Compactifications of subschemes of integral models of Hodge-type Shimura Varieties with Parahoric level structures
Shengkai Mao
TL;DR
The paper develops a comprehensive framework for well-positioned subschemes on the special fibers of Kisin-Pappas and Pappas-Rapoport integral models of Hodge-type Shimura varieties with connected parahoric level. It provides axiomatic compactifications, defines well-positioned subsets and Igusa-extended concepts, and proves that central leaves, Newton strata, KR, and EKOR strata (and their level-pullbacks) are well-positioned, with stable boundary behavior under Hecke actions and change of parahorics. By integrating Siegel embeddings, local model diagrams, and Hodge tensors, the authors obtain detailed boundary descriptions and affine partial compactifications for many strata, including the minimal EKOR and Igusa strata. The work generalizes prior results of Lan and Stroh to broader parahoric settings, enhances the boundary theory of EO/KR/EKOR strata in Hodge-type contexts, and establishes robust functoriality and compactification properties that are foundational for further geometric and arithmetic applications. The results have significant implications for understanding mod-p geometry of Shimura varieties and for the study of nearby cycles, stratifications, and torsor structures at the boundary, with concrete corollaries on affine boundary components and boundary independence from Siegel embeddings.
Abstract
We prove that central leaves, Igusa varieties, Newton strata, Kottwitz-Rapoport Strata, Ekedahl-Kottwitz-Oort-Rapoport strata on the special fiber of a Kisin-Pappas integral model of a Hodge-type Shimura variety with connected parahoric level structure are well-positioned, generalizing some work of Kai-wen Lan and Benoit Stroh.