A characterization of Oeljeklaus-Toma manifolds in locally conformally K\"{a}hler geometry

TL;DR

This paper characterizes Oeljeklaus-Toma manifolds within locally conformally Kähler geometry, linking their structure to solvable Lie groups with specific metrics and lattices, and explaining the number-theoretic origins of their construction.

Contribution

It establishes a characterization of Oeljeklaus-Toma manifolds via solvable Lie groups with invariant non-Vaisman locally conformally Kähler metrics and lattices.

Findings

Oeljeklaus-Toma manifolds arise from certain solvable Lie groups.

Number-theoretic considerations are fundamental in their construction.

The characterization explains the geometric and algebraic structure of these manifolds.

Abstract

We show that for a certain class of solvable Lie groups, if they admit a left-invariant non-Vaisman locally conformally K\"{a}hler metric and a lattice, they must arise from the construction of Oeljeklaus-Toma manifolds. This result provides a natural explanation for why number-theoretic considerations play a role in the construction of Oeljeklaus-Toma manifolds.