A toroidal resolution for the bad reduction of some Shimura varieties

TL;DR

This paper constructs a canonical log-smooth toroidal resolution for the bad reduction of certain Shimura varieties, leveraging Lafforgue's compactification and a reduction principle, enabling semi-stable resolutions over extended p-adic rings.

Contribution

It introduces a new canonical resolution method for bad reductions of Shimura varieties of specific types, extending the toolkit for understanding their singularities.

Findings

Constructed a canonical log-smooth toroidal resolution for bad reduction

Enabled derivation of semi-stable resolutions over extended p-adic rings

Applied Lafforgue's compactification to Shimura varieties

Abstract

Borrowing a reduction principle to a recent preprint of G. Faltings (toroidal resolution of some matrix singularities, 1999), we use Lafforgue's compactification of PGL_r^{N+1}/PGL_r to construct a canonical log-smooth toroidal resolution for the bad reduction in a prime p of Shimura varieties of unitary and symplectic type with parahoric level structures at p. Using this result, non-canonical semi-stable resolutions over Z_p[p^{1/\nu}] can be derived.