Let H be an arrangement of hyperplanes in R and PervpC,Hq be the category of perverse sheaves on C smooth with respect to the stratification given by complexified flats of H. We give a description of PervpC,Hq in terms of “matrix diagrams”, i.e., diagrams formed by vector spaces EA,B labelled by pairs pA,Bq of real faces of H (of all dimensions) or, equivalently, by the cells iA ` B of a natural cell decomposition of C. A matrix diagram is formally similar to a datum describing a constructible (non-perverse) sheaf but with the direction of one half of the arrows reversed.
{"title":"Perverse Sheaves over Real Hyperplane Arrangements II","authors":"M. Kapranov, V. Schechtman","doi":"10.4171/prims/58-4-5","DOIUrl":"https://doi.org/10.4171/prims/58-4-5","url":null,"abstract":"Let H be an arrangement of hyperplanes in R and PervpC,Hq be the category of perverse sheaves on C smooth with respect to the stratification given by complexified flats of H. We give a description of PervpC,Hq in terms of “matrix diagrams”, i.e., diagrams formed by vector spaces EA,B labelled by pairs pA,Bq of real faces of H (of all dimensions) or, equivalently, by the cells iA ` B of a natural cell decomposition of C. A matrix diagram is formally similar to a datum describing a constructible (non-perverse) sheaf but with the direction of one half of the arrows reversed.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43542553","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 article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(ngeq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
{"title":"The Category $mathcal O$ for Lie Algebras of Vector Fields (I): Tilting Modules and Character Formulas","authors":"Feifei Duan, B. Shu, Yufeng Yao","doi":"10.4171/prims/56-4-3","DOIUrl":"https://doi.org/10.4171/prims/56-4-3","url":null,"abstract":"In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(ngeq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/56-4-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46201998","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":"Absolute Schauder Decompositions and Linearization of Holomorphic Mappings of Bounded Type","authors":"G. Botelho, V. V. Fávaro, J. Mujica","doi":"10.4171/PRIMS/55-3-6","DOIUrl":"https://doi.org/10.4171/PRIMS/55-3-6","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-3-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48660351","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 construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, we show that the extended affine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups.
{"title":"Groups of Extended Affine Lie Type","authors":"S. Azam, A. Parsa","doi":"10.4171/PRIMS/55-3-5","DOIUrl":"https://doi.org/10.4171/PRIMS/55-3-5","url":null,"abstract":"We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, we show that the extended affine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-3-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41321305","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":"Period Map of Triple Coverings of P$^2$ and Mixed Hodge Structures","authors":"Keiji Matsumoto, T. Terasoma","doi":"10.4171/PRIMS/55-3-2","DOIUrl":"https://doi.org/10.4171/PRIMS/55-3-2","url":null,"abstract":"","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PRIMS/55-3-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47825372","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 $R$ and $S$ be rings and $_Romega_S$ a semidualizing bimodule. We study when the double functor $Tor^S_i(omega, Ext^i_{R}(omega,-))$ preserves epimorphisms and the double functor $Ext_{R}^i(omega, Tor_i^{S}(omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $omega$.
{"title":"Cograde Conditions and Cotorsion Pairs","authors":"Xi Tang, Zhaoyong Huang","doi":"10.4171/prims/56-3-2","DOIUrl":"https://doi.org/10.4171/prims/56-3-2","url":null,"abstract":"Let $R$ and $S$ be rings and $_Romega_S$ a semidualizing bimodule. We study when the double functor $Tor^S_i(omega, Ext^i_{R}(omega,-))$ preserves epimorphisms and the double functor $Ext_{R}^i(omega, Tor_i^{S}(omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $omega$.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47593947","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}
Motivated by the problem for the existence of Kahler-Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not big. Cheltsov, Rubinstein and Zhang proved it affirmatively in dimension $2$ with irreducible boundaries except for the type $(operatorname{I.9B.}n)$ with $1leq nleq 6$. Unfortunately, recently, Fujita, Liu, Suss, Zhang and Zhuang showed the non-K-polystability for some members of type $(operatorname{I.9B.}1)$ and for some members of type $(operatorname{I.9B.}2)$. In this article, we show that Cheltsov--Rubinstein's problem is true for all of the remaining cases. More precisely, we explicitly compute the delta-invariant for asymptotically log del Pezzo surfaces of type $(operatorname{I.9B.}n)$ for all $ngeq 1$ with small cone angles. As a consequence, we finish Cheltsov--Rubinstein's problem in dimension $2$ with irreducible boundaries.
