Differential Geometry Seminar (2025)

A characterization of Oeljeklaus-Toma manifolds in LCK geometry

Oeljeklaus–Toma (OT) manifolds are known as examples of complex manifolds that do not admit a Kähler metric, and they are regarded as higher-dimensional analogues of Inoue surfaces. OT manifolds are solvmanifolds constructed from number-theoretic data, and some of them admit locally conformally Kähler (LCK) metrics. In this way, a large number of examples of solvmanifolds equipped with LCK metrics have been obtained, and OT manifolds have been actively studied as important examples in LCK geometry. Although their construction may seem intricate, aside from some simple examples, OT manifolds are the only known solvmanifolds admitting LCK metrics. In this talk, I will show that if a certain class of solvmanifolds admits an LCK metric, then it is essentially an OT manifold. Since number theory naturally arises from geometric constraints in our setting, this result suggests that number-theoretic arguments are indispensable in the construction of certain classes of solvmanifolds. This talk is based on the preprint arXiv:2502.12500.