Let $G$ be a connected reductive affine algebraic group defined over $mathbb C$ and $mathfrak g$ its Lie algebra. We study the monodromy map from the space of $mathfrak g$-differential systems on a compact connected Riemann surface $Sigma$ of genus $g ,geq, 2$ to the character variety of $G$-representations of the fundamental group of $Sigma$. If the complex dimension of $G$ is at least three, we show that the monodromy map is an immersion at the generic point.
{"title":"The Monodromy Map from Differential Systems to the Character Variety Is Generically Immersive","authors":"Indranil Biswas, Sorin Dumitrescu","doi":"10.4171/prims/59-4-5","DOIUrl":"https://doi.org/10.4171/prims/59-4-5","url":null,"abstract":"Let $G$ be a connected reductive affine algebraic group defined over $mathbb C$ and $mathfrak g$ its Lie algebra. We study the monodromy map from the space of $mathfrak g$-differential systems on a compact connected Riemann surface $Sigma$ of genus $g ,geq, 2$ to the character variety of $G$-representations of the fundamental group of $Sigma$. If the complex dimension of $G$ is at least three, we show that the monodromy map is an immersion at the generic point.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pro-$p$ Criterion of Good Reduction of Punctured Elliptic Curves","authors":"Wojciech Porowski","doi":"10.4171/prims/59-4-6","DOIUrl":"https://doi.org/10.4171/prims/59-4-6","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.
{"title":"Trace- and Improved Data-Processing Inequalities for von Neumann Algebras","authors":"Stefan Hollands","doi":"10.4171/prims/59-4-1","DOIUrl":"https://doi.org/10.4171/prims/59-4-1","url":null,"abstract":"We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Galois Action on the Inertia Stack of Moduli Spaces of Curves","authors":"Benjamin Collas, Sylvain Maugeais","doi":"10.4171/prims/59-4-2","DOIUrl":"https://doi.org/10.4171/prims/59-4-2","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantum Dilogarithm Identities Arising from the Product Formula for the Universal R-Matrix of Quantum Affine Algebras","authors":"Masaru Sugawara","doi":"10.4171/prims/59-4-4","DOIUrl":"https://doi.org/10.4171/prims/59-4-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary","authors":"Takeo Ohsawa","doi":"10.4171/prims/59-4-3","DOIUrl":"https://doi.org/10.4171/prims/59-4-3","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show the existence of group-theoretic sections of certain geometrically pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves over $p$-adic local fields which are non-geometric, i.e., which do not arise from rational points. Among these quotients is the geometrically metabelian arithmetic fundamental group.
{"title":"Further Examples of Non-geometric Sections of Arithmetic Fundamental Groups","authors":"Mohamed Saïdi","doi":"10.4171/prims/59-3-5","DOIUrl":"https://doi.org/10.4171/prims/59-3-5","url":null,"abstract":"We show the existence of group-theoretic sections of certain geometrically pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves over $p$-adic local fields which are non-geometric, i.e., which do not arise from rational points. Among these quotients is the geometrically metabelian arithmetic fundamental group.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Enhanced ind-sheaves provide a suitable framework for the irregular Riemann–Hilbert correspondence. In this paper, we give some precision on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $mathcal{M}$ be a holonomic algebraic $mathcal{D}$-module on the affine line, and denote by ${^{mathsf{L}}}mathcal{M}$ its Fourier–Laplace transform. For a point $a$ on the affine line, denote by $ell_a$ the corresponding linear function on the dual affine line. Then the vanishing cycles of $mathcal{M}$ at $a$ are isomorphic to the graded component of degree $ell_a$ of the Stokes filtration of ${^{mathsf{L}}}mathcal{M}$ at infinity.
增强的尾轴为不规则黎曼-希尔伯特对应提供了一个合适的框架。本文给出了一维增强反常对象的附近环和消失环的精度。作为应用，我们给出了以下事实的拓扑证明。设$mathcal{M}$是仿射直线上的一个完整代数$mathcal{D}$-模，用${^{mathsf{L}}}mathcal{M}$表示它的傅里叶-拉普拉斯变换。对于仿射线上的点$a$，用$ well _a$表示对应的对偶仿射线上的线性函数。那么$mathcal{M}$在$a$处的消失周期与${^{mathsf{L}}}mathcal{M}$在无穷远处的Stokes滤波的阶次$ well _a$同构。
{"title":"Enhanced Nearby and Vanishing Cycles in Dimension One and Fourier Transform","authors":"Andrea D’Agnolo, Masaki Kashiwara","doi":"10.4171/prims/59-3-4","DOIUrl":"https://doi.org/10.4171/prims/59-3-4","url":null,"abstract":"Enhanced ind-sheaves provide a suitable framework for the irregular Riemann–Hilbert correspondence. In this paper, we give some precision on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $mathcal{M}$ be a holonomic algebraic $mathcal{D}$-module on the affine line, and denote by ${^{mathsf{L}}}mathcal{M}$ its Fourier–Laplace transform. For a point $a$ on the affine line, denote by $ell_a$ the corresponding linear function on the dual affine line. Then the vanishing cycles of $mathcal{M}$ at $a$ are isomorphic to the graded component of degree $ell_a$ of the Stokes filtration of ${^{mathsf{L}}}mathcal{M}$ at infinity.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is a continuation of a previous work on the extended Green–Naghdi system. We prolong the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Confining the work to the one-dimensional case, wellposedness and consistency with respect to initial data and parameters are proved, taking into account the effect of surface tension. The results are local, but long term in the sense of dependence upon initial data. As a conclusion, our solution remains close to the exact solution of the full Euler system with a better (smaller) precision and therefore the full justification of the models.
{"title":"Full Justification for the Extended Green–Naghdi System for an Uneven Bottom with/without Surface Tension","authors":"Bashar Khorbatly, Samer Israwi","doi":"10.4171/prims/59-3-6","DOIUrl":"https://doi.org/10.4171/prims/59-3-6","url":null,"abstract":"This paper is a continuation of a previous work on the extended Green–Naghdi system. We prolong the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Confining the work to the one-dimensional case, wellposedness and consistency with respect to initial data and parameters are proved, taking into account the effect of surface tension. The results are local, but long term in the sense of dependence upon initial data. As a conclusion, our solution remains close to the exact solution of the full Euler system with a better (smaller) precision and therefore the full justification of the models.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a holonomic $mathbb{C}[z,z^{-1}]langle partial_zrangle$-module $mathbb{M}$, following Loeser and Sabbah (Comment. Math. Helv. $mathbf{66}$ (1991), 458–503), one can consider its Mellin transform, which is a difference system on the affine line over $mathbb{C}$. In this note we prove a stationary phase formula, which shows that its formal behavior at infinity is determined by the local germs defined by $mathbb{M}$ at its singular points.
{"title":"Formal Stationary Phase for the Mellin Transform of a $mathcal D$-Module","authors":"Ricardo García López","doi":"10.4171/prims/59-3-2","DOIUrl":"https://doi.org/10.4171/prims/59-3-2","url":null,"abstract":"Given a holonomic $mathbb{C}[z,z^{-1}]langle partial_zrangle$-module $mathbb{M}$, following Loeser and Sabbah (Comment. Math. Helv. $mathbf{66}$ (1991), 458–503), one can consider its Mellin transform, which is a difference system on the affine line over $mathbb{C}$. In this note we prove a stationary phase formula, which shows that its formal behavior at infinity is determined by the local germs defined by $mathbb{M}$ at its singular points.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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