SKT and Tamed Symplectic structures on solvmanifolds

TL;DR

This paper investigates conditions under which solvmanifolds admit SKT metrics and taming symplectic forms, providing negative results for certain classes and linking to non-Kähler examples.

Contribution

It establishes new restrictions on the existence of taming symplectic forms on solvmanifolds with specific complex structures, extending understanding of their geometric properties.

Findings

Certain solvmanifolds cannot admit taming symplectic forms unless they are Kähler.

Invariance conditions on complex structures prevent taming symplectic forms in many cases.

Non-Kähler manifolds by Oeljeklaus and Toma do not admit taming symplectic forms.

Abstract

We study the existence of strong K\"ahler 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 $G/\Gamma$ 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.