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 < 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 nH n with contained in the norm-one group of a maximal order of a quaternion algebra over a real number field. 11F06, 22E40

Lower bounds for covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$

“…The quaternionic Shimura varieties with χ(X Γ ) = 1 are of particular interest as they give examples of smooth projective varieties with the same Betti numbers as (P 1 ) n , see [8], [7]. The corollary implies that the number of automorphisms of such a variety is bounded by 60.…”

“…Therefore we need more precise informations on maximal elements in C(k, B). Such results on maximal elements of C(k, B), where B is a quaternion algebra over a general number field and their covolumes are obtained by A. Borel [5] (see also [8] and [7] for the case of lattices in G n ). According to Borel we have the following result on the minimal value of χ on C(k, B).…”

“…We now follow the steps of [6] to deal with classes C(k, B) where m ≤ 23 and [k ′ B : k] ≤ 2 and make now use of the inequality (7) with K = k ′ B . Again, using the estimates for T (0, 1) as in [6] but replacing 3 2…”