On a generalized Calabi-Yau equation
[Sur l’équation de Calabi-Yau généralisée]
Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1595-1615.

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.

DOI : 10.5802/aif.2566
Classification : 53C07, 53D05, 58J99
Keywords: Calabi-Yau equation, symplectic form, almost complex structure, Hermitian metric, Nijenhuis tensor, pseudo holomorphic function
Mot clés : équation de Calabi-Yau, forme symplectique, structur presque complexe, métrique Hermitienne, tenseur de Nijenhuis, fonction speudo holomorphe
Wang, Hongyu 1 ; Zhu, Peng 2

1 Yangzhou University School of Mathematical Science Yangzhou, Jiangsu 225002 (P. R. China)
2 School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225002, (P. R. China)
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Wang, Hongyu; Zhu, Peng. On a generalized Calabi-Yau equation. Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1595-1615. doi : 10.5802/aif.2566. http://www.numdam.org/articles/10.5802/aif.2566/

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

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

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

[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 | Zbl

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

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

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

[8] McDuff, D.; Salamon, D. Introduction to symplectic topology, Oxford University Press, 1998 | MR | Zbl

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

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

[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., Volume 76 (2007), pp. 317-349 | MR | Zbl

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

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

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

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