Solvable Models for Kodaira Surfaces

Abstract

In this work, we study families of compact spaces which are of the form \({G/\Lambda_{k,i}}\) for G the oscillator group and \({\Lambda_{k,i} < G}\) a lattice. The solvmanifolds \({G/\Lambda_{k,i}}\) are not pairwise diffeomorphic and one has the coverings \({G \to M_{k, 0} \to M_{k, \pi} \to M_{k, \pi/2}}\) for \({k \in \mathbb{Z}}\) . We compute their cohomologies and minimal models. Each manifold Mk, 0 is diffeomorphic to a Kodaira–Thurston manifold, i.e., a compact quotient \({S^1 \times {\rm H}_3 (\mathbb{R}) /\Gamma_k}\) where \({\Gamma_k}\) is a lattice of the real three-dimensional Heisenberg group \({{\rm H}_3 (\mathbb{R})}\) . Furthermore, any Mk, 0 provides an example of a solvmanifold whose cohomology does not depend on the Lie algebra only. We explain some geometrical aspects of those compact spaces, to show how to distinguish them (by invariant complex, symplectic and metric structures). For instance, no invariant symplectic structure of G can be induced to the any quotient.