The Kodaira dimension of even-dimensional ball quotients

TL;DR

This paper proves finiteness results for certain Hermitian lattices leading to non-general type ball quotients and shows that higher-dimensional even-dimensional ball quotients are generally of general type, using automorphic forms and Arthur's formula.

Contribution

It establishes finiteness of Hermitian lattices with non-general type quotients and proves that high-dimensional ball quotients are always of general type, employing automorphic forms and Arthur's multiplicity formula.

Findings

Finiteness of Hermitian lattices with non-general type quotients for large n.

Higher-dimensional ball quotients (n > 207) are always of general type.

Construction of a nontrivial cusp form of weight n on the complex ball.

Abstract

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for an odd discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\mathrm{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.