In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in P 3 which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics.
7 pages, Comments welcomeInternational audienceWe study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map $\wedge^{2}H^{1}(S,\mathbb{C})\to H^{2}(S,\mathbb{C})$ and we discuss the problem related to the so-called Lagrangian surfaces. We obtain that this surface $S$ has maximal Picard number and has no higher genus fibrations. We compute that its Albanese variety $A$ is isomorphic to $(\mathbb{C}/\mathbb{Z}[\alpha])^{7}$, for $\alpha=e^{2i\pi/3}$
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
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