On endomorphisms of arrangement complements

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 tropical variety $\operatorname {trop}(\Omega )$. This generalizes a result in [Compos. Math. 149 (2013), pp. 1211–1224], 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 proof in [Compos. Math. 149 (2013), pp. 1211–1224] is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization proved in the present paper relies on matroids and tropical geometry.