Locally conformally Kähler metrics on certain non-Kählerian surfaces
A Hermitian metric \(\omega\) on a complex manifold \(M\) is called locally conformally Kähler (lcK) if there is an open cover \(\{U_j\}\) of \(M\) together with smooth functions \(f_j\) on \(U_j\) such that \(e^{f_j}\omega\) is Kähler on \(U_j\), for all \(j\). Clearly every Kähler manifold is lcK, but the converse is false, and a very interesting problem is to determine exactly which non-Kähler manifolds admit lcK metrics. The first interesting case is that of compact complex non-Kähler surfaces. Compact complex non-Kähler surfaces are not completely classified but there are many examples (and also a conjecture that these are all the possible ones). Some of these are known to admit lcK metrics (Hopf surfaces), and others do not (some Inoue surfaces). In this paper the author proves that every Enoki surface admits a lcK metric. He also considers the more general case of Kato surfaces, and he proves that the property of admitting lcK metrics is stable under small deformations. In a subsequent paper the author proved that indeed all Kato surfaces admit lcK metrics, thus giving in particular a much simplified proof of the main result of this paper.
scientific article; zbMATH DE number 5664550