an:02162468
Zbl 1071.32017
Oeljeklaus, Karl; Toma, Matei
Non-K??hler compact complex manifolds associated to number fields
EN
Ann. Inst. Fourier 55, No. 1, 161-171 (2005).
00115362
2005
j
32J18 32M17 11H56
compact complex manifold; algebraic number field; locally conformal K??hler metric
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 K??hler. 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 K??hler 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 K??hler structure and \(b_{2i+1}\in 2\mathbb N, i\geq 0\), are necessarily K??hler.
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.
Eberhard Oeljeklaus (Bremen)