A Liouville theorem on an analytic space
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?
scientific article; zbMATH DE number 469063