The Kodaira dimension of even-dimensional ball quotients

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $\mathscr{O}_E$ 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 e