School of Mathematical Sciences

Non-homeomorphic Galois-conjugate Beauville surfaces menu

Non-homeomorphic Galois-conjugate Beauville surfaces

Gabino Gonzalez-Diez (Madrid)
Mon, 06/06/2011 - 17:30
Seminar series: 

It is known that although general field automorphisms of C are highly dis-
continuous the Betti numbers of a complex projective variety are the same as
those of the variety obtained by transforming the points by an arbitrary
field automorphism. In dimension 1 this is enough to ensure that the two varieties
are homeomorphic. However in 1964 Serre gave an example of two Galois conju-
gate but not homeomorphic varieties. Several other examples followed afterwards.
Recently Catanese has introduced a class of rigid surfaces (Beauville surfaces) defined
over the field of algebraic numbers that should provide a fertile source of
examples of this phenomenon.
In this talk I will present an explicit construction of a such surface which has the
property that its orbit under the action of the absolute Galois group
consists of two surfaces with non isomorphic fundamental groups. It turns out
that this is "the first" Beauville surface for which this phenomenon occurs. If time
permits I shall attempt to outline a proof Catanese's rigidity results within the
framework of Geometric Group Theory.