Varieties of general type with the same Betti numbers as $\mathbb{P}^1\times \mathbb{P}^1\times\cdots\times \mathbb{P}^1$

Abstract

We study quotients $\Gamma \setminus {ℍ}^n$ of the $n$–fold product of the upper half-plane $ℍ$ by irreducible and torsion-free lattices $\Gamma <{PSL}_{\mathfrak{2}}{\left(ℝ\right)}^n$ with the same Betti numbers as the $n$–fold product ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ of projective lines. Such varieties are called fake products of projective lines or fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$. These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ is finite (independently of $n$), we give examples of fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^{\mathfrak{4}}$ and show that for $n>\mathfrak{4}$ there are no fake ${\left({\mathbb{P}}^{\mathfrak{1}}\right)}^n$ of the form $\Gamma \setminus {ℍ}^n$ with $\Gamma$ contained in the norm-one group of a maximal order of a quaternion algebra over a real number field.