We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.
{"title":"Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics","authors":"I. Belegradek, David Gonz'alez-'Alvaro","doi":"10.5802/aif.3471","DOIUrl":"https://doi.org/10.5802/aif.3471","url":null,"abstract":"We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41953854","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 study rational Lagrangian immersions in a cotangent bundle, based on the microlocal theory of sheaves. We construct a sheaf quantization of a rational Lagrangian immersion and investigate its properties in Tamarkin category. Using the sheaf quantization, we give an explicit bound for the displacement energy and a Betti/cup-length estimate for the number of the intersection points of the immersion and its Hamiltonian image by a purely sheaf-theoretic method.
{"title":"Sheaf quantization and intersection of rational Lagrangian immersions","authors":"Tomohiro Asano, Yuichi Ike","doi":"10.5802/aif.3554","DOIUrl":"https://doi.org/10.5802/aif.3554","url":null,"abstract":"We study rational Lagrangian immersions in a cotangent bundle, based on the microlocal theory of sheaves. We construct a sheaf quantization of a rational Lagrangian immersion and investigate its properties in Tamarkin category. Using the sheaf quantization, we give an explicit bound for the displacement energy and a Betti/cup-length estimate for the number of the intersection points of the immersion and its Hamiltonian image by a purely sheaf-theoretic method.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41502102","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 deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is an universal constant (positive integer) $N=N(g)$ that depends only on $g$ that enjoys the following properties. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.
{"title":"Tate classes on self-products of Abelian varieties over finite fields","authors":"Y. Zarhin","doi":"10.5802/aif.3483","DOIUrl":"https://doi.org/10.5802/aif.3483","url":null,"abstract":"We deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is an universal constant (positive integer) $N=N(g)$ that depends only on $g$ that enjoys the following properties. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41587911","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}
In this paper, we construct a scattering theory for classical massive Dirac fields near the "double" horizon of an extreme Kerr-de Sitter blackhole. Our main tool is the existence of a conjugate operator in the sense of Mourre theory. Additionally, despite the fact that effects of the rotation are 'amplified' near the double horizon, we show that one can still reduce our study to a 1-dimensional problem through an appropriate decomposition of the Hilbert space.
{"title":"Scattering theory for Dirac fields near an extreme Kerr–de Sitter black hole","authors":"Jack Borthwick","doi":"10.5802/aif.3553/","DOIUrl":"https://doi.org/10.5802/aif.3553/","url":null,"abstract":"In this paper, we construct a scattering theory for classical massive Dirac fields near the \"double\" horizon of an extreme Kerr-de Sitter blackhole. Our main tool is the existence of a conjugate operator in the sense of Mourre theory. Additionally, despite the fact that effects of the rotation are 'amplified' near the double horizon, we show that one can still reduce our study to a 1-dimensional problem through an appropriate decomposition of the Hilbert space.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46747874","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}
Laurenctiu Maxim, Laurenctiu Puaunescu, Mihai Tibùar
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.
{"title":"Vanishing cohomology and Betti bounds for complex projective hypersurfaces","authors":"Laurenctiu Maxim, Laurenctiu Puaunescu, Mihai Tibùar","doi":"10.5802/aif.3486","DOIUrl":"https://doi.org/10.5802/aif.3486","url":null,"abstract":"We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48292832","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}
Let $D$ be a strictly pseudoconvex domain in $C^N$ and $X$ a pure-dimensional non-reduced subvariety that behaves well at $partial D$. We provide $L^p$-estimates of extensions of holomorphic functions defined on $X$.
{"title":"L p -estimates of extensions of holomorphic functions defined on a non-reduced subvariety","authors":"M. Andersson","doi":"10.5802/aif.3586","DOIUrl":"https://doi.org/10.5802/aif.3586","url":null,"abstract":"Let $D$ be a strictly pseudoconvex domain in $C^N$ and $X$ a pure-dimensional non-reduced subvariety that behaves well at $partial D$. We provide $L^p$-estimates of extensions of holomorphic functions defined on $X$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47573647","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 study definable sets, groups, and fields in the theory $T_infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($mathbb{N}times mathbb{Z},leq_{lex}$)-valued dimension on definable sets in $T_infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_infty$, we define a dimension on sets definable in $T^{RCF}_infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.
