Some properties of Grauert type surfaces

TL;DR

This paper explores properties of Grauert type surfaces, showing restrictions on compact curves, connectedness of level sets, and illustrating the necessity of double covers for certain constructions.

Contribution

It establishes new properties of Grauert type surfaces, including limitations on compact curves and connectedness of level sets, and provides an example requiring double covers.

Findings

Only negative compact curves can be contained in Grauert type surfaces.

Level sets of the pluriharmonic function are connected.

An example demonstrates the necessity of double covers in the construction.

Abstract

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.