Domains with a continuous exhaustion in weakly complete surfaces
- PDF / 262,494 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 73 Downloads / 183 Views
Mathematische Zeitschrift
Domains with a continuous exhaustion in weakly complete surfaces Samuele Mongodi1 · Zbigniew Slodkowski2 Received: 8 April 2018 / Accepted: 23 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In previous works, Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and D ⊆ X a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X ; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X . Mathematics Subject Classification 32C40 · 32E05 · 32U10
1 Introduction In [8–10], Tomassini and the authors initiated the study of the geometry of weakly complete spaces; the question had been raised by Tomassini and the second author in [15], where they also defined the minimal kernel of a weakly complete space, an essential tool in the investigations presented in [9,10]. In that first attempt, complex surfaces which admit a real-analytic plurisubharmonic exhaustion function were considered and it was proved that they fall into one of the following three classes: (1) modifications of Stein spaces of dimension 2, (2) complex surfaces which map properly onto an open Riemann surface (possibly singular), (3) surfaces of Grauert type.
B
Samuele Mongodi [email protected] Zbigniew Slodkowski [email protected]
1
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi, 9, 20133 Milan, Italy
2
Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA
123
S. Mongodi, Z. Slodkowski
A surface of Grauert type is a complex surface endowed with a smooth plurisubharmonic exhaustion function whose regular level sets are Levi-flat hypersurfaces, foliated with dense complex leaves; some examples are discussed in [9, Section 2] and their geometry is studied to some depth in [10]. In [6], the first author considered, inside such surfaces, open domains with an exhaustion which could be, a priori, only smooth and proved that they actually admit a real-analytic exhaustion. This paper aims to study the case of an open domain, inside a complex surface with a global real-analytic exhaustion, which admits only a continuous plurisubharmonic exhaustion function, thus generalizing the results of [6] to the continuous case. We obtain the following result. Theorem 1.1 Let X be a complex surface which admits a real-analytic plurisubharmonic exhaustion function and consider a domain D ⊆ X which admits a continuous plurisubharmonic exhaustion function. Then, one of the following holds t
Data Loading...
