This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.
{"title":"A proof of A. Gabrielov’s rank theorem","authors":"André Belotto da Silva, Octave Curmi, G. Rond","doi":"10.5802/jep.173","DOIUrl":"https://doi.org/10.5802/jep.173","url":null,"abstract":"This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123592940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples due to the first author. Our new family of counterexamples is now built on the parametrix construction from our earlier work. Interestingly enough, it is sharp in at least some regions of phase space.
{"title":"New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain","authors":"Oana Ivanovici, G. Lebeau, F. Planchon","doi":"10.5802/jep.168","DOIUrl":"https://doi.org/10.5802/jep.168","url":null,"abstract":"We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples due to the first author. Our new family of counterexamples is now built on the parametrix construction from our earlier work. Interestingly enough, it is sharp in at least some regions of phase space.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132615030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We prove first results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. We focus in this paper in the finite state space case.
{"title":"Monotone solutions for mean field games master equations: finite state space and optimal stopping","authors":"Charles Bertucci","doi":"10.5802/JEP.167","DOIUrl":"https://doi.org/10.5802/JEP.167","url":null,"abstract":"We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We prove first results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. We focus in this paper in the finite state space case.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130476451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n ≥ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension ≤ n − 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.
{"title":"Hyperbolicity and specialness of symmetric powers","authors":"Benoît Cadorel, F. Campana, Erwan Rousseau","doi":"10.5802/jep.185","DOIUrl":"https://doi.org/10.5802/jep.185","url":null,"abstract":"Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n ≥ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension ≤ n − 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131646291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish P=W and PI=WI conjectures for character varieties with structural group $mathrm{GL}_n$ and $mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-etale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.
{"title":"P=W conjectures for character varieties with symplectic resolution","authors":"Camilla Felisetti, Mirko Mauri","doi":"10.5802/jep.196","DOIUrl":"https://doi.org/10.5802/jep.196","url":null,"abstract":"We establish P=W and PI=WI conjectures for character varieties with structural group $mathrm{GL}_n$ and $mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-etale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"550 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132588865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schrodinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.
{"title":"Critical time for the observability of Kolmogorov-type equations","authors":"J'er'emi Dard'e, Julien Royer","doi":"10.5802/jep.160","DOIUrl":"https://doi.org/10.5802/jep.160","url":null,"abstract":"This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schrodinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125510936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].
{"title":"On the rational motivic homotopy category","authors":"F. D'eglise, J. Fasel, Adeel A. Khan, F. Jin","doi":"10.5802/JEP.153","DOIUrl":"https://doi.org/10.5802/JEP.153","url":null,"abstract":"We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126423722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.
{"title":"Body of constant width with minimal area in a given annulus","authors":"A. Henrot, I. Lucardesi","doi":"10.5802/JEP.150","DOIUrl":"https://doi.org/10.5802/JEP.150","url":null,"abstract":"In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133502271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure with finite mass on $Omega$. Then we solve the Holder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. Namely, we show that this equation admits a unique Holder continuous solution on $Omega$ with a given Holder continuous boundary values if it admits a Holder continuous subsolution on $Omega$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Holder continuous $m$-subharmonic function on $Omega$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $Omega$ with an (explicit) exponent $tau > 1$.
{"title":"The Hölder continuous subsolution theorem for complex Hessian equations","authors":"A. Benali, A. Zeriahi","doi":"10.5802/JEP.133","DOIUrl":"https://doi.org/10.5802/JEP.133","url":null,"abstract":"Let $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure with finite mass on $Omega$. Then we solve the Holder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. Namely, we show that this equation admits a unique Holder continuous solution on $Omega$ with a given Holder continuous boundary values if it admits a Holder continuous subsolution on $Omega$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Holder continuous $m$-subharmonic function on $Omega$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $Omega$ with an (explicit) exponent $tau > 1$.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128261541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that generically, the degeneracies of a family of Hermitian matrices depending on three parameters have a conical structure. Our result applies to the study of topological phases of matter. It implies that adiabatic deformations of two-dimensional topological insulators come generically with Dirac-like propagating currents, whose total conductivity equals the chiral number of conical points.
{"title":"Ubiquity of conical points in topological insulators","authors":"A. Drouot","doi":"10.5802/JEP.152","DOIUrl":"https://doi.org/10.5802/JEP.152","url":null,"abstract":"We show that generically, the degeneracies of a family of Hermitian matrices depending on three parameters have a conical structure. Our result applies to the study of topological phases of matter. It implies that adiabatic deformations of two-dimensional topological insulators come generically with Dirac-like propagating currents, whose total conductivity equals the chiral number of conical points.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132497982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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