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Automorphisms and quotients of quaternionic fake quadrics. (English) Zbl 1292.14018
A \(\mathbb Q\)-homology quadric surface is a normal projective algbraic surfaces with the same Betti number as the quadric surface in \(\mathbb P^3\), i.e., \(b_1=1\) and \(b_2=2\). The article under review is devoted to study the so-called quaternionic fake quadrics, i.e., quadric surfaces of general type of the form \(\Gamma \setminus \mathbb H \times \mathbb H\) with \(\Gamma\) a cocompact irreducible arithmetic lattices in \(\text{Aut}(\mathbb H) \times \text{Aut}(\mathbb H)\), where \(\mathbb H\) is the complex upper half plane. The authors study the possible automorphism group of such a surface, provide examples, and obtain the minimal desingularization of the quotient of a quaternionic fake quadrics by a group of automorphisms, some of which give new examples of surfaces of general type with \(q=p_g=0\).
Reviewer: Xin Lu (Mainz)

MSC:
14G35 Modular and Shimura varieties
14J10 Families, moduli, classification: algebraic theory
14J29 Surfaces of general type
14J50 Automorphisms of surfaces and higher-dimensional varieties
11F06 Structure of modular groups and generalizations; arithmetic groups
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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