Automorphisms of Drinfeld half-spaces over a finite field (Q2841773)

Let \(k\) be a finite field and \(V\) be a finite-dimensional vector space over \(k\). Recall that the Drinfeld half-plane \(\Omega(V)\) is defined as the complement of all \(k\)-rational hyperplanes in the projective space \(\mathbb{P}(V)\). The authors study the automorphisms of \(\Omega(V)\) and prove that they all extend to automorphisms of \(\mathbb{P}(V)\): \(\mathrm{Aut}_{k}(\Omega(V)) = \mathrm{PGL}(V)\).NEWLINENEWLINEThe proof uses ideas from geometry and valuation theory. The authors first introduce the space \(X\) obtained from \(\mathbb{P}(V)\) by blowing up all \(k\)-rational linear subspaces. Every such linear subspace \(L\) naturally gives rise to a hypersurface \(E_{L}\) of \(X\), hence to a discrete valuation \(\mathrm{ord}_{L}\) on the function field \(\kappa(V)\) of \(X\) (which is also that of \(\mathbb{P}(V)\) and \(\Omega(V)\)). The set of such valuations \(\Gamma(V)\) is naturally endowed with a simplicial structure (because the set of linear subspaces of \(\mathbb{P}(V)\) is). The fact that an automorphism \(\varphi\) of \(\Omega(V)\) extends is encoded in the induced map \(\varphi^\ast\) on \(\Gamma(V)\): \(\varphi\) extends to \(X\) (resp. \(\mathbb{P}(V)\)) if, and only if, \(\varphi^\ast\) preserves \(\Gamma(V)\) and its simplicial structure (resp. preserves the subset of \(\Gamma(V)\) defined by hyperplanes).NEWLINENEWLINEThe proof of the result now goes in two steps. The first one involves \(k\)-analytic spaces in the sense of Berkovich, where \(k\) is endowed with the trivial absolute value. The authors show that the simplicial set \(\Gamma(V)\) may be realized as the 1-skeleton of a conical complex \(\mathfrak{S}(V)\) in the analytification \(\Omega(V)^{\mathrm{an}}\) of \(\Omega(V)\). This complex \(\mathfrak{S}(V)\) is associated to the toroidal embedding \(\Omega(V) \to X\) by the construction of [\textit{A. Thuillier}, Manuscr. Math. 123, No. 4, 381--451 (2007; Zbl 1134.14018)] and may be described in terms of maximal points in \(\Omega(V)^{\mathrm{an}}\). As a consequence, it is preserved by automorphisms of \(\Omega(V)\), hence \(\Gamma(V)\) is too.NEWLINENEWLINEBy the result quoted above, automorphisms of \(\Omega(V)\) extend to \(X\) and in particular preserve the set of linear subspaces of \(\mathbb{P}(V)\). To conclude, it is enough to show that hyperplanes are sent to hyperplanes. This is the second step and it is carried out by computations using Chow groups and intersection theory.NEWLINENEWLINENote that the result of the paper was previously known to hold when \(k\) is a non-Archimedean local field. (In this case \(\Omega(V)\) needs to be defined as a \(k\)-analytic variety). The proofs by \textit{V. G. Berkovich} [C. R. Acad. Sci., Paris, Sér. I 321, No. 9, 1127--1132 (1995; Zbl 0856.14007)] and \textit{G. Alon} [``Automorphisms of products of Drinfeld half planes'', unpublished] use completely different methods based on embedding Bruhat-Tits buildings into Berkovich spaces.