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

الأصل · EN

We study quotients Γ Hⁿ of the n-fold product of the upper half plane H by irreducible and torsion-free lattices Γ< PSL₂(R)ⁿ with the same Betti numbers as the n-fold product (P¹)ⁿ of projective lines. Such varieties are called fake products of projective lines or fake (P¹)ⁿ. These are higher dimensional analogs of fake quadrics. In this paper we show that the number of fake (P¹)ⁿ is finite (independently of n), we give examples of fake (P¹)⁴ and show that for n>4 there are no fake (P¹)ⁿ of the form Γ Hⁿ with Γ contained in the norm-1 group of a maximal order of a quaternion algebra over a real number field.