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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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