We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also deserve independent interest themselves. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.
{"title":"Derived Hom Spaces in Rigid Analytic Geometry","authors":"Mauro Porta, Tony Yue Yu","doi":"10.4171/prims/57-3-7","DOIUrl":"https://doi.org/10.4171/prims/57-3-7","url":null,"abstract":"We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also deserve independent interest themselves. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47764098","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}
Let X be a variety over a field k and let X∞ be its space of arcs. We study the embedding dimension of the completion A^ of the local ring of X∞ at P where P is the stable point defined by a divisorial valuation ν on X. Assuming char k = 0, we prove that the embedding dimension of A^ is equal to k + 1 where k is the Mather discrepancy of X with respect to ν. We also obtain that the dimension of A^ has as lower bound the Mather-Jacobian log-discrepancy of X with respect to ν. For X normal and complete intersection, we prove as a consequence that points P of codimension one in X ∞ have discrepancy k ≤ 0.
{"title":"Mather Discrepancy as an Embedding Dimension in the Space of Arcs","authors":"H. Mourtada, Ana J. Reguera","doi":"10.4171/PRIMS/54-1-4","DOIUrl":"https://doi.org/10.4171/PRIMS/54-1-4","url":null,"abstract":"Let X be a variety over a field k and let X∞ be its space of arcs. We study the embedding dimension of the completion A^ of the local ring of X∞ at P where P is the stable point defined by a divisorial valuation ν on X. Assuming char k = 0, we prove that the embedding dimension of A^ is equal to k + 1 where k is the Mather discrepancy of X with respect to ν. We also obtain that the dimension of A^ has as lower bound the Mather-Jacobian log-discrepancy of X with respect to ν. For X normal and complete intersection, we prove as a consequence that points P of codimension one in X ∞ have discrepancy k ≤ 0.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-1-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47010888","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":"Almost Sure Well-Posedness of Fractional Schrödinger Equations with Hartree Nonlinearity","authors":"Gyeongha Hwang","doi":"10.4171/PRIMS/54-1-1","DOIUrl":"https://doi.org/10.4171/PRIMS/54-1-1","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-1-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43364588","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}
Fil: Angiono, Ivan Ezequiel. Universidad Nacional de Cordoba; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina
线程:Angiono, Ivan Ezequiel。科尔多瓦国立大学;阿根廷。国家科学技术研究委员会。Conicet科技中心-科尔多瓦。数学研究和研究中心。科尔多瓦国立大学。数学研究与研究中心;阿根廷
{"title":"A Quantum Version of the Algebra of Distributions of SL$_2$","authors":"I. Angiono","doi":"10.4171/PRIMS/54-1-5","DOIUrl":"https://doi.org/10.4171/PRIMS/54-1-5","url":null,"abstract":"Fil: Angiono, Ivan Ezequiel. Universidad Nacional de Cordoba; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-1-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45880867","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":"Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves","authors":"A. Dimca, Gabriel Sticlaru","doi":"10.4171/PRIMS/54-1-6","DOIUrl":"https://doi.org/10.4171/PRIMS/54-1-6","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-1-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41648766","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 consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists a magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} neq 0$, then generically $operatorname{dim} operatorname{Ker} H = infty$. If $b_{0} = 0$, then for each $m in {mathbb N} cup { infty }$, we construct almost periodic $b$ such that $operatorname{dim} operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.
{"title":"Spectral Properties of 2D Pauli Operators with Almost-Periodic Electromagnetic Fields","authors":"J. Bony, Nicolás Espinoza, G. Raikov","doi":"10.4171/PRIMS/55-3-1","DOIUrl":"https://doi.org/10.4171/PRIMS/55-3-1","url":null,"abstract":"We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists a magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} neq 0$, then generically $operatorname{dim} operatorname{Ker} H = infty$. If $b_{0} = 0$, then for each $m in {mathbb N} cup { infty }$, we construct almost periodic $b$ such that $operatorname{dim} operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-3-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45577106","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}
As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $ k $ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariants
{"title":"Milnor–Orr Invariants from the Kontsevich Invariant","authors":"Takefumi Nosaka","doi":"10.4171/prims/56-1-7","DOIUrl":"https://doi.org/10.4171/prims/56-1-7","url":null,"abstract":"As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $ k $ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariants","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-1-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46746574","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}
Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds.
{"title":"Integrable Deformations and Degenerations of Some Irregular Singularities","authors":"C. Sabbah","doi":"10.4171/PRIMS/57-3-2","DOIUrl":"https://doi.org/10.4171/PRIMS/57-3-2","url":null,"abstract":"Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44741685","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 develop a theory of moduli of Galois representations. More generally, for an object in a rather general class A of non-commutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) “f-A scheme”. Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the “difference” between the strong and week versions of the finiteness conjecture of Fontaine-Mazur is filled in by the finiteness conjecture of Khare-Moon.
{"title":"Moduli of Galois Representations","authors":"Y. Taguchi","doi":"10.4171/PRIMS/53-4-1","DOIUrl":"https://doi.org/10.4171/PRIMS/53-4-1","url":null,"abstract":"We develop a theory of moduli of Galois representations. More generally, for an object in a rather general class A of non-commutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) “f-A scheme”. Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the “difference” between the strong and week versions of the finiteness conjecture of Fontaine-Mazur is filled in by the finiteness conjecture of Khare-Moon.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/53-4-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44139508","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 give a sufficient condition for a class of tame compactified Landau-Ginzburg models in the sense of Katzarkov-Kontsevich-Pantev to satisfy some versions of their conjectures. We also give examples which satisfy the condition. The relations to the quantum D-modules of Fano manifolds and the original conjectures are explained in Appendices.
{"title":"Hodge–Tate Conditions for Landau–Ginzburg Models","authors":"Yota Shamoto","doi":"10.4171/PRIMS/54-3-2","DOIUrl":"https://doi.org/10.4171/PRIMS/54-3-2","url":null,"abstract":"We give a sufficient condition for a class of tame compactified Landau-Ginzburg models in the sense of Katzarkov-Kontsevich-Pantev to satisfy some versions of their conjectures. We also give examples which satisfy the condition. The relations to the quantum D-modules of Fano manifolds and the original conjectures are explained in Appendices.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/54-3-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47931895","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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