Propagation of algebraic dependence of meromorphic mappings (Q5950103)

The article under review contains criteria for the propagation of algebraic dependence of meromorphic mappings and several applications of them in the following setting. Let \(\pi:X\rightarrow \mathbb C^{m}\) be a finite analytic covering space of degree \(s\) and let \(f_{1},\dots,f_{l}\) be dominant meromorphic mappings from \(X\) into a projective algebraic manifold \(M\). Assume that their inverse images of some given divisor \(D\) in \(M\) are the same (up to a certain multiplicity) and that they satisfy the same algebraic relation on any irreducible component of this subvariety. The author introduces a line bundle \(L\) on \(M\), related to \([D]\) by a technical condition. If the bundle \(L\otimes K_{M}\) is big and if every \(f_{i}\) separates the general fiber of \(\pi\), then \(f_{1},\dots,f_{l}\) are algebraically dependent on \(M\). The following uniqueness theorem is one of the applications: Let \(M\) be a smooth elliptic curve, \(l=2\), \(f_{1}\), \(f_{2}\) holomorphic and nonconstant, and \(a_{1},\dots,a_{d}\) distinct points in \(M\) such that \(f^{*}_{1}(a_{i})\) and \(f^{*}_{2}(a_{i})\) have the same components of multiplicity one, \(1\leq i\leq d\). If \(d>16s-12\), then \(f_{1}=f_{2}\).NEWLINENEWLINENEWLINEThe article is related to results of \textit{J. Noguchi} [Hiroshima Math. J. 6, 265-280 (1976; Zbl 0338.32016)] and \textit{S. J. Drouilhet} [Ill. J. Math. 26, 492-502 (1982; Zbl 0493.32023)]. There are no proofs; the author refers to an unpublished paper by himself for details.