Mathematische Zeitschrift

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

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



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 \(\mathbb R\).


Rational surfaces Automorphism groups Real structures Real forms Galois cohomology 

Mathematics Subject Classification

14J26 14J50 14P05 12G05 



The author is grateful to Frédéric Mangolte for asking him this question, and also for his advice. We want to thank Jérémy Blanc, Serge Cantat, Stéphane Druel, Viatcheslav Kharlamov and Stéphane Lamy for useful discussions.


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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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