A six dimensional compact symplectic solvmanifold without Kähler structures (Q1917016)

The authors conjecture that a compact solvmanifold has a Kähler structure if and only if it is a finite quotient of a complex torus. The purpose of the paper is to construct a compact symplectic (non-nilpotent) solvmanifold \(M^6=\Gamma/G\) of dimension six for which either the Hard Lefschetz Theorem does not hold or the minimal model of \(M\) is not formal. Hence \(M\) does not admit a Kähler structure. They show that the minimal model of \(M^6\) is not formal by proving that there are nontrivial (quadruple) Massey products and remark that all (triple) Massey products of \(M^6\) vanish. For \(G\) a real Lie group of dimension \(2n\) they show that the space of left invariant almost complex structures on \(G\) has dimension \(2n^2\) and \(G\) has no left invariant complex structures.