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.

# Non-homeomorphic Galois-conjugate Beauville surfaces

Speaker:

Gabino Gonzalez-Diez (Madrid)

Date/Time:

Mon, 06/06/2011 - 17:30

Room:

M103

Seminar series: