L'espace de modules des courbes stables de Deligne et Mumford est une compactification de l'espace de modules des courbes lisses de genre $geq 2$, parametrant certaines courbes nodales. C'est un outil puissant pour l'etude des courbes algebriques. Des analogues en dimension superieure ont ete construits par Kollar, Shepherd-Barron et Alexeev en dimension 2, et par Viehweg dans le cas des varietes lisses. Nous expliquerons les idees recentes ayant permis la construction de ces espaces de modules en general, notamment le theoreme de reduction stable en dimension superieure, qui reflete leur compacite.
{"title":"Exposé Bourbaki 1158 : Réduction stable en dimension supérieure (d'après Kollár, Hacon-Xu, ...)","authors":"Olivier Benoist","doi":"10.24033/ast.1137","DOIUrl":"https://doi.org/10.24033/ast.1137","url":null,"abstract":"L'espace de modules des courbes stables de Deligne et Mumford est une compactification de l'espace de modules des courbes lisses de genre $geq 2$, parametrant certaines courbes nodales. C'est un outil puissant pour l'etude des courbes algebriques. Des analogues en dimension superieure ont ete construits par Kollar, Shepherd-Barron et Alexeev en dimension 2, et par Viehweg dans le cas des varietes lisses. Nous expliquerons les idees recentes ayant permis la construction de ces espaces de modules en general, notamment le theoreme de reduction stable en dimension superieure, qui reflete leur compacite.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47796416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.
{"title":"Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes","authors":"V. Delecroix, É. Goujard, P. Zograf, A. Zorich","doi":"10.24033/AST.1107","DOIUrl":"https://doi.org/10.24033/AST.1107","url":null,"abstract":"We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43305588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an introduction of a book called "strong regularity", to appear at Ast'erisque, containing: 1) Yoccoz' proof of Jakobson theorem www.college-de-france.fr/media/jean-christophe-yoccoz/UPL7416254474776698194_Jakobson_jcy.pdf 2) Berger's proof of the abundance of non-uniformly hyperbolic H'enon like endomorphisms arxiv.org/abs/0903.1473 It gives an overview of the main examples and conjectures of non-uniformly hyperbolic set for low dimensional dynamical systems. It compares the proofs of parameter selections based on the concept of binding with those based on the one of strong regularity.
{"title":"Strong regularity","authors":"P. Berger, J. Yoccoz","doi":"10.24033/ast.1076","DOIUrl":"https://doi.org/10.24033/ast.1076","url":null,"abstract":"This is an introduction of a book called \"strong regularity\", to appear at Ast'erisque, containing: 1) Yoccoz' proof of Jakobson theorem www.college-de-france.fr/media/jean-christophe-yoccoz/UPL7416254474776698194_Jakobson_jcy.pdf 2) Berger's proof of the abundance of non-uniformly hyperbolic H'enon like endomorphisms arxiv.org/abs/0903.1473 It gives an overview of the main examples and conjectures of non-uniformly hyperbolic set for low dimensional dynamical systems. It compares the proofs of parameter selections based on the concept of binding with those based on the one of strong regularity.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41329936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the emph{dual Roth type} condition is a condition on the emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) emph{distributional} limit shapes, which are constructed in a similar way to the emph{limit shapes} of Birkhoff sums associated in cite{MMY3} to functions which correspond to positive Lyapunov exponents.
{"title":"On Roth type conditions, duality and central Birkhoff sums for i.e.m.","authors":"S. Marmi, C. Ulcigrai, J. Yoccoz","doi":"10.24033/ast.11111","DOIUrl":"https://doi.org/10.24033/ast.11111","url":null,"abstract":"We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the emph{dual Roth type} condition is a condition on the emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) emph{distributional} limit shapes, which are constructed in a similar way to the emph{limit shapes} of Birkhoff sums associated in cite{MMY3} to functions which correspond to positive Lyapunov exponents.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47836477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Progrès récents concernant le programme de Zimmer [d’après A. Brown, D. Fisher et S. Hurtado]","authors":"Serge Cantat","doi":"10.24033/ast.1080","DOIUrl":"https://doi.org/10.24033/ast.1080","url":null,"abstract":"","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68829013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flexibilité en géométrie de contact en grande dimension (d’après Borman, Eliashberg et Murphy)","authors":"Patrick MASSOT","doi":"10.24033/ast.1069","DOIUrl":"https://doi.org/10.24033/ast.1069","url":null,"abstract":"","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"51 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68828404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Anti-gravity à la Carlotto-Schoen (after Carlotto and Schoen)","authors":"Piotr T Chruściel","doi":"10.24033/ast.1058","DOIUrl":"https://doi.org/10.24033/ast.1058","url":null,"abstract":"","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68827869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Le problème de Riemann-Hilbert dans le cas irrégulier (d’après des travaux de D’Agnolo, Kashiwara, Mochizuki et Schapira)","authors":"Stéphane GUILLERMOU","doi":"10.24033/ast.1066","DOIUrl":"https://doi.org/10.24033/ast.1066","url":null,"abstract":"","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68828358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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