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

Petsche, Clayton; Szpiro, Lucien; Tepper, Michael

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

Authors

Petsche, Clayton

Szpiro, Lucien

Tepper, Michael

Description

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 $ϕ:{\mathbb P}^N_K\to{\mathbb P}^N_K$ 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 ${\mathcal L}$.
Changes in this version: moved some preliminary material on non-archimedean fields to section 2; clarified the geometric proof of Theorem 1; replaced our proof of Prop. 2(c)--which had a gap in it--with a reference to the proof by Fakhruddin; corrected several small errors and typos, and added some new references