We study the existence of strong Kahler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures J on solvmanifolds G / 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 / Gamma cannot admit any symplectic form taming the complex structure J, unless GIP is Kahler. As a consequence, we show that the family of non-lahler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.
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