Oeljeklaus-Toma manifolds and arithmetic invariants

TL;DR

Oeljeklaus-Toma manifolds are complex manifolds derived from number fields, with their geometric and topological properties intricately linked to number-theoretic invariants like discriminants and regulators, revealing deep connections between geometry and arithmetic.

Contribution

This paper investigates the relationship between OT manifolds' geometric invariants and number-theoretic data, highlighting new insights into their structure and properties.

Findings

Fundamental group often determines the underlying number field.

Volume of the LCK metric relates to discriminant and regulator.

OT manifolds are never Kähler but admit LCK metrics.

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 K\"ahler, but carry an LCK metric (locally conformally K\"ahler). 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.