Work

The connected components of affine Deligne–Lusztig varieties

Year: 2025

Type: article

Abstract: We compute the connected components of arbitrary parahoric level affine Deligne–Lusztig varieties and local Shimura varieties, thus resolving a folklore conjecture raised in (He in Some results on affine Deligne–Lusztig varieties. YouTube video, 2018; Zhou in Duke Math. J. 169(15):2937–3031, 2020) in full generality (even for non-quasisplit groups). We achieve this by relating them to the connected components of infinite level moduli spaces of $p$ p -adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites . Along the way, we give a $p$ p -adic Hodge-theoretic characterization of HN-irreducibility. As applications, we obtain many results on the geometry of integral models of Shimura varieties of Hodge type at arbitrary stabilizer-parahoric levels. In particular, we deduce new CM lifting results on integral models of Shimura varieties for quasisplit groups at parahoric levels that arise as stabilizer Bruhat–Tits group schemes.

Source: Inventiones mathematicae

DOI: 10.1007/s00222-025-01386-1

Authors: Ian Gleason, Dong Gyu Lim, Yujie Xu

Institutions: University of Bonn, University of California, Berkeley, Columbia University

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: National Science Foundation, Deutsche Forschungsgemeinschaft, Rheinische Friedrich-Wilhelms-Universität Bonn, Leibniz-Gemeinschaft