Mathematische Zeitschrift

, Volume 282, Issue 3–4, pp 1127–1136 | Cite as

Real structures on rational surfaces and automorphisms acting trivially on Picard groups

Article

Abstract

In this article, we prove that any complex smooth rational surface X which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if X cannot be obtained by blowing up \(\mathbb P^2_{\mathbb C}\) at \(r\ge 10\) points). In particular, we prove that the group \(\mathrm {Aut\,}^{\#}X\) of complex automorphisms of X which act trivially on the Picard group of X is a linear algebraic group defined over \(\mathbb R\).

Keywords

Rational surfaces Automorphism groups Real structures Real forms Galois cohomology 

Mathematics Subject Classification

14J26 14J50 14P05 12G05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.LAREMA, UMR CNRS 6093Université d’AngersAngers Cedex 01France

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