Open Access
Translator Disclaimer
June 2012 Tamed Symplectic forms and Strong Kahler with torsion metrics
Nicola Enrietti, Anna Fino, Luigi Vezzoni
J. Symplectic Geom. 10(2): 203-223 (June 2012).

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.

Citation

Download Citation

Nicola Enrietti. Anna Fino. Luigi Vezzoni. "Tamed Symplectic forms and Strong Kahler with torsion metrics." J. Symplectic Geom. 10 (2) 203 - 223, June 2012.

Information

Published: June 2012
First available in Project Euclid: 7 June 2012

zbMATH: 1248.53070
MathSciNet: MR2926995

Rights: Copyright © 2012 International Press of Boston

JOURNAL ARTICLE
21 PAGES


SHARE
Vol.10 • No. 2 • June 2012
Back to Top