دانلود کتاب Some properties of the Cremona group

Abstract. We recall some properties, unfortunately not all, of the Cremonagroup.We first begin by presenting a nice proof of the amalgamated productstructure of the well-known subgroup of the Cremona group made up of thepolynomial automorphisms of C2. Then we deal with the classification ofbirational maps and some applications (Tits alternative, non-simplicity...)Since any birational map can be written as a composition of quadraticbirational maps up to an automorphism of the complex projective plane,we spend time on these special maps. Some questions of group theory areevoked: the classification of the finite subgroups of the Cremona group andrelated problems, the description of the automorphisms of the Cremonagroup and the representations of some lattices in the Cremona group.The description of the centralizers of discrete dynamical systems is animportant problem in real and complex dynamic, we describe the state ofthe art for this problem in the Cremona group.Let S be a compact complex surface which carries an automorphism fof positive topological entropy. Either the Kodaira dimension of S is zeroand f is conjugate to an automorphism on the unique minimal model ofS which is either a torus, or a K3 surface, or an Enriques surface, or S isa non-minimal rational surface and f is conjugate to a birational map ofthe complex projective plane. We deal with results obtained in this lastcase: construction of such automorphisms, dynamical properties (rotationdomains...).