On compact Kähler surfaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 287-302.

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.

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.

     author = {Buchdahl, Nicholas},
     title = {On compact {K\"ahler} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {287--302},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {1},
     year = {1999},
     doi = {10.5802/aif.1674},
     zbl = {0926.32025},
     mrnumber = {2000f:32029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1674/}
AU  - Buchdahl, Nicholas
TI  - On compact Kähler surfaces
JO  - Annales de l'Institut Fourier
PY  - 1999
DA  - 1999///
SP  - 287
EP  - 302
VL  - 49
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1674/
UR  - https://zbmath.org/?q=an%3A0926.32025
UR  - https://www.ams.org/mathscinet-getitem?mr=2000f:32029
UR  - https://doi.org/10.5802/aif.1674
DO  - 10.5802/aif.1674
LA  - en
ID  - AIF_1999__49_1_287_0
ER  - 
%0 Journal Article
%A Buchdahl, Nicholas
%T On compact Kähler surfaces
%J Annales de l'Institut Fourier
%D 1999
%P 287-302
%V 49
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1674
%R 10.5802/aif.1674
%G en
%F AIF_1999__49_1_287_0
Buchdahl, Nicholas. On compact Kähler surfaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 287-302. doi : 10.5802/aif.1674. http://archive.numdam.org/articles/10.5802/aif.1674/

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

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

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

[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

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

Cited by Sources: