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A note about invariant SKT structures and generalized Kähler structures on flag manifolds

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  21 March 2012

Dmitri V. Alekseevsky  and
Liana David
Dmitri V. Alekseevsky
Affiliation:
School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK (d.aleksee@ed.ac.uk)
Liana David
Affiliation:
Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, Calea Grivitei nr. 21, 010702 Bucharest, Romania (liana.david@imar.ro)
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Abstract

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We prove that any invariant strong Kähler structure with torsion (SKT structure) on a flag manifold M = G/K of a semi-simple compact Lie group G is Kähler. As an application we describe invariant generalized Kähler structures on M.

Keywords

SKT structures(generalized) Kahler structuresflag manifold

MSC classification

Secondary: 53D18: Generalized geometries (à la Hitchin)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 543 - 549
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Alekseevsky, D. V., Flag manifolds, Zb. Rad. (Beogr.) 6(14) (1997), 3–35.Google Scholar
2.Alekseevsky, D. V. and Perelomov, A. M., Invariant Kähler–Einstein metrics on compact homogeneous spaces, Funct. Analysis Applic. 20 (1986), 171–182.CrossRefGoogle Scholar
3.Enrietti, N., Fino, A. and Vezzoni, L., Tamed Symplectic forms and SKT metrics, J. Symplectic Geom., in press.Google Scholar
4.Fernandez, M., Fino, A., Ugarte, L. and Villacampa, R., Strong Kähler with torsion structures from almost contact manifolds, Pac. J. Math. 249 (2011), 49–75.CrossRefGoogle Scholar
5.Fino, A., Parton, M. and Salamon, S., Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), 317–340.CrossRefGoogle Scholar
6.Fino, A. and Tomassini, A., Blow ups and resolutions of strong Kähler with torsion metrics, Adv. Math. 221 (2009), 914–935.CrossRefGoogle Scholar
7.Fino, A. and Tomassini, A., On astheno-Kähler metrics, J. Lond. Math. Soc. 83 (2011), 290–308.CrossRefGoogle Scholar
8.Gates, S. J., Hull, C. M. and Rocek, M., Twisted multiplets and new supersymmetric nonlinear σ models, Nucl. Phys. B248 (1984), 157–186.CrossRefGoogle Scholar
9.Gualtieri, M., Generalized complex geometry, DPhil thesis, University of Oxford (2003).Google Scholar
10.Gualtieri, M., Generalized Kähler geometry, preprint (math.DG/1007.3485).Google Scholar
11.Helgason, S., Differential geometry: Lie groups and symmetric spaces (Academic Press, New York, 1978).Google Scholar
12.Hitchin, N. G., Generalized Calabi–Yau manifolds, Q. J. Math. 54 (2003), 281–308.CrossRefGoogle Scholar
13.Lindström, U., Rocek, M., von Unge, R. and Zabzine, M., Generalized Kähler geometry and manifest = (2,2) supersymmetric nonlinear sigma-models, J. High Energy Phys. 07 (2005) 067.CrossRefGoogle Scholar
14.Wang, H. C., Closed manifolds with homogeneous complex structures, Am. J. Math. 76 (1954), 1–32.CrossRefGoogle Scholar