Non-Kähler compact complex manifolds associated to number fields.
type: Item
instance of: scholarly article
title: Non-Kähler compact complex manifolds associated to number fields.
zbMATH Open document ID: 1071.32017
author: Karl Oeljeklaus
author: Matei Toma
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116182
review text: The authors consider a class of compact complex manifolds \(X\) which they construct as quotients of \(\mathbb H^s\times\mathbb C^t\) by properly discontinuous group actions, \(s,t>0\). These manifolds have the property that \(b_1(X)=s\leq \dim H^1(X,{\mathcal O}_X)\), hence they are not Kähler. Moreover \[ H^0(X,\Omega^1_X)=H^0(X,K^{\otimes k}_X)=\{0\} \] for all \(k\geq 1\). The quotients \(X_{s,1}\) of \(\mathbb H^s\times\mathbb C\) admit locally conformal Kähler metrics. In particular, the example \(X_{2,1}\) with Betti numbers \(b_1=b_5=2\), \(b_3=0\), \(b_{2i}=1\), \(0\leq i\leq 3\), answers the question whether compact complex manifolds with locally conformal Kähler structure and \(b_{2i+1}\in 2\mathbb N, i\geq 0\), are necessarily Kähler. The construction is based on well known facts from geometric number theory: Let \({\mathcal O}_K\) be the ring of integers of the algebraic number field \(K\) and \({\mathcal O}_K^*\) the group of units in \({\mathcal O}_K\). Assume that \(K\) admits \(s\) embeddings \(\rho_1, \dots,\rho_s\) of \(K\) into \(\mathbb R\) and \(2t\) non-real embeddings \(\sigma_1, \overline{\sigma}_1, \dots,\sigma_t, \overline{\sigma}_t\) into \(\mathbb C\). \({\mathcal O}_K\) can be realized as a lattice of rank \(s+2t=[K:\mathbb Q]\) in \(\mathbb C^s\times\mathbb C^t\) via the injection \(\tau:K\rightarrow \mathbb C^{s+t}, \tau(a):=(\rho_1(a),\dots,\rho_s(a),\sigma_1(a),\dots,\sigma_t(a))\), operating on \(\mathbb C^s\times\mathbb C^t\) by translations and leaving \(\mathbb H^s\times\mathbb C^t\) invariant. The quotient \((\mathbb H^s\times\mathbb C^t)/\tau({\mathcal O}_K)\) is diffeomorph to \((\mathbb R_{>0})^s\times (S^1)^{s+2t}\). The image of the logarithmic representation \(\lambda: {\mathcal O}^*_K\rightarrow \mathbb R^{s+t}\), \( \lambda(u):=(\log| \rho_1(u)| ,\dots,\log| \rho_s(u)| ,\log| \sigma_1(u)| ^2,\dots,\log| \sigma_t(u)| ^2)\), is a lattice of maximal rank in the linear hyperplane \(\{(x_1,\dots,x_{s+t})\in\mathbb R^{s+t}\mi…
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32J18
Mathematics Subject Classification ID: 32M17
Mathematics Subject Classification ID: 11H56
zbMATH DE Number: 2162468
zbMATH Keywords: compact complex manifold
zbMATH Keywords: algebraic number field
zbMATH Keywords: locally conformal Kähler metric
MaRDI profile type: Publication
OpenAlex ID: W2261570693
cites work: Q5521595
cites work: Locally conformal Kähler geometry
cites work: On surfaces of class VII\(_0\)
cites work: Q4296949
DOI: 10.5802/AIF.2093
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162468