A characterization of hyperbolic Kato surfaces. (English) Zbl 1293.32027

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.


32J15 Compact complex surfaces
32U05 Plurisubharmonic functions and generalizations
32V40 Real submanifolds in complex manifolds


Zbl 1149.32011
Full Text: DOI Euclid