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Nonarchimedean integral geometry
Citation key BKL-Nonarchimedean-integral-geometry
Author Peter Bürgisser and Avinash Kulkarni and Antonio Lerario
Year 2022
Month 06
Abstract Let K be a nonarchimedean local field of characteristic zero with valuation ring R, for instance, K=ℚp and R=ℤp. We introduce the concept of an R-structure on a K-analytic manifold, which is akin to the notion of a Riemannian metric on a smooth manifold. An R-structure determines a norm and volume form on the tangent spaces of a K-analytic manifold, which allows one to integrate real valued functions on the manifold. We prove such integrals satisfy a nonarchimedean version of Sard's lemma and the coarea formula. We also consider R-structures on K-analytic groups and homogeneous K-analytic spaces that are compatible with the group operations. In this framework, we prove a general integral geometric formula analogous to the corresponding result over the reals. This generalizes the p-adic integral geometric formula for projective spaces recently discovered by Kulkarni and Lerario, e.g., to the setting of Grassmannians. As a first application, we compute the volume of special Schubert varieties and we outline the construction of a probabilistic nonarchimedean Schubert Calculus. As a second application we bound, and in some cases exactly determine, the expected number of zeros in K of random fewnomial systems.
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