Isotriviality is equivalent to potential good reduction for endomorphisms of PN over function fields

Let K = k(C) be the function field of a complete non-singular curve C over an arbitrary field k. The main result of this paper states that a morphism phi : P-K(N) -> P-K(N) is isotrivial if and only if it has potential good reduction at all places upsilon of K; this generalizes results of Benedetto for polynomial maps on P-K(1) and Baker for arbitrary rational maps on P-K(1). We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N = 1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of P-K(N) of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf epsilon of rank N + 1 on C decomposes as a direct sum L circle plus...circle plus L of N + 1 copies of the same invertible sheaf L. (C) 2008 Elsevier Inc. All rights reserved.