Solvable models for Kodaira surfaces

TL;DR

This paper constructs and analyzes three families of four-dimensional solvmanifolds related to Kodaira surfaces, computing their cohomology, minimal models, and geometric properties, revealing new insights into their symplectic structures and diffeomorphism types.

Contribution

It introduces new families of solvmanifolds derived from the oscillator group, computes their cohomology and minimal models, and explores their geometric and symplectic properties.

Findings

The solvmanifolds are not pairwise diffeomorphic.

Each $M_{k, 0}$ is diffeomorphic to a Kodaira--Thurston manifold.

Some manifolds admit symplectic structures invariant under different groups.

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 $G \to M_{k, 0} \to M_{k, \pi} \to M_{k, \pi/2}$ for $k\in \Z$. 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 $S^1 \times \Heis_3 (\R) /\Gamma_k$ where $\Gamma_k$ is a lattice of the real three-dimensional Heisenberg group $\Heis_3 (\R)$. 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 $\R \times\Heis_3 (\R)$ but not under the oscillator group $G$.