Comptes Rendus
no. 4
Differential geometry
A characterization of balanced manifolds

Presented by: Jean-Pierre Demailly

Lucia Alessandrini1
1 Dipartimento di Matematica e Informatica, Università di Parma, Parma, Italy
Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 345-350.

A Hermitian metric on a complex manifold is Kähler if and only if it approximates the Euclidean metric to order 2 at each point, in a suitable coordinate system. We prove here an analogous characterization of balanced metrics, namely, a Hermitian metric is balanced if and only if its fundamental form ω has closed trace and ωi,j(z) does not contain linear terms involving zi,zj,zi¯,zj¯, for each point, in a suitable coordinate system.

Une métrique hermitienne de forme fondamentale ω sur une variété complexe M est kählérienne si et seulement s'il existe un système de cordonnées z sur un voisinage de chaque point de M, tel que la composante linéaire de ωi,j(z) s'annule. On montre ici un critère de semi-kählérianité, à savoir qu'une métrique hermitienne de forme ω sur M est semi-kählérienne si et seulement s'il existe un système de cordonnées z sur un voisinage de chaque point de M, tel que la part linéaire de ωi,j(z) ne contienne pas zi,zj,zi¯,zj¯, et que la trace de ω soit fermée.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.02.004

Lucia Alessandrini 1

1 Dipartimento di Matematica e Informatica, Università di Parma, Parma, Italy 
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Lucia Alessandrini. A characterization of balanced manifolds. Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 345-350. doi : 10.1016/j.crma.2014.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.004/

[1] L. Alessandrini; G. Bassanelli Metric properties of manifolds bimeromorphic to compact Kähler spaces, J. Differ. Geom., Volume 37 (1993), pp. 95-121

[2] L. Alessandrini; G. Bassanelli Wedge product of positive currents and balanced manifolds, Tohoku Math. J., Volume 60 (2008), pp. 123-134

[3] J.-P. Demailly Complex Analytic and Differential Geometry http://www.fourier.ujf_grenoble.fr/demailly/books.html (free accessible book)

[4] P. Griffiths; J. Harris Principles of Algebraic Geometry, John Wiley & Sons, Inc., New York, 1978

[5] M.L. Michelson On the existence of special metrics in complex geometry, Acta Math., Volume 143 (1983), pp. 261-295

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