We describe the $mathfrak sl_3$ non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in work on rank-$2$ spider categories. Fontaine, Kamnitzer, Kuperberg showed that the $mathfrak sl_3$ non-elliptic webs are dual to CAT(0) diskoids in the affine building. We show that each such dual diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces the duals of the non-elliptic web basis. The convex hulls in the affine building were first introduced by Faltings and are related to tropical convexity, as discussed in work by Joswig, Sturmfels, Yu and by Zhang.
{"title":"Non-Elliptic Webs and Convex Sets in the Affine Building","authors":"Tair Akhmejanov","doi":"10.4171/dm/802","DOIUrl":"https://doi.org/10.4171/dm/802","url":null,"abstract":"We describe the $mathfrak sl_3$ non-elliptic webs in terms of convex sets in the affine building. Kuperberg defined the non-elliptic web basis in work on rank-$2$ spider categories. Fontaine, Kamnitzer, Kuperberg showed that the $mathfrak sl_3$ non-elliptic webs are dual to CAT(0) diskoids in the affine building. We show that each such dual diskoid is the intersection of the min-convex and max-convex hulls of a generic polygon in the building. Choosing a generic polygon from each of the components of the Satake fiber produces the duals of the non-elliptic web basis. The convex hulls in the affine building were first introduced by Faltings and are related to tropical convexity, as discussed in work by Joswig, Sturmfels, Yu and by Zhang.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"393 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76674219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-22DOI: 10.25537/dm.2022v27.489-518
S. Cacciatori, S. Noja, R. Re
The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of D-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frölicher) spectral sequence of supermanifolds with Kähler reduced manifold does not converge in general at page one.
{"title":"The universal de Rham / Spencer double complex on a supermanifold","authors":"S. Cacciatori, S. Noja, R. Re","doi":"10.25537/dm.2022v27.489-518","DOIUrl":"https://doi.org/10.25537/dm.2022v27.489-518","url":null,"abstract":"The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of D-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frölicher) spectral sequence of supermanifolds with Kähler reduced manifold does not converge in general at page one.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"116 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79456818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.
{"title":"On Unramified Brauer Groups of Torsors over Tori","authors":"E. Bayer-Fluckiger, R. Parimala","doi":"10.4171/dm/776","DOIUrl":"https://doi.org/10.4171/dm/776","url":null,"abstract":"In this paper we introduce a method to obtain algebraic information using arithmetic one in the study of tori and their principal homogeneous spaces. In particular, using some results of the authors with Tingyu Lee, we determine the unramified Brauer groups of some norm one tori, and their torsors.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"44 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82433921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a new description of logarithmic topological Andre-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to an etale descent formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. We also summarize and clarify analogous results relating notions of formal etaleness defined in terms of ordinary THH and TAQ.
{"title":"On the relationship between logarithmic TAQ and logarithmic THH","authors":"T. Lundemo","doi":"10.4171/dm/839","DOIUrl":"https://doi.org/10.4171/dm/839","url":null,"abstract":"We provide a new description of logarithmic topological Andre-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to an etale descent formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. We also summarize and clarify analogous results relating notions of formal etaleness defined in terms of ordinary THH and TAQ.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"39 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85899990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the $K$-theory of its groupoid $C^*$-algebra.
{"title":"Matui's AH conjecture for graph groupoids","authors":"P. Nyland, E. Ortega","doi":"10.4171/dm/853","DOIUrl":"https://doi.org/10.4171/dm/853","url":null,"abstract":"We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the $K$-theory of its groupoid $C^*$-algebra.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"56 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83887357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-14DOI: 10.25537/dm.2021v26.713-731
Tommi Brander, Jarkko Siltakoski
We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using a Muntz-Szasz theorem after reducing the problem to determining a function from its $L^p$-norms.
{"title":"Recovering a variable exponent","authors":"Tommi Brander, Jarkko Siltakoski","doi":"10.25537/dm.2021v26.713-731","DOIUrl":"https://doi.org/10.25537/dm.2021v26.713-731","url":null,"abstract":"We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using a Muntz-Szasz theorem after reducing the problem to determining a function from its $L^p$-norms.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"50 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73613669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove several results of concentration for eigenfunctions in Toeplitz quantization. With mild assumptions on the regularity, we prove that eigenfunctions are $O(exp(-cN^{delta}))$ away from the corresponding level set of the symbol, where N is the inverse semiclassical parameter and $0 < delta < 1$ depends on the regularity. As an application, we prove a precise bound for the free energy of spin systems at high temperatures, sharpening a result of Lieb.
{"title":"Fractional Exponential Decay in the Forbidden Region for Toeplitz Operators","authors":"Alix Deleporte","doi":"10.4171/dm/778","DOIUrl":"https://doi.org/10.4171/dm/778","url":null,"abstract":"We prove several results of concentration for eigenfunctions in Toeplitz quantization. With mild assumptions on the regularity, we prove that eigenfunctions are $O(exp(-cN^{delta}))$ away from the corresponding level set of the symbol, where N is the inverse semiclassical parameter and $0 < delta < 1$ depends on the regularity. As an application, we prove a precise bound for the free energy of spin systems at high temperatures, sharpening a result of Lieb.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"8 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89064000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-14DOI: 10.25537/DM.2020V25.483-525
A. Hochenegger, C. Meachan
Given an exact functor between triangulated categories which admits both adjoints and whose cotwist is either zero or an autoequivalence, we show how to associate a unique full triangulated subcategory of the codomain on which the functor becomes either Frobenius or spherical, respectively. We illustrate our construction with examples coming from projective bundles and smooth blowups. This work generalises results about spherical subcategories obtained by Martin Kalck, David Ploog and the first author.
在三角化范畴之间给出一个精确函子，该函子既允许伴随，且其cotwist为零或自等价，我们给出了如何关联上域上唯一的满三角化子范畴，该子范畴上的函子分别成为Frobenius或球形。我们用来自投影束和平滑膨胀的例子来说明我们的构造。本文推广了Martin Kalck, David Ploog和第一作者关于球面子范畴的结果。
{"title":"Frobenius and Spherical Codomains and Neighbourhoods","authors":"A. Hochenegger, C. Meachan","doi":"10.25537/DM.2020V25.483-525","DOIUrl":"https://doi.org/10.25537/DM.2020V25.483-525","url":null,"abstract":"Given an exact functor between triangulated categories which admits both adjoints and whose cotwist is either zero or an autoequivalence, we show how to associate a unique full triangulated subcategory of the codomain on which the functor becomes either Frobenius or spherical, respectively. We illustrate our construction with examples coming from projective bundles and smooth blowups. This work generalises results about spherical subcategories obtained by Martin Kalck, David Ploog and the first author.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"85 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73099922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Norm-Compatible Systems of Galois Cohomology Classes for $mathbf{GSp}_6$","authors":"Antonio Cauchi, Joaquín Rodrigues Jacinto","doi":"10.4171/dm/767","DOIUrl":"https://doi.org/10.4171/dm/767","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"33 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81104171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Comparison Theory of Distance Spheres along Geodesics","authors":"Reinhard Brocks","doi":"10.4171/dm/798","DOIUrl":"https://doi.org/10.4171/dm/798","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"8 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87526281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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