Journal of Symplectic Geometry

Tamed Symplectic forms and Strong Kahler with torsion metrics

Nicola Enrietti, Anna Fino, and Luigi Vezzoni

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Abstract

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial\bar{\partial}$-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if $M$ is a nilmanifold (not a torus) endowed with an invariant complex structure $J$, then $(M,J)$ does not admit any symplectic form taming $J$. Moreover, we show that if a nilmanifold $M$ endowed with an invariant complex structure $J$ admits an SKT metric, then $M$ is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

Article information

Source
J. Symplectic Geom., Volume 10, Number 2 (2012), 203-223.

Dates
First available in Project Euclid: 7 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1339096435

Mathematical Reviews number (MathSciNet)
MR2926995

Zentralblatt MATH identifier
1248.53070

Citation

Enrietti, Nicola; Fino, Anna; Vezzoni, Luigi. Tamed Symplectic forms and Strong Kahler with torsion metrics. J. Symplectic Geom. 10 (2012), no. 2, 203--223. https://projecteuclid.org/euclid.jsg/1339096435


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