A Liouville theorem on an analytic space
type: Item
instance of: scholarly article
title: A Liouville theorem on an analytic space
zbMATH Open document ID: 0788.32004
author: Kensho Takegoshi
published in: Journal of the Mathematical Society of Japan
publication date: 1994-05-26T00:00:00Z
review text: Let \(M\) be a reduced analytic space of dimension \(m\) which possesses a non-degenerate (at some nonsingular point of \(M)\) \(d\)-closed positive current \(\omega \circ f\) bidegree \((m-1,m-1)\). If there exists an unbounded exhaustion function whose level hypersurfaces satisfy a slow volume growth condition relative to \(\omega\), then the image of any nonconstant meromorphic map from \(M\) into the projective space \(\mathbb{P}_ n\) is shown to interest almost all hyperplanes in \(\mathbb{P}_ n\). Moreover, if in addition \(\omega\) is nondegenerate almost everywhere, then \(M\) admits no nonconstant negative smooth subharmonic functions relative to \(\omega\). The following problem is set up: if the projective volume \(n(A,r)\) of an irreducible analytic subset \(A \subset \mathbb{C}^ n\) satisfies the condition \[ \lim_{r \to \infty} {\log n(A,r) \over \log r}=0, \] then, does \(A\) admit no nonconstant holmorphic functions?
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32H04
zbMATH DE Number: 469063
zbMATH Keywords: Liouville theorem
zbMATH Keywords: reduced analytic space
zbMATH Keywords: positive current
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04520301
OpenAlex ID: W2063618947
DOI: 10.2969/JMSJ/04520301
description: scientific article; zbMATH DE number 469063