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

Džambić, Amir

doi:10.2140/gt.2015.19.2257

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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

DOI: 10.2140/gt.2015.19.2257

Abstract

We study quotients $Γ ∖ ℍ^{n}$ of the $n$ –fold product of the upper half-plane $ℍ$ by irreducible and torsion-free lattices $Γ < {PSL}_{2} {(ℝ)}^{n}$ with the same Betti numbers as the $n$ –fold product ${(ℙ^{1})}^{n}$ of projective lines. Such varieties are called fake products of projective lines or fake ${(ℙ^{1})}^{n}$ . These are higher-dimensional analogs of fake quadrics. In this paper we show that the number of fake ${(ℙ^{1})}^{n}$ is finite (independently of $n$ ), we give examples of fake ${(ℙ^{1})}^{4}$ and show that for $n > 4$ there are no fake ${(ℙ^{1})}^{n}$ of the form $Γ ∖ ℍ^{n}$ with $Γ$ 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

References

Publication

Received: 31 January 2014

Revised: 4 July 2014

Accepted: 15 October 2014

Published: 29 July 2015

Proposed: Walter Neumann

Seconded: Ronald Stern, Simon Donaldson

Authors

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

Volume 19, issue 4 (2015)