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Uniruledness of some low-dimensional ball quotients

Yota Maeda

Abstract

We define reflective modular forms on complex balls and use a method of Gritsenko and Hulek to show that some ball quotients of dimensions 3, 4 and 5 are uniruled. We give examples of Hermitian lattices over the rings of integers of imaginary quadratic fields $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ for which the associated ball quotients are uniruled. Our examples include the moduli space of 8 points on $\mathbb{P}^1$. Moreover, we find that some of their Satake-Baily-Borel compactifications are rationally chain connected modulo certain cusps.

Uniruledness of some low-dimensional ball quotients

Abstract

We define reflective modular forms on complex balls and use a method of Gritsenko and Hulek to show that some ball quotients of dimensions 3, 4 and 5 are uniruled. We give examples of Hermitian lattices over the rings of integers of imaginary quadratic fields and for which the associated ball quotients are uniruled. Our examples include the moduli space of 8 points on . Moreover, we find that some of their Satake-Baily-Borel compactifications are rationally chain connected modulo certain cusps.

Paper Structure

This paper contains 6 sections, 20 theorems, 59 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Ball quotients
  3. Uniruledness criterion
  4. Applications to the uniruledness of ball quotients
  5. Reflective modular forms on $\mathscr{D}_{L_Q}$
  6. Examples of Hermitian lattices

Key Result

Theorem 1.1

Let $F$ be an imaginary quadratic field and $(L,\langle\ ,\ \rangle)$ be a Hermitian lattice over $\mathscr{O}_F$ of signature $(1,n)$ for $n>1$. Let $a,k > 0$ be positive integers satisfying $k > a(n+1)$. If there exists a non-zero reflective modular form $F_{a,k}\in M_k(\Gamma,\chi)$ of weight $k$

Theorems & Definitions (42)

  • Theorem 1.1: Theorem \ref{['uniruledness']}
  • Theorem 1.2: Theorem \ref{['Kodaira_dim_app']}
  • Theorem 1.3: Theorem \ref{['Kodaira_dim_app2']}
  • Theorem 1.4: Theorem \ref{['Kodaira_dim_app3']}
  • Theorem 1.5: Theorem \ref{['Kodaira_dim_app4']}
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 32 more