Products of curves as ball quotients
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929539253272576 |
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| author | Stover, Matthew |
| author_facet | Stover, Matthew |
| contents | For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $Γ< \mathrm{PU}(2,1)$ such that the ball quotient $Γ\backslash \mathbb{B}^2$ 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 $κ\le 0$, namely that they admit deformations $V^\prime$ such that there is a compact ball quotient $Γ\backslash \mathbb{B}^2$ with a rational map $Γ\backslash \mathbb{B}^2 \dashrightarrow V^\prime$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_05699 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Products of curves as ball quotients Stover, Matthew Geometric Topology Algebraic Geometry For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $Γ< \mathrm{PU}(2,1)$ such that the ball quotient $Γ\backslash \mathbb{B}^2$ 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 $κ\le 0$, namely that they admit deformations $V^\prime$ such that there is a compact ball quotient $Γ\backslash \mathbb{B}^2$ with a rational map $Γ\backslash \mathbb{B}^2 \dashrightarrow V^\prime$. 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. |
| title | Products of curves as ball quotients |
| topic | Geometric Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2312.05699 |