Examples of compact Lefschetz solvmanifolds. (Q1812483)

A symplectic manifold \((M^{2m},\omega)\) is called a Lefschetz manifold if the mapping \(\wedge\omega^{m-1}: H^1_{DR}\to H^{2m-1}_{DR}\) on \(M\) is an isomorphism. By a solvmanifold is meant a homogeneous space \(G/\Gamma\) where \(G\) is a simply connected solvable Lie group and \(\Gamma\) is a lattice. A solvable Lie algebra \({\mathfrak g}\) is said to be completely solvable if \(\text{ad}(X)\) has only real eigenvalues for every \(X\in{\mathfrak g}\). The purpose of this paper is to present examples of higher-dimensional completely solvable Lie groups which admit lattices and compact Lefschetz solvmanifolds. It extends also to the construction of compact symplectic solvmanifolds with the property known as Hard-Lefschetz's.