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Fano Shimura varieties with mostly branched cusps

Yota Maeda, Yuji Odaka

Abstract

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.

Fano Shimura varieties with mostly branched cusps

Abstract

We prove that the Satake-Baily-Borel compactification of certain Shimura varieties are Fano varieties, Calabi-Yau varieties or have ample canonical divisors with mild singularities. We also prove some variants statements, give applications and discuss various examples including new ones, for instance, the moduli spaces of unpolarized (log) Enriques surfaces.

Paper Structure

This paper contains 18 sections, 26 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main results and proofs
  3. Convention and Notation
  4. Special reflective modular forms
  5. Main general results and proofs
  6. Modular varieties with big anti-canonical classes
  7. Examples of Fano and K-ample cases
  8. Siegel modular cases
  9. Orthogonal modular cases, Part I
  10. Orthogonal modular cases, Part II
  11. Preparation for unitary case - Hermitian lattice
  12. Unramifiedness of unitary modular varieties
  13. Unitary modular cases, Part I - Fano cases
  14. Unitary modular cases, Part II - with ample canonical class
  15. More examples
  16. ...and 3 more sections

Key Result

Lemma 2.3

In the orthogonal case $G=O(2,n)$ (resp., in the unitary case $G=U(1,n)$), the canonical weight $c$ in the sense of § Notation is $n$ (resp., $n+1$).

Theorems & Definitions (60)

  • Lemma 2.3: cf., Freitag, GHS
  • proof
  • Theorem 2.4: Birational properties
  • proof
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Corollary 2.8: Boundary structure for Fano Shimura varieties
  • proof
  • Lemma 2.9: Log canonical centers
  • ...and 50 more