Isotriviality is equivalent to potential good reduction for endomorphisms of [formula omitted] over function fields

Abstract

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 φ : P K N → P K N 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 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 E of rank N + 1 on C decomposes as a direct sum L ⊕ ⋯ ⊕ L of N + 1 copies of the same invertible sheaf L .