Classes of non-Kähler, locally conformally Kähler manifolds
The author considers a problem posed by \textit{S. Dragomir} and \textit{L. Ornea} in [Locally Conformal Kähler Manifolds, Birkhäuser (1998; Zbl 0887.53001)], whether there exist non-compact complex, locally conformally Kähler manifolds which are not Kähler? Let \(A=[a_{ij}]\in \text{SL}(n, \mathbb Z)\) be a matrix with eigenvalues \(\lambda_ 1<\lambda_ 2<\dots <\lambda_{n-1}<1<\lambda_ n\), \(v_ j\), \(j=1,2, \dots, n\) the corresponding eigenvectors and \(D=\log A\). For these data the author defines a domain \(V=\{\sum_{i=1}^{n}\alpha_ iv_ i\); \(\alpha_ i\in \mathbb C\) and \({\mathfrak {Im}}\;\alpha_ i>0 \text{ for } i=1,2,\dots, n-1\}\) and for \(K=\mathbb Z, \mathbb R, \mathbb C\) denotes the group \(G_ K=K\times K^ n\) with the multiplication \[ (z,b)\cdot (z_ 0, b_ 0)=\left( z+z_ 0, e^{z_ 0D}b+b_ 0\right). \] Thus we obtain a manifold \(\Omega = (\mathbb C\times V)/G_{\mathbb Z}\). Moreover, if we put \[ \exp (z_ 1, z_ 2,\dots, z_ n)=\left(e^{-2i\pi z_ 1}, e^{-2i\pi z_ 2},\dots, e^{-2i\pi z_ n}\right), \qquad z_ i\in\mathbb C,\qquad i=1,2, \dots, n \] then we can consider a group \(\langle \widetilde {g_ A}\rangle\) generated by an automorphism \(\widetilde {g_ A}: \mathbb C\times \exp(V)\mapsto \mathbb C\times \exp(V)\), where \(\widetilde {g_ A}(z,w_ 1, \dots, w_ n)=\left(z+1, w_ 1^{a_{11}}w_ 2^{a_{12}}\dots w_ n^{a_{1n}}, \dots, w_ 1^{a_{n1}}w_ 2^{a_{n2}}\dots w_ n^{a_{nn}}\right).\) From the above considerations we obtain a manifold \(\exp(V)/\langle {g_ A}\rangle\), where \(\langle {g_ A}\rangle\) is the restriction of \(\langle \widetilde {g_ A}\rangle\) to \(\exp(V)\). The manifolds \(\Omega\) and \(\exp(V)/\langle {g_ A}\rangle\) are non-compact complex, locally conformally Kähler manifolds and non-Kähler. The proof heavily depends on the results of \textit{J.-J. Loeb} [Ann. Inst. Fourier 35, No. 4, 59--97 (1985; Zbl 0563.32013)].
scientific article; zbMATH DE number 2093998