Non-homeomorphic Galois conjugate Beauville structures on

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• 作者：G. Gonzá ; lez-Diez ; ¹ ; ^{gabino.gonzalez@uam.es} ; D. Torres-Teigell¹ ; ² ; ^{david.torres@uam.es}
• 关键词：14J25 ; 14H30 ; 20H10
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：1 April, 2012
• 卷：229
• 期：6
• 页码：3096-3122
• 文件大小：296 K

摘要

Catanese始s rigidity results for surfaces isogenous to a product of curves indicate that Beauville surfaces should provide a fertile source of examples of Galois conjugate varieties that are not homeomorphic, a phenomenon discovered by J.P. Serre in the sixties.

In this paper, we construct Beauville surfaces with group for , and curves , such that the orbit of S under the action of the absolute Galois group contains non-homeomorphic conjugate surfaces. When the orbit consists exactly of two surfaces that have non-isomorphic fundamental groups, and the curves , have genera 8 and 49, which is shown to be the minimum for which there is a pair of non-homeomorphic Galois conjugate Beauville surfaces. As p grows the orbits contain an arbitrarily large number of non-homeomorphic surfaces.

Along the way we prove a metric rigidity theorem for Beauville surfaces which provides an elementary proof of the part of Catanese始s theory needed to prove our results.