Reflective obstructions of unitary modular varieties

TL;DR

This paper provides a quantitative estimate of reflective obstructions in unitary modular varieties, showing they are small in higher dimensions, which aids in understanding their geometric properties and relates to the existence of certain cusp forms.

Contribution

It offers the first quantitative bounds on reflective obstructions for unitary modular varieties and links these bounds to the construction of small-weight cusp forms.

Findings

Reflective obstructions are negligible in dimensions greater than 138.

The study reduces to constructing small-weight cusp forms.

Partial proof of finiteness of Hermitian lattices with reflective modular forms.

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.