Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in question have uniformly bounded area and genus on each of the compact regions (but not necessarily global bounds), we prove a subsequence converges in an exhaustive Gromov sense.
{"title":"Exhaustive Gromov compactness for pseudoholomorphic curves","authors":"Joel W. Fish, H. Hofer","doi":"10.24033/ast.11101","DOIUrl":"https://doi.org/10.24033/ast.11101","url":null,"abstract":"Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in question have uniformly bounded area and genus on each of the compact regions (but not necessarily global bounds), we prove a subsequence converges in an exhaustive Gromov sense.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45419561","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}
1.2. Dans l’article [Dr], qui reste pour moi aussi mystérieux qu’il y a 31 ans, Drinfeld calcule le nombre de points fixes de φ : E → E. Un Q̄l-faisceau L0 de rang un sur Spec(Fq) est déterminé à isomorphisme près par l’unité λ de Q̄l telle que le Frobenius géométrique Fr ∈ Gal(F/Fq) agisse par multiplication par λ sur la fibre de L0 au point géométrique Spec(F). La Fq-torsion de F0 sur X0 par L0 est le produit tensoriel avec l’image inverse de L0 sur X0. Par abus de langage, on dira aussi “Fq-torsion par λ”. Drinfeld utilise que la classe d’isomorphie d’un Q̄l-faisceau lisse F sur X est fixe par Frob si et seulement si F est l’image inverse d’un Q̄l-faisceau F0 sur X0, et que, si F
1.2. [博士]条中,仍然是神秘的对我来说也有31岁,φDrinfeld定点数计算:E→E .一个Q̄l-faisceau L0级别上一个Spec (Fq)决心isomorphisme近Qλ股̄l这样的几何Frobenius滞后∈(F / Fr Fq)采取行动通过乘法λL0纤维上的几何Spec (F)段。F0在X0上乘以L0的扭转fq是L0在X0上逆像的张量积。为了滥用语言，我们也会说“Fq-torsion by λ”。Drinfeld利用下课,一个Q d’isomorphiēl-faisceau光滑F对X是固定由Frob当且仅当Q F逆形象是一个̄l-faisceau F (X0),并说,如果上fo
{"title":"Comptage de faisceaux $l$-adiques","authors":"P. Deligne","doi":"10.24033/AST.963","DOIUrl":"https://doi.org/10.24033/AST.963","url":null,"abstract":"1.2. Dans l’article [Dr], qui reste pour moi aussi mystérieux qu’il y a 31 ans, Drinfeld calcule le nombre de points fixes de φ : E → E. Un Q̄l-faisceau L0 de rang un sur Spec(Fq) est déterminé à isomorphisme près par l’unité λ de Q̄l telle que le Frobenius géométrique Fr ∈ Gal(F/Fq) agisse par multiplication par λ sur la fibre de L0 au point géométrique Spec(F). La Fq-torsion de F0 sur X0 par L0 est le produit tensoriel avec l’image inverse de L0 sur X0. Par abus de langage, on dira aussi “Fq-torsion par λ”. Drinfeld utilise que la classe d’isomorphie d’un Q̄l-faisceau lisse F sur X est fixe par Frob si et seulement si F est l’image inverse d’un Q̄l-faisceau F0 sur X0, et que, si F","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68830862","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}
Using the combinatorics of two interpenetrating face centered cubic lattices together with the part of calculus naturally encoded in combinatorial topology, we construct from first principles a lattice model of 3D incompressible hydrodynamics on triply periodic three space. Actually the construction applies to every dimension, but has special duality features in dimension three.
{"title":"Lattice Hydrodynamics","authors":"D. Sullivan","doi":"10.24033/ast.11106","DOIUrl":"https://doi.org/10.24033/ast.11106","url":null,"abstract":"Using the combinatorics of two interpenetrating face centered cubic lattices together with the part of calculus naturally encoded in combinatorial topology, we construct from first principles a lattice model of 3D incompressible hydrodynamics on triply periodic three space. Actually the construction applies to every dimension, but has special duality features in dimension three.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45038681","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}
Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the"characteristic polynomial"of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded"Chow ring"$A(M)$ for which they prove analogs of the Poincar'e duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.
有限拟阵是表达线性独立性概念的组合结构。1964年，G.-C.Rota猜想拟阵$M$的“特征多项式”的系数，即其系数枚举其给定秩的子集的多项式，形成对数凹序列。K.Adiprasto，J.Huh et E.Katz已经用一些方法证明了这一猜想，这些方法虽然完全是组合的，但受到了代数几何的启发。从拟阵$M$的Bergman扇出发，他们定义了一个分次的“Chow环”$a（M）$，并证明了其类似于庞加莱对偶、Hard-Lefschetz定理和Hodge-Riemann关系。所寻求的对数凹性不等式类似于Khovanskii-Teissier不等式。
{"title":"Relations de Hodge-Riemann et combinatoire des matroïdes","authors":"Antoine Chambert-Loir","doi":"10.24033/ast.1088","DOIUrl":"https://doi.org/10.24033/ast.1088","url":null,"abstract":"Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the\"characteristic polynomial\"of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded\"Chow ring\"$A(M)$ for which they prove analogs of the Poincar'e duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44670660","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 consider unitary polynomials $P in Z[X]$ whose roots $(x_i)$ belong to a given compact $K$ of $C$. To such a polynomial we associate the measure $mu_P$ on $K$ which is the mean value of the Dirac measures $delta_{x_i}$. What are the limits of the measures $mu_P$ when $P$ varies ? In particular, what are their supports? We give partial answers to such questions, especially when $K$ is contained in $R$.
