A characterization of hyperbolic Kato surfaces (Q2438277)

The main result of the article states that Kato surfaces are the only connected compact complex surfaces without nonconstant meromorphic functions which admit a Green function with non-empty polar set. More precise: A surface \(S\) as above is a (not necessarily minimal) Kato surface if there exists an infinite cyclic covering \(\pi:\tilde{S}\rightarrow S\) and a negative plurisubharmonic function \(F:\tilde{S}\rightarrow[-\infty,0)\) with the property that \(F\circ\varphi=\lambda\cdot F\) for some \(\lambda >0\), \(\varphi\) a generator of the group of deck transformations, and moreover that \(dd^cF\) is supported on a non-empty analytic subset \(Z\subset\tilde{S}\). The leaves of the foliation of \(S\) induced by \(F\) are non-compact Levi-flat real hypersurfaces and the author shows by using \(Z\not=\emptyset\) that they can be approximated by compact strictly pseudoconvex hypersurfaces which are not homologous to zero. By a former result of the author [Ann. Inst. Fourier 58, No. 5, 1723--1732 (2008; Zbl 1149.32011)] this implies that \(S\) is a Kato surface. The article was submitted after the author had passed away.