Sur les groupes de transformations birationnelles des surfaces

Abstract

Nous étudions les groupes de type fini agissant par transformations birationnelles sur les surfaces complexes compactes kählériennes. Nous montrons (a) que le groupe des transformations birationnelles d’une surface satisfait l’alternative de Tits, (b) que les actions birationnelles de groupes de Kazhdan sur les surfaces sont toutes birationnellement conjuguées à des actions homographiques sur le plan projectif et (c) que si $f$ et $g$ sont deux transformations birationnelles de surfaces qui commutent, alors ou bien $f$ préserve un pinceau de courbes, ou bien l’un des itérés $g^m$ de $g,$ $m>0,$ coïncide avec un itéré $f^n$ de $f,$ $n\in \bf{Z}$.
Let $S$ be any compact complex kähler surface and $\sf{Bir}(S)$ the group of birational transformations of $S.$ We study the structure of finitely generated subgroups of $\sf{Bir}(S)$ and prove three main results: (a) $\sf{Bir}(S)$ satisfies the Tits Alternative, (b) if a group with Kazhdan property (T) acts on a compact kähler surface by birational transformations, then the action is conjugate to an action by linear projective transformations on the projective plane, and (c) if $f$ is an element of $\sf{Bir}(S)$ which does not preserve any pencil of curves and if $g$ commutes with $f$ then a positive iterate of $g$ coincides with an iterate of $f.$ The second statement provides a positive answer to Zimmer’s conjecture for birational actions on surfaces.

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      AUTHOR = {Dolgachev, Igor V.},
      TITLE = {Rational surfaces with a pencil of elliptic curves},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {30},
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      TITLE = {Distances hilbertiennes invariantes sur un espace homogène},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
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      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
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      author={Ghys, {É}tienne and Harpe, Pierre de~la Harpe},
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      SERIES={Progr. Math.},
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      AUTHOR = {Lieberman, David I.},
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      VENUE={{S}ém. {F}ran\c cois {N}orguet, 1975--1977},
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    @book {Margulis:book, MRKEY = {1090825},
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Authors

Serge Cantat

Université de Rennes I
Campus de Beaulieu
35042 Rennes cedex
France