Solvable models for Kodaira surfaces

2011/11/10 by Sergio Console, Console, Sergio, Gabriela P. Ovando +3
Mathematics · #22E25 #22E40 #53C30 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:22E25 #msc:22E40 #msc:53C30

paper · pdf · doi:10.48550/arxiv.1111.2417

arxiv created 2011/11/10 · openalex publication_date 2011/11/10 · arxiv updated 2011/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider three families of lattices on the oscillator group G, which is an almost nilpotent not completely solvable Lie group, giving rise to coverings G → M_{k, 0} → M_{k, π} → M_{k, π/2} for k∈ \Z. We show that the corresponding families of four dimensional solvmanifolds are not pairwise diffeomorphic and we compute their cohomology and minimal models. In particular, each manifold M_{k, 0} is diffeomorphic to a Kodaira--Thurston manifold, i.e. a compact quotient S¹ × \Heis₃ (\R) /Γ_k where Γ_k is a lattice of the real three-dimensional Heisenberg group \Heis₃ (\R). We summarize some geometric aspects of those compact spaces. In particular, we note that any M_{k, 0} provides an example of a solvmanifold whose cohomology does not depend on the Lie algebra only and which admits many symplectic structures that are invariant by the group \R ×\Heis₃ (\R) but not under the oscillator group G.

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