Units in number fields satisfying a multiplicative relation with application to Oeljeklaus-Toma manifolds
Let \(K\) be a real number field with signature \((r,s)\). Dirichlet's unit theorem states that the group \(U_{K}\) of units of the ring of integers of \(K\) is the direct product of the group \(\{-1,1\}\) and a group with rank \(r+s-1\). Suppose \(s\geq 1\). According to [\textit{K. Oeljeklaus} and \textit{M. Toma}, Ann. Inst. Fourier 55, No. 1, 161--171 (2005; Zbl 1071.32017)], a subgroup \(U\) of \(U_{K}\) is said to be admissible for \(K\) if \(U\) has \(r\) generators, say \( u_{1},\dots,u_{r}\), such that \[ \sigma_{i}(u_{j})>0,\text{ for all } (i,j)\in \{1,\dots,r\}^{2},\text{ and }\det [\log \sigma_{i}(u_{j})]_{1\leq i,j\leq r}\neq 0, \] where \(\sigma_{1},\dots,\sigma_{r}\) denote the embeddings of \(K\) into \( \mathbb{R}\). In the paper under review, the author shows that for any real normal field \(N\) of degree \(r\) there are a cubic extension \(K\) of \(N\), with signature \( (r,r)\), and a group \(U\), admissible for \(K\), such that \[ \sigma_{i}(u)\left\vert \sigma_{i+r}(u)\right\vert^{2}=1,\text{ for all } (u,i)\in U\times \{1,\dots,r\}, \] where the real embedding \(\sigma_{i}\) and the complex embedding \(\sigma_{i+r}\) are extensions to \(K\) of the same automorphism of \(N\). This theorem, together with a recent result of \textit{A. Otiman} [``Special Hermitian metrics on Oeljeklaus-Toma manifolds'', Preprint, \url{arXiv:2009.02599}], implies that the corresponding Oeljeklaus-Toma manifold \(X(K,U)\) has a pluriclosed metric.
scientific article; zbMATH DE number 7369090
scientific article