The canonical foliation of a compact generalized Hopf manifold
A locally conformal Kähler manifold \((M,J,g)\) with parallel Lee form \(\omega\) is called a compact generalized Hopf manifold. Set \(\theta=-\omega\circ J\) and denote by \(B\) and \(A\) the dual vector fields corresponding to \(\omega\) and \(\theta\), respectively. Then \(A=JB\). The distribution generated by \(A\) and \(B\) is completely integrable and defines a foliation \(\mathcal F\), which is called the canonical foliation. The author proves the following. (1) One can canonically lift the foliation to any holomorphic line bundle \(F\) on \(M\). Moreover, holomorphic sections of \(F\) are foliated one. (2) The canonical foliation \(\mathcal F\) has a compact leaf. Moreover, he also provides the description of the geometric structure of a compact generalized Hopf manifold on which the canonical foliation is regular.
scientific article; zbMATH DE number 1339069