#### Volume 19, issue 4 (2015)

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

### Amir Džambić

Geometry & Topology 19 (2015) 2257–2276
##### 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 ${\left({ℙ}^{\mathfrak{1}}\right)}^{n}$ of projective lines. Such varieties are called fake products of projective lines or fake ${\left({ℙ}^{\mathfrak{1}}\right)}^{n}$. These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake ${\left({ℙ}^{\mathfrak{1}}\right)}^{n}$ is finite (independently of $n$), we give examples of fake ${\left({ℙ}^{\mathfrak{1}}\right)}^{\mathfrak{4}}$ and show that for $n>\mathfrak{4}$ there are no fake ${\left({ℙ}^{\mathfrak{1}}\right)}^{n}$ 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.

##### Keywords
Varieties of general type, Betti numbers, arithmetic groups, quaternion algebras
##### Mathematical Subject Classification 2010
Primary: 11F06, 22E40