Units in number fields satisfying a multiplicative relation with application to Oeljeklaus-Toma manifolds
type: Item
instance of: scholarly article
title: Units in number fields satisfying a multiplicative relation with application to Oeljeklaus-Toma manifolds
zbMATH Open document ID: 1472.11275
published in: Results in Mathematics
publication date: 2021-07-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 7369090
zbMATH Keywords: units
zbMATH Keywords: totally real Galois extensions
zbMATH Keywords: irreducibility
zbMATH Keywords: Oeljeklaus-Toma manifolds
zbMATH Keywords: pluriclosed metrics
author: A. Dubickas
reviewed by: Toufik Zaïmi
MaRDI profile type: Publication
full work available at URL: s00025-021-01396-4
OpenAlex ID: W3141537097
cites work: Holomorphic Line Bundles Over Domains in Cousin Groups and the Algebraic Dimension of Oeljeklaus-Toma Manifolds
cites work: Oeljeklaus-Toma manifolds and arithmetic invariants
cites work: Nonreciprocal units in a number field with an application to Oeljeklaus-Toma manifolds
cites work: On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles
cites work: On the existence of balanced and SKT metrics on nilmanifolds
cites work: Elementary criteria for irreducibility of \(f(X^r)\)
cites work: De Rham and twisted cohomology of Oeljeklaus-Toma manifolds
cites work: Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds
cites work: On the non-existence of static pluriclosed metrics on non-Kähler minimal complex surfaces
cites work: Q2770573
cites work: Problems in Algebraic Number Theory
cites work: An explicit formula for the fundamental units of a real pure sextic number field and its Galois closure
cites work: Q4254917
cites work: Non-Kähler compact complex manifolds associated to number fields.
cites work: Oeljeklaus-Toma manifolds admitting no complex subvarieties
cites work: Flat affine subvarieties in Oeljeklaus-Toma manifolds
cites work: Q2937074
cites work: Q4488096
cites work: LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker's theorem
cites work: The Chern–Ricci Flow on Oeljeklaus–Toma Manifolds
DOI: 10.1007/S00025-021-01396-4
Recommended article: Nonreciprocal units in a number field with an application to Oeljeklaus-Toma manifolds
Recommended article: Flat affine subvarieties in Oeljeklaus-Toma manifolds
Recommended article: Oeljeklaus-Toma manifolds and arithmetic invariants
Recommended article: Curves on the Oeljeklaus-Toma manifolds
Recommended article: On locally conformally Kähler metrics on Oeljeklaus-Toma manifolds
Recommended article: An elementary proof for a theorem of Thomas and Vasquez
Recommended article: LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker's theorem
Recommended article: Note on an index formula of elliptic units in a ring class field
Recommended article: Oeljeklaus-Toma manifolds admitting no complex subvarieties
Recommended article: Momente der Klassenzahlen binärer quadratischer Formen mit ganzalgebraischen Koeffizienten
description: scientific article; zbMATH DE number 7369090