Tamed symplectic structures on compact solvmanifolds of completely solvable type

TL;DR

This paper proves that compact solvmanifolds of completely solvable type admit a symplectic form taming an invariant complex structure only if they are complex tori, extending previous results from nilmanifolds.

Contribution

It generalizes the characterization of taming symplectic forms from nilmanifolds to all compact solvmanifolds of completely solvable type.

Findings

A compact solvmanifold admits a taming symplectic form iff it is a complex torus.

The result extends previous work on nilmanifolds.

Provides a complete characterization of taming symplectic structures in this setting.

Abstract

A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a K\"ahler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.