A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called an irreducible component of the Hodge locus) for such variations is defined over $overline{mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $overline{mathbb{Q}}$ to the case of special points.
{"title":"On the fields of definition of Hodge loci","authors":"Bruno Klingler, Anna Otwinowska, David Urbanik","doi":"10.24033/asens.2555","DOIUrl":"https://doi.org/10.24033/asens.2555","url":null,"abstract":"A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called an irreducible component of the Hodge locus) for such variations is defined over $overline{mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $overline{mathbb{Q}}$ to the case of special points.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135647333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2leq nleq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that $(M,g)$ is thick at infinity, i.e. any connected finite volume complete minimal hypersurface is closed. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface if $(M,g)$ has finite volume, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result of Irie-Marques-Neves when $(M,g)$ is shrinking to zero at infinity.
{"title":"A dichotomy for minimal hypersurfaces in manifolds thick at infinity","authors":"Antoine SONG","doi":"10.24033/asens.2550","DOIUrl":"https://doi.org/10.24033/asens.2550","url":null,"abstract":"Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2leq nleq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that $(M,g)$ is thick at infinity, i.e. any connected finite volume complete minimal hypersurface is closed. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. \u0000Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface if $(M,g)$ has finite volume, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result of Irie-Marques-Neves when $(M,g)$ is shrinking to zero at infinity.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135739912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $overline{mathbb{A}}:=(mathbb{R}/mathbb{Z})times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $mu$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measure $mu$ cannot be an endpoint of this segment. We also study the case of homeomorphisms such that $mathbb{A}=(mathbb{R}/mathbb{Z})times (0,1)$ is a region of instability of $f$. We show that, if the rotation numbers of the restriction of $f$ to the boundary components lies in the interior of the rotation set of $f$, then $f$ has uniformly bounded deviations from its rotation set. Finally, by combining this last result and recent work on realization of rotation vectors for annular continua, we obtain that if $f$ is any area-preserving homeomorphism of $overline{mathbb{A}}$ isotopic to the identity, then for every real number $rho$ in the rotation set of $f$, there exists an associated Aubry-Mather set, that is, a compact $f$-invariant set such that every point in this set has a rotation number equal to $rho$. This extends a result by P. Le Calvez previously known only for diffeomorphisms.
本文从旋转理论的角度，利用新提出的表面同胚强迫理论，研究了同胚同位素闭合环的同胚。我们的第一个结果是闭环上所谓的强形式Boyland猜想的一个解:假设$f$是$overline{mathbb{A}}:=(mathbb{R}/mathbb{Z})times [0,1]$的同胚，它是恒等式的同位素，并且保留了一个完全支持的Borel概率测度$mu$。证明了如果$f$的旋转集是一个非平凡段，则测度$mu$的旋转数不能是这个段的端点。我们也研究了同胚的情况，使得$mathbb{A}=(mathbb{R}/mathbb{Z})times (0,1)$是$f$的一个不稳定区域。我们证明，如果$f$对边界分量的限制的旋转数位于$f$的旋转集的内部，则$f$与其旋转集具有一致有界的偏差。最后，结合最近关于环形连续体旋转向量实现的工作，我们得到了如果$f$是$overline{mathbb{A}}$同位素对恒等式的任意保面积同胚，那么对于$f$的旋转集中的每一个实数$rho$，存在一个关联的Aubry-Mather集合，即紧致$f$ -不变集合，使得该集合中每一点的旋转数都等于$rho$。这扩展了P. Le Calvez先前只知道微分同态的结果。
{"title":"Applications of forcing theory to homeomorphisms of the closed annulus","authors":"Jonathan Conejeros, Fabio Armando Tal","doi":"10.24033/asens.2552","DOIUrl":"https://doi.org/10.24033/asens.2552","url":null,"abstract":"This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $overline{mathbb{A}}:=(mathbb{R}/mathbb{Z})times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $mu$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measure $mu$ cannot be an endpoint of this segment. We also study the case of homeomorphisms such that $mathbb{A}=(mathbb{R}/mathbb{Z})times (0,1)$ is a region of instability of $f$. We show that, if the rotation numbers of the restriction of $f$ to the boundary components lies in the interior of the rotation set of $f$, then $f$ has uniformly bounded deviations from its rotation set. Finally, by combining this last result and recent work on realization of rotation vectors for annular continua, we obtain that if $f$ is any area-preserving homeomorphism of $overline{mathbb{A}}$ isotopic to the identity, then for every real number $rho$ in the rotation set of $f$, there exists an associated Aubry-Mather set, that is, a compact $f$-invariant set such that every point in this set has a rotation number equal to $rho$. This extends a result by P. Le Calvez previously known only for diffeomorphisms.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135648201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.
