Varieties of general type with the same Betti numbers as ℙ1 × ℙ1 ×⋯ × ℙ1

Džambić, Amir¹

¹ Institut für Mathematik, Johann Wolfgang Goethe Universität, Robert-Mayer-Str. 6-8, D-60325 Frankfurt am Main, Germany

Geometry & topology, Tome 19 (2015) no. 4, pp. 2257-2276.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2257

Classification : 11F06, 22E40
Keywords: Varieties of general type, Betti numbers, arithmetic groups, quaternion algebras

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