Local models of Shimura varieties, I. Geometry and combinatorics

TL;DR

This paper surveys the theory of local models of Shimura varieties, focusing on their geometry, combinatorics, and connections to other algebraic varieties, providing an overview of recent developments over the past 15 years.

Contribution

It offers a comprehensive overview of local models of Shimura varieties, highlighting their geometric and combinatorial properties and their relations to various algebraic structures.

Findings

Summary of geometric properties of local models

Connections to nilpotent orbit closures and affine Schubert varieties

Overview of recent results in the last 15 years

Abstract

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.