Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined. Comment: 28 pages

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Special cycles on toroidal compactifications of orthogonal Shimura varieties

Mathematische Annalen ◽

10.1007/s00208-021-02271-x ◽

2021 ◽

Author(s):

Jan Hendrik Bruinier ◽

Shaul Zemel

Keyword(s):

Shimura Varieties ◽

Green Functions ◽

Toroidal Compactifications ◽

Generating Series ◽

Automorphic Green Functions

AbstractWe determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

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Algebraic Constructions of Minimal Compactifications

Arithmetic Compactifications of PEL-Type Shimura Varieties ◽

10.23943/princeton/9780691156545.003.0007 ◽

2013 ◽

Author(s):

Kai-Wen Lan

Keyword(s):

Large Class ◽

Automorphic Forms ◽

Algebraic Construction ◽

Local Structures ◽

Jacobi Expansion ◽

Integral Bases ◽

Toroidal Compactifications ◽

Jacobi Expansions ◽

Algebraic Constructions ◽

Moduli Problems

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.

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Algebraic Constructions of Toroidal Compactifications

Arithmetic Compactifications of PEL-Type Shimura Varieties ◽

10.23943/princeton/9780691156545.003.0006 ◽

2013 ◽

Author(s):

Kai-Wen Lan

Keyword(s):

General Construction ◽

Formal Models ◽

Algebraic Models ◽

Algebraic Construction ◽

Algebraic Stacks ◽

Local Structures ◽

Correct Formulation ◽

Toroidal Compactifications ◽

Multiplicative Type ◽

Algebraic Constructions

This chapter explains the algebraic construction of toroidal compactifications. For this purpose the chapter utilizes the theory of toroidal embeddings for torsors under groups of multiplicative type. Based on this theory, the chapter begins the general construction of local charts on which degeneration data for PEL structures are tautologically associated. The next important step is the description of good formal models, and good algebraic models approximating them. The correct formulation of necessary properties and the actual construction of these good algebraic models are the key to the gluing process in the étale topology. In particular, this includes the comparison of local structures using certain Kodaira–Spencer morphisms. As a result of gluing, this chapter obtains the arithmetic toroidal compactifications in the category of algebraic stacks. The chapter is concluded by a study of Hecke actions on towers of arithmetic toroidal compactifications.

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