A generalization to the q-convex case of a theorem of Fornæss and Narasimhan

Abstract

Fornaess and Narasimhan proved (in [8, Thm. 5.3.1]) that, for any complex space X, the identity WPSH(X) = PSH(X) holds, where WPSH(X) denotes the weakly plurisubharmonic functions on X and PSH(X) denotes, as usual, the plurisubharmonic functions on X. When X has no singularities, this identity is clear. For the singular case, however, the inclusion WPSH(X) ⊆ PSH(X) is no longer trivial; one must find locally a plurisubharmonic extension to the ambient space of an embedding of X. In this paper we give another proof for this identity (Theorem 3.3). It is shorter and easier and has the advantage that it can be generalized to q-plurisubharmonic functions (Theorem 4.16). However it has the disadvantage that it works only for continuous functions. The q-plurisubharmonic functions were introduced by Hunt and Murray in [10] (see also [9]), but we will call here q-plurisubharmonic what they call (q − 1)-plurisubharmonic. We also obtain a generalization of a theorem of Siu [16]; namely, we show (Lemma 4.18) that every q-complete subspace with corners of a complex spaceX admits a neighborhood in X that is q-complete with corners. This will be needed in the proof of our main result. The results and proofs of this paper have been announced in [13]. This paper is part of the author’s doctoral thesis written in Wuppertal. I thank Prof. M. Colţoiu and Prof. K. Diederich for many helpful discussions during the whole time of preparing my thesis. I thank the Department of Mathematics of the University of Wuppertal for providing me a nice working atmosphere.

Locations

• The Michigan Mathematical Journal - View - PDF