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

Abstract

We study quotients nH n of the n-fold product of the upper half plane H by irreducible and torsion-free lattices < PSL2(R) n with the same Betti numbers as the n-fold product (P 1 ) n of projective lines. Such a variety is called fake product of projective lines or fake (P 1 ) n . These are higher dimensional analogs of fake quadrics. In this paper we give examples of fake (P 1 ) 4 and show that there are no higher dimensional fake (P 1 ) n of the form nH n with contained in the norm-1 group of a maximal order of a quaternion algebra over a real number eld.