2023/12/09 by Matthew Stover, Stover, Matthew · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2312.05699
openalex publication_date 2023/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any g1, g2 ≥ 0, this paper shows that there is a cocompact lattice Γ< PU(2,1) such that the ball quotient Γ\backslash \mathbbB2 is birational to a product C1 × C2 of smooth projective curves Cj of genus gj. The only prior examples were ℙ1 × ℙ1, 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 κ≤ 0, namely that they admit deformations V^′ such that there is a compact ball quotient Γ\backslash \mathbbB2 with a rational map Γ\backslash \mathbbB2 \dashrightarrow V^′. Often the proof gives the stronger conclusion that V^′ 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.