Products of curves as ball quotients

Abstract: For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $\Gamma < \mathrm{PU}(2,1)$ such that the ball quotient $\Gamma \backslash \mathbb{B}^2$ is birational to a product $C_1 \times C_2$ of smooth projective curves $C_j$ of genus $g_j$. The only prior examples were $\mathbb{P}^1 \times \mathbb{P}^1$, due to Deligne--Mostow and rediscovered by many others, and a lesser-known product of elliptic curves whose existence follows from work of Hirzebruch. Combined with related new examples, this answers the rational variant of a question of Gromov in the positive for surfaces of Kodaira dimension $\kappa \le 0$, namely that they admit deformations $V^\prime$ such that there is a compact ball quotient $\Gamma \backslash \mathbb{B}^2$ with a rational map $\Gamma \backslash \mathbb{B}^2 \dashrightarrow V^\prime$. Often the proof gives the stronger conclusion that $V^\prime$ is birational to a ball quotient orbifold. It also follows that every simply connected $4$-manifold is dominated by a complex hyperbolic manifold. All examples considered in this paper are shown to be arithmetic, and even arithmeticity of Hirzebruch's example appears to be new.