{"title":"Intermediate Jacobians and rationality over arbitrary fields","authors":"Olivier Benoist, Olivier Wittenberg","doi":"10.24033/asens.2549","DOIUrl":"https://doi.org/10.24033/asens.2549","url":null,"abstract":"We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"202 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135647332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Petr Dunin-Barkowski, Reinier Kramer, Alexandr Popolitov, Sergey Shadrin
We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called $r$-ELSV formula, as well as its orbifold generalization, the $qr$-ELSV formula, proposed recently in [KLPS17].
{"title":"Loop equations and a proof of Zvonkine's $qr$-ELSV formula","authors":"Petr Dunin-Barkowski, Reinier Kramer, Alexandr Popolitov, Sergey Shadrin","doi":"10.24033/asens.2553","DOIUrl":"https://doi.org/10.24033/asens.2553","url":null,"abstract":"We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called $r$-ELSV formula, as well as its orbifold generalization, the $qr$-ELSV formula, proposed recently in [KLPS17].","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135647331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {em critical regularity} and of {em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^alpha$ diffeomorphisms of a 1-manifold $M$, not embeddable into $mathrm{Diff}^beta(M)$ for any $beta > alpha > 1$.
{"title":"Reconstructing maps out of groups","authors":"Kathryn Mann, Maxime Wolff","doi":"10.24033/asens.2551","DOIUrl":"https://doi.org/10.24033/asens.2551","url":null,"abstract":"We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {em critical regularity} and of {em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^alpha$ diffeomorphisms of a 1-manifold $M$, not embeddable into $mathrm{Diff}^beta(M)$ for any $beta > alpha > 1$.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135648199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.
{"title":"Motivic cohomology of quaternionic Shimura varieties and level raising","authors":"Rong ZHOU","doi":"10.24033/asens.2554","DOIUrl":"https://doi.org/10.24033/asens.2554","url":null,"abstract":"We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135744194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a reduced scheme $X$, smooth, geometrically connected, separated and of finite type over $k$ and a smooth proper morphism $f:Yrightarrow X$. The main result of this paper is that there are "lots" of closed points $xin X$ such that the fibre of $f$ at $x$ has the same geometric Picard rank as the generic fibre. If $X$ is a curve we show that this is true for all but finitely many $k$-rational points. In characteristic zero, these results have been proved by Andre (existence) and Cadoret-Tamagawa (finiteness) using Hodge theoretic methods. To extend the argument in positive characteristic we use the variational Tate conjecture in crystalline cohomology, the comparison between different $p$-adic cohomology theories and independence techniques. The result has applications to the Tate conjecture for divisors, uniform boundedness of Brauer groups, proper families of projective varieties and to the study of families of hyperplane sections of smooth projective varieties.
{"title":"Specialization of Néron-Severi groups in positive characteristic","authors":"Emiliano Ambrosi","doi":"10.24033/asens.2542","DOIUrl":"https://doi.org/10.24033/asens.2542","url":null,"abstract":"Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a reduced scheme $X$, smooth, geometrically connected, separated and of finite type over $k$ and a smooth proper morphism $f:Yrightarrow X$. The main result of this paper is that there are \"lots\" of closed points $xin X$ such that the fibre of $f$ at $x$ has the same geometric Picard rank as the generic fibre. If $X$ is a curve we show that this is true for all but finitely many $k$-rational points. In characteristic zero, these results have been proved by Andre (existence) and Cadoret-Tamagawa (finiteness) using Hodge theoretic methods. To extend the argument in positive characteristic we use the variational Tate conjecture in crystalline cohomology, the comparison between different $p$-adic cohomology theories and independence techniques. The result has applications to the Tate conjecture for divisors, uniform boundedness of Brauer groups, proper families of projective varieties and to the study of families of hyperplane sections of smooth projective varieties.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and Serhiyenko--Sherman-Bennett--Williams.
{"title":"Positroid varieties and cluster algebras","authors":"Pavel Galashin, Thomas Lam","doi":"10.24033/asens.2545","DOIUrl":"https://doi.org/10.24033/asens.2545","url":null,"abstract":"We show that the coordinate ring of an open positroid variety coincides with the cluster algebra associated to a Postnikov diagram. This confirms conjectures of Postnikov, Muller--Speyer, and Leclerc, and generalizes results of Scott and Serhiyenko--Sherman-Bennett--Williams.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adrien Boulanger, Pierre Mathieu, Cagri Sert, Alessandro Sisto
Let $Gamma$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $mu$ a probability measure on $Gamma$ whose support generates a non-elementary subsemigroup. Under the assumption that $mu$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $mu$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.
{"title":"Large deviations for random walks on Gromov-hyperbolic spaces","authors":"Adrien Boulanger, Pierre Mathieu, Cagri Sert, Alessandro Sisto","doi":"10.24033/asens.2546","DOIUrl":"https://doi.org/10.24033/asens.2546","url":null,"abstract":"Let $Gamma$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $mu$ a probability measure on $Gamma$ whose support generates a non-elementary subsemigroup. Under the assumption that $mu$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $mu$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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