Tubular neighborhoods of local models

TL;DR

This paper proves that certain local models of p-adic shtukas are unibranch and normal, confirming parts of the Scholze–Weinstein conjecture for small primes, using general topological methods and a generalized nearby cycles theorem.

Contribution

It establishes the unibranch property of v-sheaf local models and proves normality with reduced special fiber for scheme-theoretic local models, advancing the Scholze–Weinstein conjecture.

Findings

V-sheaf local models are unibranch.

Scheme-theoretic local models are normal with reduced special fiber.

Generalized nearby cycles comparison theorem for v-sheaves.

Abstract

We show that the v-sheaf local models of moduli spaces of $p$-adic shtukas are unibranch. In particular, this proves that the scheme-theoretic local models defined in our joint work with Ansch\"{u}tz and Richarz are always normal with reduced special fiber, thereby establishing the remaining cases of the geometric part of the Scholze$-$Weinstein conjecture when $p \leq 3$. Our methods are general, topological, and simplify Zhu's proof of the coherence conjecture. As a technical input, we generalize a comparison theorem of nearby cycles of Huber to the v-sheaf setup.