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Non-Kähler compact complex manifolds associated to number fields. (English) Zbl 1071.32017
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 \(\{(x_1,\dots,x_{s+t})\in\mathbb R^{s+t}\mid \sum_{i=1}^{s+t} x_i=0\,\}\) and \(\{(\log| \rho_1(u)| ,\dots,\log| \rho_s(u)| )\mid u\in U\,\}\) is a lattice of rank \(s\) in \(\mathbb R^s\) for suitable subgroups \(U\) of \({\mathcal O}^*_K\). Combining these representations yields a properly discontinuous action of the semidirect product \(U\ltimes {\mathcal O}_K\) on \(\mathbb H^s\times\mathbb C^t\), and the quotient \(X=X(K,U)\) is diffeomorph to a fiber bundle over \((S^1)^s\) with \((S^1)^{s+2t}\) as fiber.

32J18 Compact complex \(n\)-folds
32M17 Automorphism groups of \(\mathbb{C}^n\) and affine manifolds
11H56 Automorphism groups of lattices
Full Text: DOI Numdam EuDML
[1] Number theory, (1966), Academic Press, New York-London · Zbl 0145.04902
[2] Locally conformal Kähler geometry, (1998), Birkhäuser, Boston · Zbl 0887.53001
[3] On surfaces of class \(VII_0,\) Invent. Math., 24, 269-310, (1974) · Zbl 0283.32019
[4] An introduction to homological algebra, (1994), Cambridge · Zbl 0797.18001
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