Varieties of general type with the same Betti numbers as mathbb\n P¹\× mathbb P¹\×\…\× mathbb P¹

2014/11/12 by Amir Džambić, Džambić, Amir
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Algebraic Geometry and Number Theory #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.1411.3384

Abstract

We study quotients \Γ backslash mathbb Hⁿ of the n-fold product of\nthe upper half plane mathbb H by irreducible and torsion-free lattices\n\Γ < PSL₂( mathbb R)ⁿ with the same Betti numbers as the n-fold\nproduct ( mathbb P¹)ⁿ of projective lines. Such varieties are called fake\nproducts of projective lines or fake ( mathbb P¹)ⁿ. These are higher\ndimensional analogs of fake quadrics. In this paper we show that the number of\nfake ( mathbb P¹)ⁿ is finite (independently of n), we give examples of\nfake ( mathbb P¹)⁴ and show that for n>4 there are no fake ( mathbb\nP¹)ⁿ of the form \Γ backslash mathbb Hⁿ with \Γ contained in\nthe norm-1 group of a maximal order of a quaternion algebra over a real number\nfield.\n

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