Tamed Symplectic forms and SKT metrics

Details Tamed Symplectic forms and SKT metrics Tamed Symplectic forms and SKT metrics

2010-02-16

arxiv.org/abs/1002.3099v3

Mathematics

Differential Geometry

19 pages. Final version of the paper "Hermitian-Symplectic structures and SKT metrics". To appear in J. Symplectic Geom

Scientific paper

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial $-closed, i.e. to strong K\"ahler with torsion (${\rm 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 K\"ahler 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 ${\rm SKT}$ metric, then $M$ is at most 2-step. As a consequence we classify 8-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

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