函数域上${\mathbb P}^N$的自同态的等平凡性等价于潜在的好约化
Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields
摘要：
设$K=k(C)$是任意域$k$上的完全非奇异曲线$C$的函数域。本文的主要结果是态射$ϕ:{\mathbb P}^N_K\to{\mathbb P}^N_K$是等平凡的当且仅当它在$K$的所有位置$v$都有潜在的好约化，这推广了Benedetto关于${\mathbb P}^1_K$上多项式映射和Baker关于${\mathbb P}^1_K$上任意有理映射的结果.我们提供了两个证明：第一个使用代数几何和几何不变理论，它是新的，即使在情况下N=1。第二个证明使用非阿基米德分析和动力学，它更直接地推广了Benedetto和Baker的证明。我们还将提供两个应用程序。第一个陈述了次数至少为2的${\mathbb P}^N_K$的自同态是等平凡的当且仅当它有一个等平凡的代数。第二个给出了一个动力学准则是否（基变化后）的局部自由相干层${\mathcal E}$的秩$N+1$ $C$分解为直接和${\mathcal L}\oplus...\oplus{\mathcal L}$的$N+1$副本相同的可逆层${\mathcal L}$。
英文摘要：
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 ${\mathcal L}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
