The Kodaira dimension of even-dimensional ball quotients
Shuji Horinaga, Yota Maeda, Takuya Yamauchi
TL;DR
<3-5 sentence high-level summary> The paper addresses the Kodaira dimension of ball quotients arising from U(1,n) with even n>12 and imaginary quadratic fields of odd discriminant. It develops a representation-theoretic framework based on Arthur’s multiplicity formula to construct low-weight cusp forms of weight n and to control obstructions to general type. By combining cusp form existence (A), reflective obstructions (B) via volume computations, and elliptic considerations (C), the authors prove finiteness results for non-general-type quotients and provide explicit general-type ranges, including discriminant-kernel refinements. This approach bridges automorphic representation theory with birational geometry, yielding new general-type criteria for higher-dimensional ball quotients.
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.