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

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 \(\mathbb R\).

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Notes

  1. This is equivalent to “antiholomorphic” if X is projective.

  2. So that one can obtain finiteness for smooth projective curves.

  3.  In our work, we will have to distinguish two kinds of rational surfaces: a rational surface is called basic if it dominates \(\mathbb P^2\) and non-basic otherwise.

  4. This result is probably well-known but we prove it in the course of proving the second part of this Theorem.

  5.  In fact, the following proposition is slightly simpler than the original one, which replaces the hypothesis of the finiteness of \(H^1(G,A_b)\) for every b by the weaker hypothesis of the finiteness of \(\mathrm {Im\,}(H^1(G,A_b)\rightarrow H^1(G,B_b)) \simeq (g^*)^{-1}(\{g^*(b)\})\) for every b.

  6. Remember that p is the natural morphism \(\mathrm {Aut\,}X\rightarrow \mathrm {O}(\mathrm {Pic\,}X)\), which we have seen in the Introduction.

  7. In fact, one can prove directly owing to the definitions that \(H^1(G,\mathbb {Z})\) is finite independently of the action of G, without using the deep Theorem 2.4.

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Acknowledgments

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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Correspondence to Mohamed Benzerga.

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Benzerga, M. Real structures on rational surfaces and automorphisms acting trivially on Picard groups. Math. Z. 282, 1127–1136 (2016). https://doi.org/10.1007/s00209-015-1581-x

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Keywords

  • Rational surfaces
  • Automorphism groups
  • Real structures
  • Real forms
  • Galois cohomology

Mathematics Subject Classification

  • 14J26
  • 14J50
  • 14P05
  • 12G05