Cheltsov和Rubinstein受到Kahler-Einstein边度量存在性问题的启发，在反对数正则因子不大的情况下，推测了具有小锥角的渐近对数Fano变型的k -多稳定性。Cheltsov, Rubinstein和Zhang在除$1leq nleq 6$型$(operatorname{I.9B.}n)$外的不可约边界的维度$2$上肯定地证明了它。不幸的是，最近Fujita, Liu, Su ss, Zhang和Zhuang证明了$(operatorname{I.9B.}1)$型的一些成员和$(operatorname{I.9B.}2)$型的一些成员的非k -polystability。在这篇文章中，我们证明了Cheltsov—Rubinstein问题对所有剩余的情况都是正确的。更精确地说，我们显式地计算了对于所有具有小锥角的$ngeq 1$型的渐近log del Pezzo曲面$(operatorname{I.9B.}n)$的δ不变量。因此，我们完成了Cheltsov—Rubinstein的边界不可约维问题$2$。
{"title":"On Uniform K-Stability for Some Asymptotically log del Pezzo Surfaces","authors":"Kento Fujita","doi":"10.4171/prims/58-1-6","DOIUrl":"https://doi.org/10.4171/prims/58-1-6","url":null,"abstract":"Motivated by the problem for the existence of Kahler-Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not big. Cheltsov, Rubinstein and Zhang proved it affirmatively in dimension $2$ with irreducible boundaries except for the type $(operatorname{I.9B.}n)$ with $1leq nleq 6$. Unfortunately, recently, Fujita, Liu, Suss, Zhang and Zhuang showed the non-K-polystability for some members of type $(operatorname{I.9B.}1)$ and for some members of type $(operatorname{I.9B.}2)$. In this article, we show that Cheltsov--Rubinstein's problem is true for all of the remaining cases. More precisely, we explicitly compute the delta-invariant for asymptotically log del Pezzo surfaces of type $(operatorname{I.9B.}n)$ for all $ngeq 1$ with small cone angles. As a consequence, we finish Cheltsov--Rubinstein's problem in dimension $2$ with irreducible boundaries.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47522358","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 characteristic zero, the residual order constitutes, after the local multiplicity, the second key invariant for the resolution of singularities. It is defined as the order of the coefficient ideal in a local hypersurface of maximal contact, minus the exceptional multiplicities. It does not increase under blowup in permissible centers as long as the local multiplicity remains constant. In positive characteristic, however, the residual order (defined now as the maximum over all smooth local hypersurfaces) may increase under blowup. In the article we analyze in detail the circumstances when this happens. This may help to develop a modification of the residual order which does work in positive characteristic.
{"title":"Characterizing the Increase of the Residual Order under Blowup in Positive Characteristic","authors":"H. Hauser, Stefan Perlega","doi":"10.4171/prims/55-4-7","DOIUrl":"https://doi.org/10.4171/prims/55-4-7","url":null,"abstract":"In characteristic zero, the residual order constitutes, after the local multiplicity, the second key invariant for the resolution of singularities. It is defined as the order of the coefficient ideal in a local hypersurface of maximal contact, minus the exceptional multiplicities. It does not increase under blowup in permissible centers as long as the local multiplicity remains constant. In positive characteristic, however, the residual order (defined now as the maximum over all smooth local hypersurfaces) may increase under blowup. In the article we analyze in detail the circumstances when this happens. This may help to develop a modification of the residual order which does work in positive characteristic.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/prims/55-4-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44771384","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 minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.
我们证明了对数正则对的极小模型是存在的，假设光滑变种的极小模型存在。
{"title":"On the Existence of Minimal Models for Log Canonical Pairs","authors":"Vladimir Lazi'c, Nikolaos Tsakanikas","doi":"10.4171/PRIMS/58-2-3","DOIUrl":"https://doi.org/10.4171/PRIMS/58-2-3","url":null,"abstract":"We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41481660","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 p be a prime number. In the present paper, we study geometrically pro-p arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a p-adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro-p arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space. 2010 Mathematics Subject Classification: Primary 14H30; Secondary 14H10.
{"title":"The Semi-absolute Anabelian Geometry of Geometrically Pro-$p$ Arithmetic Fundamental Groups of Associated Low-Dimensional Configuration Spaces","authors":"Kazumi Higashiyama","doi":"10.14989/DOCTOR.K21544","DOIUrl":"https://doi.org/10.14989/DOCTOR.K21544","url":null,"abstract":"Let p be a prime number. In the present paper, we study geometrically pro-p arithmetic fundamental groups of low-dimensional configuration spaces associated to a given hyperbolic curve over an arithmetic field such as a number field or a p-adic local field. Our main results concern the group-theoretic reconstruction of the function field of certain tripods (i.e., copies of the projective line minus three points) that lie inside such a configuration space from the associated geometrically pro-p arithmetic fundamental group, equipped with the auxiliary data constituted by the collection of decomposition groups determined by the closed points of the associated compactified configuration space. 2010 Mathematics Subject Classification: Primary 14H30; Secondary 14H10.","PeriodicalId":54528,"journal":{"name":"Publications of the Research Institute for Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43434915","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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