{"title":"Distribution asymptotique des valeurs propres des endomorphismes de Frobenius d'après Abel, Chebyshev, Robinson,...","authors":"Jean-Pierre Serre","doi":"10.24033/ast.1090","DOIUrl":"https://doi.org/10.24033/ast.1090","url":null,"abstract":"We consider unitary polynomials $P in Z[X]$ whose roots $(x_i)$ belong to a given compact $K$ of $C$. To such a polynomial we associate the measure $mu_P$ on $K$ which is the mean value of the Dirac measures $delta_{x_i}$. What are the limits of the measures $mu_P$ when $P$ varies ? In particular, what are their supports? We give partial answers to such questions, especially when $K$ is contained in $R$.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41335811","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}
The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p$. We introduce a category of cochain complexes equipped with an endomorphism $F$ (of underlying graded abelian groups) satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L eta_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison in $A Omega$-cohomology theory.
{"title":"Revisiting the de Rham-Witt complex","authors":"B. Bhatt, J. Lurie, A. Mathew","doi":"10.24033/ast.1146","DOIUrl":"https://doi.org/10.24033/ast.1146","url":null,"abstract":"The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p$. \u0000We introduce a category of cochain complexes equipped with an endomorphism $F$ (of underlying graded abelian groups) satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L eta_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison in $A Omega$-cohomology theory.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42912789","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}
Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccoz's complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.
{"title":"On quasi-invariant curves","authors":"Ricardo P'erez-Marco","doi":"10.24033/AST.11113","DOIUrl":"https://doi.org/10.24033/AST.11113","url":null,"abstract":"Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccoz's complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43602960","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}
Laurent Fargues, J. Fontaine, préface de Pierre Colmez
Dans ce travail nous definissons et etudions la courbe fondamentale en theorie de Hodge p-adique. Nous demontrons un theoreme de classification des fibres vectoriels sur celle-ci et nous en deduisons de nouvelles preuves des deux theoremes fondamentaux de la theorie de Hodge p-adique: faiblement admissible implique admissible et e theoreme de la monodromie p-adique.
{"title":"Courbes et fibrés vectoriels en théorie de Hodge p-adique","authors":"Laurent Fargues, J. Fontaine, préface de Pierre Colmez","doi":"10.24033/AST.1056","DOIUrl":"https://doi.org/10.24033/AST.1056","url":null,"abstract":"Dans ce travail nous definissons et etudions la courbe fondamentale en theorie de Hodge p-adique. Nous demontrons un theoreme de classification des fibres vectoriels sur celle-ci et nous en deduisons de nouvelles preuves des deux theoremes fondamentaux de la theorie de Hodge p-adique: faiblement admissible implique admissible et e theoreme de la monodromie p-adique.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":"406 1","pages":"1-382"},"PeriodicalIF":1.1,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68827666","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}
B. Fayad, Jean-Pierre Marco, D. Sauzin, Jean-Pierre Marco
We give examples of symplectic diffeomorphisms of R^6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit.
我们给出了R^6的辛微分同态的例子，其原点是一个吸引轨道的非共振椭圆不动点。
{"title":"Attracted by an elliptic fixed point","authors":"B. Fayad, Jean-Pierre Marco, D. Sauzin, Jean-Pierre Marco","doi":"10.24033/AST.11118","DOIUrl":"https://doi.org/10.24033/AST.11118","url":null,"abstract":"We give examples of symplectic diffeomorphisms of R^6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48417152","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}
If $f : S' to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_otimes : mathcal{H}_{bullet}(S')to mathcal{H}_{bullet}(S)$, where $mathcal{H}_bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_otimes : mathcal{S}mathcal{H}(S') to mathcal{S}mathcal{H}(S)$, where $mathcal{S}mathcal{H}(S)$ is the $mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.
{"title":"Norms in motivic homotopy theory","authors":"Tom Bachmann, Marc Hoyois","doi":"10.24033/ast.1147","DOIUrl":"https://doi.org/10.24033/ast.1147","url":null,"abstract":"If $f : S' to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal \"norm\" functor $f_otimes : mathcal{H}_{bullet}(S')to mathcal{H}_{bullet}(S)$, where $mathcal{H}_bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_otimes : mathcal{S}mathcal{H}(S') to mathcal{S}mathcal{H}(S)$, where $mathcal{S}mathcal{H}(S)$ is the $mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.","PeriodicalId":55445,"journal":{"name":"Asterisque","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42046211","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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