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The stack of $G$-zips is a Mori dream space

Jean-Stefan Koskivirta

Abstract

We first extend previous results of the author with T. Wedhorn and W. Goldring regarding the existence of $μ$-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of $G$-zips of Pink--Wedhorn--Ziegler. Then, we define and study the Cox ring of the stack of $G$-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of $G$-zips and study its properties. We prove that it is finitely generated in certain cases.

The stack of $G$-zips is a Mori dream space

Abstract

We first extend previous results of the author with T. Wedhorn and W. Goldring regarding the existence of -ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of -zips of Pink--Wedhorn--Ziegler. Then, we define and study the Cox ring of the stack of -zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of -zips and study its properties. We prove that it is finitely generated in certain cases.
Paper Structure (24 sections, 45 theorems, 120 equations)

This paper contains 24 sections, 45 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Definition
  3. Group-theoretical notation
  4. Shimura varieties
  5. Vector bundles on quotient stacks
  6. Picard group of a quotient stack
  7. Brylinski–-Kostant filtration
  8. The space of global sections
  9. Global sections of line bundles
  10. Mori dream spaces
  11. Graded rings and monoids
  12. First properties
  13. Main result
  14. The ring of automorphic forms
  15. Ring of automorphic forms on Shimura varieties
  16. ...and 9 more sections

Key Result

Theorem 2.3.1

Theorems & Definitions (68)

  • Theorem 2.3.1: Pink-Wedhorn-Ziegler-zip-data
  • Proposition 2.6.1: Koskivirta-Wedhorn-Hasse
  • Proposition 3.2.1
  • Theorem 3.2.2
  • Proposition 3.3.1
  • proof
  • Corollary 3.3.2
  • proof
  • Proposition 3.3.3
  • proof
  • ...and 58 more