Lattices in almost abelian Lie groups with locally conformal K\"ahler or symplectic structures

TL;DR

This paper investigates the existence of lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures, revealing dimension-specific existence results and providing explicit examples.

Contribution

It establishes that lattices with these structures exist only in dimension 4 for Kähler and in all even dimensions for symplectic cases, with explicit examples provided.

Findings

Lattices with locally conformal Kähler structures exist only in dimension 4.

Lattices with locally conformal symplectic structures exist in all even dimensions.

Explicit examples of such lattices are constructed in various dimensions.

Abstract

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal K\"ahler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, while in the latter case we provide examples of such Lie groups admitting lattices in any even dimension.