Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields

TL;DR

This paper proves that for endomorphisms of projective space over function fields, isotriviality is equivalent to potential good reduction at all places, extending prior results and providing two different proofs.

Contribution

It establishes the equivalence between isotriviality and potential good reduction for endomorphisms of projective space over function fields, with new proofs applicable even in the case N=1.

Findings

Isotriviality iff potential good reduction at all places

An endomorphism of degree ≥ 2 is isotrivial iff it has an isotrivial iterate

A dynamical criterion for sheaf decomposition on curves

Abstract

Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\phi:{\mathbb P}^N_K\to{\mathbb P}^N_K$ 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 ${\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\mathbb P}^1_K$. 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 ${\mathbb P}^N_K$ 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 ${\mathcal E}$ of rank $N+1$ on $C$ decomposes as a direct sum ${\mathcal L}\oplus...\oplus{\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\mathcal L}$.