Algebraic dependence theorems for meromorphic mappings defined on analytic covering spaces
by
Yoshihiro Aihara
Numazu College of Technology, Shizuoka 410-8501, Japan

Let  \pi:X --> C^m  be a finite analytic covering space and  M  a projective algebraic manifold. Let  f₁, ... , f_l  be meromorphic mappings from  X  into  M. Suppose that they have the same inverse images of given divisors on  M. In this talk, we give conditions under which  f₁, ... , f_l  are algebraically related and discuss their applications to uniqueness problem of meromorphic mappings from the point of view of value distribution theory. In particular, we give criteria for algebraic dependence under a condition on the existence of meromorphic mappings separating the fibers of  \pi:X --> C^m. Roughly our result says that if these mappings satisfy the same algebraic relation at all points of the set of the inverse images of divisors and if the given divisors are sufficiently ample, then they must satisfy this relationship identically.

We give here a definition of algebraic dependence of meromorphic mappings. We set  M^l = M × ... ×M (l-times). For meromorphic mappings  f₁, ... , f_l :X --> M, we define a meromorphic mapping  f₁ × ... ×f_l :X --> M^l  by

(f1 × ... ×fl)(z) = (f1(z), ... , fl(z)),    z in X - (I(f1) \cup ... \cup I(fl)),

where  I(f_j)  are the indeterminacy loci of  f_j. A proper algebraic subset  \Sigma  of  M^l  is said to be decomposable if, for some positive integer  s  not grater than  l, there exist positive integers  l₁, ... , l_s  with  l = l₁ + ... + l_s  and algebraic subsets  \Sigma_j subset or equal M^l_j  such that  \Sigma = \Sigma₁ × ... ×\Sigma_s. We denote by  B  the ramification divisor of  \pi:X --> C^m.

Definition 1.  Let  S  be an analytic subset of  X. Nonconstant meromorphic mappings  f₁, ... , f_l:X --> M  are said to be algebraically dependent on  S  if there exists a proper algebraic subset  \Sigma  of  M^l  such that  (f₁ × ... ×f_l)(S) subset or equal \Sigma  and  \Sigma  is not decomposable. In this case, we also say that  f₁, ... , f_l  are \Sigma-related on  S.

By making use of Nevanlinna theory, we obtain criteria for dependence and can deduce some unicity theorems for meromorphic mappings from these criteria.

Date received: August 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-30.