On a generalized Calabi-Yau equation
[Sur l’équation de Calabi-Yau généralisée]
Wang, Hongyu ; Zhu, Peng
Annales de l'Institut Fourier, Tome 60 (2010), p. 1595-1615 / Harvested from Numdam
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En travaillant sur l’équation de Calabi-Yau généralisée proposée par Gromov pour des variétés presque-Kalhériennes fermées, nous étendons le résultat de la non-existence prouvé en dimension complexe 2, à des dimensions arbitraires.

Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-Kähler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension 2.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/aif.2566
Classification:  53C07,  53D05,  58J99
Mots clés: équation de Calabi-Yau, forme symplectique, structur presque complexe, métrique Hermitienne, tenseur de Nijenhuis, fonction speudo holomorphe 
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Wang, Hongyu; Zhu, Peng. On a generalized Calabi-Yau equation. Annales de l'Institut Fourier, Tome 60 (2010) pp. 1595-1615. doi : 10.5802/aif.2566. http://gdmltest.u-ga.fr/item/AIF_2010__60_5_1595_0/

[1] Audin, M.; Gauduchon, P. Symplectic and almost complex manifolds, in Holomorphic curves in symplectic geometry, Ed. by M. Audin and J. Lafontaine, Progress in Math., Tome 117 (1994), pp. 41-76 | MR 1274926

[2] Calabi, E. The space of Kähler metrics, in Proceeding of the International Congress of Mathematicians Tome 2 (1954), p. 206-207

[3] Delanoë, P. Sur l’analogue presque-complexe de l’équation de Calabi-Yau, Osaka J. Math., Tome 33 (1996), pp. 829-846 | MR 1435456 | Zbl 0878.53030

[4] Donaldson, S. K. Two forms on four manifolds and elliptic equations, in Inspired by S. S. Chern, World Sci. Publ., Hackensack, NJ (2006), pp. 153-172 (in A memorial volume in honor of a great mathematician, Ed. by P. A. Griffiths) | MR 2313334 | Zbl 1140.58018

[5] Ehresmann, C.; Libermann, P. Sur les structures presque hermitiennes isotropes, C. R. Acad. Sci. Paris, Tome 232 (1951), pp. 1281-1283 | MR 40790 | Zbl 0042.15904

[6] Gauduchon, P. Hermitian connections and Dirac operators, Bull. Un. Mat. Ital., Tome B 11 (no. 2, suppl.) (1997), pp. 257-288 | MR 1456265 | Zbl 0876.53015

[7] Gromov, M. Pseudoholomorphic curves in symplectic manifolds, Invent. Math., Tome 82 (1985), pp. 307-347 | Article | MR 809718 | Zbl 0592.53025

[8] Mcduff, D.; Salamon, D. Introduction to symplectic topology, Oxford University Press (1998) | MR 1698616 | Zbl 0844.58029

[9] Moser, J. On the volume elements of a manifold, Trans. AMS, Tome 120 (1965), pp. 286-294 | Article | MR 182927 | Zbl 0141.19407

[10] Tosatti, V.; Weinkove, B.; Yau, S. Y. Taming symplectic forms and the Calabi-Yau equation, Proc. London Math. Soc., Tome 97 (2008), pp. 401-424 | Article | MR 2439667 | Zbl 1153.53054

[11] Wang, H.; Zhu, P. Calabi-Yau equation on closed symplectic 4-manifolds (in preparation)

[12] Weinkove, B. The Calabi-Yau equation on almost Kähler four manifolds, J. Diff. Geom., Tome 76 (2007), pp. 317-349 | MR 2330417 | Zbl 1123.32015

[13] Weyl, H. The Classical groups, Princeton University Press, Princeton Mathematical Series, Tome 1 (1973) | MR 1488158 | Zbl 1024.20502

[14] Yau, S. T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math., Tome 31 (1978) no. 3, pp. 339-411 | Article | MR 480350 | Zbl 0369.53059

[15] Zhu, P. On almost Hermitian manifolds, Yangzhou University (2008) (Ph. D. Thesis) | Zbl 1199.53050