Oeljeklaus-Toma manifolds and arithmetic invariants

Abstract

Oeljeklaus-Toma (OT) manifolds are certain compact complex manifolds built from number fields. Conversely, we show that the fundamental group often pins down the number field uniquely. We relate the first homology to some interesting ideal. OT manifolds are never Kähler, but carry an LCK metric (locally conformally Kähler). Oeljeklaus and Toma used them to disprove a conjecture on the Betti numbers of spaces carrying LCK metrics. Mysteriously, the volume for this metric gives the discriminant and Dirichlet regulator of the number field. Totally unlike the above topological aspects, there does not seem to be a conceptual reason why this should be expected. Is it a coincidence? We explore this and see what happens if we (entirely experimentally!) regard them as "baby siblings" of, say, hyperbolic manifolds coming from number fields. We ask the same questions and obtain similar answers, but ultimately it remains unclear whether we are chasing ghosts or not.