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arXiv · 2503.22671

Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries

Xenia Flamm·Anne Parreau
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Abstract

We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric.

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Xenia Flamm, Anne Parreau. 2025-03-28. Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries. https://arxiv.org/abs/2503.22671

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