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arXiv · 2507.22203

The Kodaira dimension of even-dimensional ball quotients

Shuji Horinaga·Yota Maeda·Takuya Yamauchi
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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.

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Shuji Horinaga, Yota Maeda, Takuya Yamauchi. 2025-07-29. The Kodaira dimension of even-dimensional ball quotients. https://arxiv.org/abs/2507.22203

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