreau, Wiesbaden, Vieweg 1994. | Zbl 0824.53064

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

[GT] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed. Berlin-Heidelberg-New York, Springer, 1983. | MR 86c:35035 | Zbl 0562.35001

[GH] P. A. Griffiths and J. Harris, Principles of Algebraic Geometry, New York, Wiley, 1987.

[HL] R. Harvey and H. B. Lawson, An intrinsic characterisation of Kähler manifolds, Invent. Math., 74 (1983), 169-198. | MR 85b:32013 | Zbl 0553.32008

[MK] J. Morrow and K. Kodaira, Complex Manifolds, Holt-Rinehart & Wilson, New York, 1971. | MR 46 #2080 | Zbl 0325.32001

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

[S1] 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

[S2] Y.-T. Siu, Review of [T] in Mathematical Reviews, (1982) MR#82k:32065.

[S3] Y.-T. Siu, Every K3 surface is Kähler, Invent. Math., 73 (1983), 139-150. | MR 84j:32036 | Zbl 0557.32004

[T] A. N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces, Invent. Math., 61 (1980), 251-265. | MR 82k:32065 | Zbl 0472.14006

[Y] S.-T. Yau, On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation, Comm. Pure Appl. Math., 31 (1978), 339-411. | MR 81d:53045 | Zbl 0369.53059