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Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties

Peihang Wu

TL;DR

The paper addresses extending principal bundles and automorphic coefficient systems from noncompact integral canonical Shimura varieties to their toroidal compactifications at hyperspecial primes. It develops a comprehensive framework of canonical models and extensions for mixed and pure Shimura data, using tensor functors, boundary tensors, and Deligne induction, and then proves the existence and uniqueness of canonical extensions for abelian-type toroidal compactifications. Central contributions include the construction of a global canonical extension $\mathscr{E}^{\mathrm{can}}_{G^c}$ with compatible de Rham data, the demonstration of local freeness of boundary quotients, and the establishment of reduction and subcanonical extension mechanisms. These results provide robust integral models for automorphic bundles and their log connections, enabling arithmetic and cohomological investigations at the boundary and offering tools for coherent and log de Rham cohomology in the abelian-type setting.

Abstract

In this paper, we construct canonical extensions of principal $\mathcal{G}^c$- (and $M^c$-)bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties with hyperspecial levels.

Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties

TL;DR

The paper addresses extending principal bundles and automorphic coefficient systems from noncompact integral canonical Shimura varieties to their toroidal compactifications at hyperspecial primes. It develops a comprehensive framework of canonical models and extensions for mixed and pure Shimura data, using tensor functors, boundary tensors, and Deligne induction, and then proves the existence and uniqueness of canonical extensions for abelian-type toroidal compactifications. Central contributions include the construction of a global canonical extension with compatible de Rham data, the demonstration of local freeness of boundary quotients, and the establishment of reduction and subcanonical extension mechanisms. These results provide robust integral models for automorphic bundles and their log connections, enabling arithmetic and cohomological investigations at the boundary and offering tools for coherent and log de Rham cohomology in the abelian-type setting.

Abstract

In this paper, we construct canonical extensions of principal - (and -)bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties with hyperspecial levels.
Paper Structure (18 sections, 38 theorems, 63 equations)

This paper contains 18 sections, 38 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Canonical models and canonical extensions of principal bundles
  3. Mixed Shimura varieties
  4. Preliminaries
  5. Canonical extensions and their canonical models
  6. Induction for torsors
  7. Canonical extensions on toroidal compactifications of integral canonical models
  8. Integral models of toroidal compactifications
  9. Canonical extensions on Hodge-type compactifications
  10. Tensors and principal bundles at the boundary
  11. Integral canonical models of principal bundles in the sense of Lovering
  12. Canonical extensions on abelian-type compactifications
  13. Statement of main theorem
  14. Proof of Theorem \ref{['thm-loc-free-quo']}
  15. Proof of main theorem
  16. ...and 3 more sections

Key Result

Proposition 1

For each $\lambda$ and each cusp label representative $\Phi_0$ with a cone $\sigma\in \Sigma_0^+(\Phi_0)$, there is a vector bundle $\mathcal{V}_{\Phi_0}(\sigma)$ on the boundary toric scheme $\mathcal{S}_{K_{\Phi_0}}(\sigma)$ and a tensor $\widetilde{s}_{\Phi_0,\lambda,\mathrm{dR}}(\sigma)\in \math

Theorems & Definitions (78)

  • Proposition
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Lov17
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 68 more