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A survey of Torelli and monodromy results for holomorphicsymplectic varieties
 In Complex and differential geometry. Conference held at Leibniz Universität
, 2009
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Continuous Families of Rational Surface Automorphisms with Positive Entropy
"... §0. Introduction. Cantat [C1] has shown that if a compact projective surface carries an automorphism of positive entropy, then it has a minimal model which is either a torus, K3, or rational (or a quotient of one of these). It has seemed that rational surfaces which carry automorphisms of positive e ..."
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§0. Introduction. Cantat [C1] has shown that if a compact projective surface carries an automorphism of positive entropy, then it has a minimal model which is either a torus, K3, or rational (or a quotient of one of these). It has seemed that rational surfaces which carry automorphisms of positive entropy are relatively rare. Indeed, the first infinite family of such rational surfaces was found only recently (see [BK1,2] and [M]). Here we will show, on
Pseudoautomorphisms of positive entropy on the blowups of products of projective spaces
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Rational surfaces with a large group of automorphisms
 J. Amer. Math. Soc
"... 2. Halphen surfaces 867 3. Coble surfaces 875 4. Gizatullin’s Theorem and Cremona special point sets of nine points 883 ..."
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2. Halphen surfaces 867 3. Coble surfaces 875 4. Gizatullin’s Theorem and Cremona special point sets of nine points 883
Mapping classes associated to mixedsign Coxeter graphs
"... In this paper, we define and study properties of generalized Coxeter mapping classes on surfaces. Like the mapping classes associated to classical Coxeter graphs studied by Thurston and Leininger, the action on first homology is conjugate to the action of the Coxeter element of the associated Coxete ..."
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In this paper, we define and study properties of generalized Coxeter mapping classes on surfaces. Like the mapping classes associated to classical Coxeter graphs studied by Thurston and Leininger, the action on first homology is conjugate to the action of the Coxeter element of the associated Coxeter system considered as a reflection group, making classification and computations of invariants easier. However, unlike in the classical case, where dilatations of pseudoAnosov examples are bounded from below by Lehmer’s number, the generalized Coxeter graphs may be used to construct pseudoAnosov mapping classes with dilatation arbitrarily close to one. We observe that the smallest dilatation orientable pseudoAnosov mapping classes of genus 2,3,4 and 5 found by Lanneau and Thiffeault can be realized as generalized Coxeter mapping classes, and that the smallest known accumulation point of normalized dilatations can be realized as the limit of normalized dilatations of a sequence of generalized Coxeter mapping classes. For the latter construction, we define nonclassical periodic Coxeter mapping classes and use them as building blocks to define twisted mapping classes. We give sufficient conditions so that a sequence of twisted mapping classes corresponds to a convergent sequence on a fibered face. 1
Some properties of the Cremona group
, 2012
"... We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the wellknown subgroup of the Cremona group made up of the polynomial automorphisms of C2. Then we deal with the classification of birational ma ..."
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We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the wellknown subgroup of the Cremona group made up of the polynomial automorphisms of C2. Then we deal with the classification of birational maps and some applications (Tits alternative, nonsimplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the classification of the finite subgroups of the Cremona group and related problems, the description of the automorphisms of the Cremona group and the representations of some lattices in the Cremona group. The description of the centralizers of discrete dynamical systems is an important problem in real and complex dynamic, we describe the state of the art for this problem in the Cremona group. Let S be a compact complex surface which carries an automorphism f of positive topological entropy. Either the Kodaira dimension of S is zero and f is conjugate to an automorphism on the unique minimal model of S which is either a torus, or a K3 surface, or an Enriques surface, or S is a nonminimal rational surface and f is conjugate to a birational map of the complex projective plane. We deal with results obtained in this last case: construction of such automorphisms, dynamical properties (rotation domains...).
Automorphisms and autoequivalences of generic analytic K3 surfaces
 Journal of Geometry and Physics
"... Abstract. This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good description for generic analytic K3 surfaces, and ar ..."
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Abstract. This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good description for generic analytic K3 surfaces, and are in fact seen to be closely interrelated. 1.