Products of curves as ball quotients

For any $g_1,g_2\ge 0$, this paper shows that there is a cocompact lattice $\mathrm{\Gamma }<\mathrm{P}\mathrm{U}(2,1)$ such that the ball quotient $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{B}}^2$ 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 $V^{\mathrm{\prime }}$ such that there is a compact ball quotient $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{B}}^2$ with a rational map $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{B}}^2⇢V^{\mathrm{\prime }}$. Often the proof gives the stronger conclusion that $V^{\mathrm{\prime }}$ 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.