Singularities of local models

TL;DR

This paper constructs and analyzes local models of Shimura varieties, focusing on their singularities, especially in wildly ramified cases, and establishes several geometric properties in characteristic settings.

Contribution

It introduces new local models for Shimura varieties, proves their reducedness and Cohen–Macaulayness, and explores their singularities using perfect geometry, especially for wildly ramified groups.

Findings

Proved reducedness of special fibers

Established Cohen–Macaulayness of local models

Analyzed singularities using perfect geometry

Abstract

We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic $2$ we construct local models, show reducedness of their special fiber, Cohen$-$Macaulayness and in equi-characteristic also (pseudo-)rationality. In mixed characteristic we conjecture their (pseudo-)rationality. This is based on the construction of parahoric group schemes over two-dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.