Birational diffeomorphisms of the sphere of finite order

Abstract

In this thesis in real algebraic geometry, we obtain the classification of conjugacy classes of elements of prime order in the group of birational diffeomorphisms of the two-dimensional sphere. More precisely, the two-dimensional sphere S(\R) is the surface defined by the equation x^2+y^2+z^2=1 in \R^3. It corresponds to the real part of the smooth complex projective surface S_\C defined by the equation x^2+y^2+z^2=w^2 in the three-dimensional complex projective space \P^3, that is to say, to the part which is contained in the chart w=1. We can study the algebraic diffeomorphisms of S(\R): birational maps from the complex surface S_\C to itself such that the coefficients that define the map are real, and the restriction to the real part S(\R) is a diffeomorphism. Those maps are called birational diffeomorphisms of the sphere and they form a group denoted by Aut(S(\R)), the subject of this thesis. The interest in this group stems from many reasons. For instance, J. Kollar and F. Mangolte showed that Aut(X(\R)) is big - that is, dense in the group of all diffeomorphisms Diff(X(\R)) - if X is a smooth real compact rational surface, as in the case of the sphere. This is in stark contrast to the group of automorphisms of the complex surface, which is finite dimensional. One motivation to classify the conjugacy classes of the group Aut(S(\R)) is that, by definition, it is contained in the group of all birational maps of the complex surface S_\C, and this latter group is isomorphic to the Cremona group, i.e. the group of birational maps of the complex projective plane. The problem of classification of conjugacy classes of elements of finite order in the Cremona group has been of interest for a lot of mathematicians. L. Bayle and A. Beauville classified the elements of order two using the tools of the minimal model program developed in dimension two by Yu. Manin and V.A. Iskovskikh. T. de Fernex generalised the classification for elements of prime order (except for one case, done by A. Beauville and J. Blanc). Because of the relation of those two groups we can relate the conjugacy classes of birational diffeomorphisms of the sphere to the conjugacy classes of elements in the Cremona group. For example, we find that the sphere features an example of a large familiy of conjugacy classes of involutions that all correspond to one single class for the complex projective plane, to name just one of the differences between the two.