Minimal kernels of weakly complete spaces |
| |
Authors: | Zibgniew Slodkowski Giuseppe Tomassini |
| |
Institution: | a Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607, USA b Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy |
| |
Abstract: | Let X be a weakly complete space i.e. X a complex space endowed with a Ck-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ1=Σ1(X) of X by the following property: x∈Σ1 if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ1 of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σc1=Σ1∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dimcX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σc1=Σ1∩{?=c} are compact spaces foliated by holomorphic curves. |
| |
Keywords: | primary 32E 32T 32U secondary 32E05 32T35 32U10 |
本文献已被 ScienceDirect 等数据库收录! |
|