On Endomorphisms of Arrangement Complements

TL;DR

This paper proves that dominant endomorphisms of arrangement complements extend to their tropical compactifications, generalizing previous automorphism extension results using matroids and tropical geometry instead of Berkovich analytic geometry.

Contribution

It generalizes the extension of automorphisms to all dominant endomorphisms of arrangement complements via tropical geometry and matroid theory.

Findings

Every dominant endomorphism extends to the tropical compactification.

Generalizes previous automorphism extension results.

Uses matroids and tropical geometry instead of Berkovich geometry.

Abstract

Let $\Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $\Omega$ extends to an endomorphism of the tropical compactification $X$ of $\Omega$ associated to the Bergman fan structure on the tropicalization of $\Omega$. This generalizes a previous result by R\'emy, Thuillier and the second author which states that every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with $X$ by results of Feichtner and Sturmfels. Whereas the previous proof is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization discussed in the present paper relies on matroids and tropical geometry.