2016/11/17 by Samuele Mongodi, Mongodi, Samuele, Zbigniew Słodkowski +3
Mathematics · #32E05 #32T35 #32U10 #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1611.05637
openalex publication_date 2016/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Remmert-Stein reduction of a holomorphically convex space. Moreover, in our classification Theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where it is necessary to do so.