Products of curves as ball quotients

Abstract

For any $g_1,g_2\ge 0$, this paper shows that there is a cocompact lattice $\hat{I}“<\mathrm{PU}(2,1)$ such that the ball quotient $\hat{I}“\backslash {\mathbb{B}}^2$ 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 $\hat{I}º\le 0$, namely that they admit deformations $V^{\prime }$ such that there is a compact ball quotient $\hat{I}“\backslash {\mathbb{B}}^2$ with a rational map $\hat{I}“\backslash {\mathbb{B}}^2⇢V^{\prime }$. 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.

Adds Theorem 1.6, proving that every simply connected 4-manifold is smoothly dominated by a compact arithmetic ball quotient, along with minor edits