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Tamed Symplectic forms and SKT metrics

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial $-closed, i.e. to strong Kähler with torsion (${\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 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 ${\rm SKT}$ 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.