Beauville surfaces without real structures, I
  Inspired by a construction by Arnaud Beauville of a surface of general type with $K^2 = 8, p_g =0$, the second author defined the Beauville surfaces as the surfaces which are rigid, i.e., they have no nontrivial deformation, and admit un unramified covering which is isomorphic to a product of curves of genus at least 2.   In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, neverless it fails to admit a real structure.   First aim of this note is to provide series of concrete examples of the second situation, respectively of the third. Second aim is to introduce a wider audience, especially group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them. 
Ingrid Bauer, Fabrizio Catanese, Fritz Grunewald (Universitaet   Bayreuth, Universitaet Bayreuth, Universitaet Duesseldorf)