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Reflective obstructions of unitary modular varieties

Yota Maeda

Abstract

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than $138$. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula.

Reflective obstructions of unitary modular varieties

Abstract

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than . Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula.
Paper Structure (40 sections, 30 theorems, 230 equations)

This paper contains 40 sections, 30 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Unitary modular varieties
  3. Main results
  4. Application I: Kodaira dimension
  5. Application II: Reflective modular forms
  6. Technical tools
  7. Outline of the proof of Theorem \ref{['mainthm:big']}
  8. Organization of the paper
  9. Dimension formula
  10. Branch divisors
  11. Reflective obstructions
  12. Prasad's volume formula
  13. Preparation
  14. Local Jordan decomposition
  15. Computation of local factors
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $L$ be a primitive Hermitian lattice over $\mathscr{O}_F$ of signature $(1,n)$ with $n>2$. Assume $(\heartsuit)$. Then, for a positive integer $a$, the line bundle $\mathscr{M}(a)$ is big if $\mathop{\mathrm{dim}}\nolimits X_L=n$ or $\theta$ is sufficiently large.

Theorems & Definitions (60)

  • Theorem 1.1: Theorem \ref{['thm:volume_conclusion_oe']}
  • Corollary 1.2: Corollary \ref{['cor:unimodular']}, Subsection \ref{['subsection:unimod_sqfree_case']}
  • Remark 1.3: Subsection \ref{['subsection:general_case']}
  • Theorem 1.4: Corollary \ref{['cor:unimodular']}, Theorem \ref{['thm:gen_type']}
  • Remark 1.5
  • Conjecture 1.6: Finiteness of Hermitian lattices admitting reflective modular forms
  • Corollary 1.7: Corollary \ref{['cor:ams']}
  • Proposition 1.8: Proposition \ref{['thm:bigness_criterion']}
  • Theorem 1.9: Theorem \ref{['thm:volume_o']}, Theorem \ref{['thm:volume_e']}
  • Remark 1.10
  • ...and 50 more