Local models of Shimura varieties, I. Geometry and combinatorics

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.

86 pages, small corrections and improvements, to appear in the "Handbook of Moduli"

References in corpus (3)

• A guide to the reduction modulo p of Shimura varieties
• Projected Richardson varieties and affine Schubert varieties
• Quiver Grassmannians and their Euler characteristics: Oberwolfach talk, May 2010

Cited by in corpus (18)

• Towards a theory of local Shimura varieties
• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
• Local models for Weil-restricted groups
• Normality and Cohen-Macaulayness of local models of Shimura varieties
• G-valued crystalline representations with minuscule p-adic Hodge type
• On the coherence conjecture of Pappas and Rapoport
• Degenerate affine Grassmannians and loop quivers
• Mustafin varieties, moduli spaces and tropical geometry
• Vertexwise criteria for admissibility of alcoves
• An alternative description of the Drinfeld p-adic half-plane
• Central leaves on Shimura varieties with parahoric reduction
• Good and semi-stable reductions of Shimura varieties
• Construction of a Rapoport-Zink space for $\mathrm{GU}(1,1)$ in the ramified $2$-adic case
• Local models of Shimura varieties and a conjecture of Kottwitz
• Regular formal moduli spaces and arithmetic transfer conjectures
• Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type
• An equivariant compactification for adjoint reductive group schemes
• EKOR strata on Shimura varieties with parahoric reduction