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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.

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