A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet another) variation on Kronecker's theorem on units. In turn, this implies that on any Oeljeklaus-Toma manifold associated to a number field with
complex embeddings and
real embeddings with
there is no LCK metric.
Key Words: Locally
conformally Kähler manifold,
number field, units.
2000 Mathematics Subject Classification: Primary: 11R27;
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