The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine — albeit from an extremely naive/non-expert point of view! — the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.
{"title":"Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations","authors":"S. Mochizuki","doi":"10.4171/PRIMS/57-1-4","DOIUrl":"https://doi.org/10.4171/PRIMS/57-1-4","url":null,"abstract":"The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine — albeit from an extremely naive/non-expert point of view! — the foundational/settheoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “inter-universal”.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44918893","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}
For a linear differential operator P on P1 with unramified irregular singular points we examine a realization of P as a confluence of singularities of a Fuchsian differential operator P̃ having the same index of rigidity as P , which we call an unfolding of P . We conjecture that this is always possible. For example, if P is rigid, this is true and the unfolding helps us to study the equation Pu = 0.
{"title":"Versal Unfolding of Irregular Singularities of a Linear Differential Equation on the Riemann Sphere","authors":"T. Oshima","doi":"10.4171/prims/57-3-6","DOIUrl":"https://doi.org/10.4171/prims/57-3-6","url":null,"abstract":"For a linear differential operator P on P1 with unramified irregular singular points we examine a realization of P as a confluence of singularities of a Fuchsian differential operator P̃ having the same index of rigidity as P , which we call an unfolding of P . We conjecture that this is always possible. For example, if P is rigid, this is true and the unfolding helps us to study the equation Pu = 0.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70902043","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}
In this paper, we develop a certain combinatorial version of the theory of Belyi cuspidalization developed by Mochizuki. Write Q ⊆ C for the subfield of algebraic numbers ∈ C. We then apply this theory of combinatorial Belyi cuspidalization to certain natural closed subgroups of the Grothendieck-Teichmüller group associated to the field of p-adic numbers [where p is a prime number] and to stably ×μ-indivisible subfields of Q, i.e., subfields for which every finite field extension satisfies the property that every nonzero divisible element in the field extension is a root of unity. 2010 Mathematics Subject Classification: Primary 14H30.
{"title":"Combinatorial Belyi Cuspidalization and Arithmetic Subquotients of the Grothendieck–Teichmüller Group","authors":"Shota Tsujimura","doi":"10.4171/prims/56-4-5","DOIUrl":"https://doi.org/10.4171/prims/56-4-5","url":null,"abstract":"In this paper, we develop a certain combinatorial version of the theory of Belyi cuspidalization developed by Mochizuki. Write Q ⊆ C for the subfield of algebraic numbers ∈ C. We then apply this theory of combinatorial Belyi cuspidalization to certain natural closed subgroups of the Grothendieck-Teichmüller group associated to the field of p-adic numbers [where p is a prime number] and to stably ×μ-indivisible subfields of Q, i.e., subfields for which every finite field extension satisfies the property that every nonzero divisible element in the field extension is a root of unity. 2010 Mathematics Subject Classification: Primary 14H30.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-4-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46938338","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":"Boundedness of Weak Fano Pairs with Alpha-Invariants and Volumes Bounded Below","authors":"Weichung Chen","doi":"10.4171/prims/56-3-4","DOIUrl":"https://doi.org/10.4171/prims/56-3-4","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45733684","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":"Algebras of Lorch Analytic Mappings Defined on Uniform Algebras","authors":"Guilherme Mauro, L. A. Moraes","doi":"10.4171/prims/56-3-1","DOIUrl":"https://doi.org/10.4171/prims/56-3-1","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49149715","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}
The generating abelian subalgebras arsing from vectors in the ergodic component of Hiai's construction of the q-deformed Araki-Woods von Neumann algebras are quasi-split.