我们研究了具有非退化对称（或交替）双线性形式的代数闭域上无穷维向量空间的理论$T_infty$中的可定义集、群和域。然后，用这个维概念作为主要工具，我们证明了在$T_infty$中可定义的所有群都是（阿贝尔代数的）-by-代数的，这特别回答了Granger的一个问题。我们得出结论，在$T_infty$中可定义的每个无限域都可定义同构于向量空间的标量域。我们导出了$T_infty$中维度的良好行为的一些其他结果，例如，任何可定义集合中的每个泛型类型都是可定义类型；每个集合都是一个可拓基；每个可定义的群都有一个可定义的连通分量。我们还考虑了$T理论^{RCF}_实闭域上具有非退化交替双线性形式或非退化对称正定双线性形式的向量空间的infty$。使用与$T_infty$情况相同的构造，我们在$T中可定义的集合上定义了一个维数^{RCF}_infty$，并用它证明了关于可定义群和域的类似结果：在$T中可定义的每个群^{RCF}_｛infty｝$is（semialgebraic by abelian）-by-semialgebraic（特别是，它是（Lie by abelian）-by-Lie），并且$T中可定义的每个域^{RCF}_｛fty｝$在标量域中是可定义的，因此它要么是实闭的，要么是代数闭的。
{"title":"Sets, groups, and fields definable in vector spaces with a bilinear form","authors":"J. Dobrowolski","doi":"10.5802/aif.3559","DOIUrl":"https://doi.org/10.5802/aif.3559","url":null,"abstract":"We study definable sets, groups, and fields in the theory $T_infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($mathbb{N}times mathbb{Z},leq_{lex}$)-valued dimension on definable sets in $T_infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. \u0000We also consider the theory $T^{RCF}_infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_infty$, we define a dimension on sets definable in $T^{RCF}_infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49282426","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}
Nous montrons un resultat de semi-continuite du groupoide de Galois d'un champs de vecteur dependant d'un parametre. Applique aux equations de Painleve, ce resultat nous permet de calculer le groupoide de Galois de ces equations pour des valeurs generales des parametres.
{"title":"Spécialisation du groupoïde de Galois d’un champ de vecteurs","authors":"G. Casale, D. Davy","doi":"10.5802/aif.3506","DOIUrl":"https://doi.org/10.5802/aif.3506","url":null,"abstract":"Nous montrons un resultat de semi-continuite du groupoide de Galois d'un champs de vecteur dependant d'un parametre. Applique aux equations de Painleve, ce resultat nous permet de calculer le groupoide de Galois de ces equations pour des valeurs generales des parametres.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46262135","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 give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation theory and is a real version of the Nishinou-Siebert approach to the tropical correspondence theorem for complex curves. In dimension two, we use similar techniques to study the counting of real log curves with Welschinger signs and we obtain a new proof of Mikhalkin's tropical correspondence theorem for Welschinger invariants.
{"title":"Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants","authors":"Hulya Arguz, Pierrick Bousseau","doi":"10.5802/aif.3507","DOIUrl":"https://doi.org/10.5802/aif.3507","url":null,"abstract":"We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation theory and is a real version of the Nishinou-Siebert approach to the tropical correspondence theorem for complex curves. In dimension two, we use similar techniques to study the counting of real log curves with Welschinger signs and we obtain a new proof of Mikhalkin's tropical correspondence theorem for Welschinger invariants.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47725033","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 classify the maximal connected algebraic subgroups of Bir(X), when X is a surface.
当X是曲面时，我们对Bir（X）的极大连通代数子群进行了分类。
{"title":"Connected algebraic groups acting on algebraic surfaces","authors":"Pascal Fong","doi":"10.5802/aif.3595","DOIUrl":"https://doi.org/10.5802/aif.3595","url":null,"abstract":"We classify the maximal connected algebraic subgroups of Bir(X), when X is a surface.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49000385","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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