SKT and Tamed Symplectic structures on solvmanifolds

We study the existence of strong K"ahler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G\mathrm{/}\mathrm{\Gamma }$ providing some negative results for some classes of solvmanifolds. In particular, we show that if either $J$ is invariant under the action of a nilpotent complement of the nilradical of $G$ or $J$ is abelian or $G$ is almost abelian (not of type (I)), then the solvmanifold $G\mathrm{/}\mathrm{\Gamma }$ cannot admit any symplectic form taming the complex structure $J$, unless $G\mathrm{/}\mathrm{\Gamma }$ is K"ahler. As a consequence, we show that the family of non-K"ahler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.