{"title":"On the Commutants of Generators of $q$-Deformed Araki–Woods von Neumann Algebras","authors":"Panchugopal Bikram, Kunal Mukherjee","doi":"10.4171/prims/58-3-1","DOIUrl":"https://doi.org/10.4171/prims/58-3-1","url":null,"abstract":"The generating abelian subalgebras arsing from vectors in the ergodic component of Hiai's construction of the q-deformed Araki-Woods von Neumann algebras are quasi-split.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46789184","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}
— In the present paper, we study Frobenius-projective structures on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of complex projective structures in the classical theory of Riemann surfaces. By means of the notion of Frobeniusprojective structures, we obtain a relationship between a certain rational function, i.e., a pseudo-coordinate, and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of indigenous bundles in the classical theory of Riemann surfaces, i.e., a Frobenius-indigenous structure. As an application of this relationship, we also prove the existence of certain Frobenius-destabilized locally free coherent sheaves of rank two.
{"title":"Frobenius-Projective Structures on Curves in Positive Characteristic","authors":"Yuichiro Hoshi","doi":"10.4171/prims/56-2-5","DOIUrl":"https://doi.org/10.4171/prims/56-2-5","url":null,"abstract":"— In the present paper, we study Frobenius-projective structures on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of complex projective structures in the classical theory of Riemann surfaces. By means of the notion of Frobeniusprojective structures, we obtain a relationship between a certain rational function, i.e., a pseudo-coordinate, and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of indigenous bundles in the classical theory of Riemann surfaces, i.e., a Frobenius-indigenous structure. As an application of this relationship, we also prove the existence of certain Frobenius-destabilized locally free coherent sheaves of rank two.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-2-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828693","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 establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.
{"title":"Subadjunction for Quasi-Log Canonical Pairs and Its Applications","authors":"O. Fujino","doi":"10.4171/prims/58-4-1","DOIUrl":"https://doi.org/10.4171/prims/58-4-1","url":null,"abstract":"We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45830635","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}
By building on a method introduced by Kashiwara and refined by Lichtin, we give upper bounds for the roots of certain b-functions associated to a regular function f in terms of a log resolution of singularities. As applications, we recover with more elementary methods a result of Budur and Saito describing the multiplier ideals of f in terms of the V-filtration of f and a result of the second named author with Popa giving a lower bound for the minimal exponent of f in terms of a log resolution.
{"title":"Upper Bounds for Roots of $b$-Functions, following Kashiwara and Lichtin","authors":"Bradley Dirks, M. Mustaţă","doi":"10.4171/prims/58-4-2","DOIUrl":"https://doi.org/10.4171/prims/58-4-2","url":null,"abstract":"By building on a method introduced by Kashiwara and refined by Lichtin, we give upper bounds for the roots of certain b-functions associated to a regular function f in terms of a log resolution of singularities. As applications, we recover with more elementary methods a result of Budur and Saito describing the multiplier ideals of f in terms of the V-filtration of f and a result of the second named author with Popa giving a lower bound for the minimal exponent of f in terms of a log resolution.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44996231","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 that when $M,N_{1},N_{2}$ are tracial von Neumann algebras with $M'cap M^{omega}$ abelian, $M'cap(Mbar{otimes}N_{1})^{omega}$ and $M'cap(Mbar{otimes}N_{2})^{omega}$ commute in $(Mbar{otimes}N_{1}bar{otimes}N_{2})^{omega}$. As a consequence, we obtain information on McDuff decompositions of $rm{II}_{1}$ factors of the form $Mbar{otimes}N$, where $M$ is a non-McDuff factor.
{"title":"On Central Sequence Algebras of Tensor Product von Neumann Algebras","authors":"Y. Hashiba","doi":"10.4171/prims/58-2-6","DOIUrl":"https://doi.org/10.4171/prims/58-2-6","url":null,"abstract":"We show that when $M,N_{1},N_{2}$ are tracial von Neumann algebras with $M'cap M^{omega}$ abelian, $M'cap(Mbar{otimes}N_{1})^{omega}$ and $M'cap(Mbar{otimes}N_{2})^{omega}$ commute in $(Mbar{otimes}N_{1}bar{otimes}N_{2})^{omega}$. As a consequence, we obtain information on McDuff decompositions of $rm{II}_{1}$ factors of the form $Mbar{otimes}N$, where $M$ is a non-McDuff factor.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42476829","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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