Tubular neighborhoods of local models

Abstract

We show that the v-sheaf local models of Scholze and Weinstein are unibranch. In particular, this proves that the scheme-theoretic local models defined in Anschütz, Gleason, Lourenço, and Richarz are always normal with reduced special fiber, thereby establishing the remaining cases of the geometric part of the Scholze–Weinstein conjecture when $p\le 3$. Our methods are general, topological, and simplify those of Zhu for tamely ramified groups in positive characteristic. As a technical input, we generalize a comparison theorem of nearby cycles of Huber to the v-sheaf setup.