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 is isotrivial if and only if it has potential good reduction at all places v of K; this generalizes results of Benedetto for polynomial maps on and Baker for arbitrary rational maps on . 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 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 E of rank N+1 on C decomposes as a direct sum L⊕⋯⊕L of N+1 copies of the same invertible sheaf L.