Geometric Methods in Algebra and Number Theory

Volume 235 of the series Progress in Mathematics pp 1-42

Beauville surfaces without real structures

  • Ingrid BauerAffiliated withDepartment of Mathematics, University of Bayreuth
  • , Fabrizio CataneseAffiliated withDepartment of Mathematics, University of Bayreuth
  • , Fritz GrunewaldAffiliated withMathematisches Institut

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Inspired by a construction by Arnaud Beauville of a surface of general type with K 2 = 8, p g = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an 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, while failing to admit a real structure.

The first aim of this note is to provide series of concrete examples of the second situation, respectively of the third.

The second aim is to introduce a wider audience, in particular group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them.