{"title":"Equivariant Functors and Sheaves","authors":"G. Vooys","doi":"10.11575/PRISM/39088","DOIUrl":"https://doi.org/10.11575/PRISM/39088","url":null,"abstract":"In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group $G$ acting on a variety $X$ are in fact equivalent. In the first part of this thesis we introduce and study equivariant categories on a quasi-projective variety $X$. These are a generalization of the equivariant derived category of Lusztig and are indexed by certain pseudofunctors that take values in the 2-category of categories. This 2-categorical generalization allow us to prove rigorously and carefully when such categories are additive, monoidal, triangulated, admit $t$-structures, among and more. We also define equivariant functors and natural transformations before using these to prove how to lift adjoints to the equivariant setting. We also give a careful foundation of how to manipulate $t$-structures on these equivariant categories for future use and with an eye towards future applications. In the final part of this thesis we prove a four-way equivalence between the different formulations of the equivariant derived category of $ell$-adic sheaves on a quasi-projective variety $X$. We show that the equivariant derived category of Lusztig is equivalent to the equivariant derived category of Bernstein-Lunts and the simplicial equivariant derived category. We then show that these equivariant derived categories are equivalent to the derived $ell$-adic category of Behrend on the algebraic stack $[G backslash X]$. We also provide an isomorphism of the simplicial equivariant derived category on the variety $X$ with the simplicial equivariant derived category on the simplicial presentation of $[G backslash X]$, as well as prove explicit equivalences between the categories of equivariant $ell$-adic sheaves, local systems, and perverse sheaves with the classical incarnations of such categories of equivariant sheaves.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129500295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Elliptic stable envelopes and 3d mirror symmetry","authors":"I. Kononov","doi":"10.7916/D8-F44Y-E433","DOIUrl":"https://doi.org/10.7916/D8-F44Y-E433","url":null,"abstract":"In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. \u0000We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety X^G of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. \u0000We formalize the notion of symplectic duality, also known as 3-dimensional mirror symmetry. We obtain a factorization theorem about the limit of elliptic stable envelopes to a wall, which generalizes the result of M.Aganagic and A.Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices etc., to actions on the K-theory of the symplectic dual variety. In the case of the Hilbert scheme of points in plane, our results imply the conjectures of E.Gorsky and A.Negut. \u0000We propose a new approach to K-theoretic quantum difference equations.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133321750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Noncommutative Riemann hypothesis","authors":"Gonçalo Tabuada","doi":"10.1090/proc/15874","DOIUrl":"https://doi.org/10.1090/proc/15874","url":null,"abstract":"In this note, making use of noncommutative $l$-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first application, we prove that the generalized Riemann hypothesis is invariant under derived equivalences and homological projective duality. As a second application, we prove the noncommutative generalized Riemann hypothesis in some new cases.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121607142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linear automorphisms of smooth hypersurfaces giving Galois points","authors":"Taro Hayashi","doi":"10.4134/BKMS.B200428","DOIUrl":"https://doi.org/10.4134/BKMS.B200428","url":null,"abstract":"Let $X$ be a smooth hypersurface $X$ of degree $dgeq4$ in a projective space $mathbb P^{n+1}$. We consider a projection of $X$ from $pinmathbb P^{n+1}$ to a plane $Hcongmathbb P^n$. This projection induces an extension of function fields $mathbb C(X)/mathbb C(mathbb P^n)$. The point $p$ is called a Galois point if the extension is Galois. In this paper, we will give a necessary and sufficient conditions for $X$ to have Galois points by using linear automorphisms.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121389392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Maximum Likelihood Estimation from a Tropical and a Bernstein--Sato Perspective","authors":"Anna-Laura Sattelberger, Robin van der Veer","doi":"10.1093/imrn/rnac016","DOIUrl":"https://doi.org/10.1093/imrn/rnac016","url":null,"abstract":"In this article, we investigate Maximum Likelihood Estimation with tools from Tropical Geometry and Bernstein--Sato theory. We investigate the critical points of very affine varieties and study their asymptotic behavior. We relate these asymptotics to particular rays in the tropical variety as well as to Bernstein--Sato ideals and give a connection to Maximum Likelihood Estimation in Statistics.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116916459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"LG/CY Correspondence Between $tt^∗$ bGeometries","authors":"Huijun Fan, T. Lan, Zong-Xin Yang","doi":"10.4208/cmr.2020-0050","DOIUrl":"https://doi.org/10.4208/cmr.2020-0050","url":null,"abstract":"The concept of $tt^*$ geometric structure was introduced by physicists (see cite{CV1, BCOV} and references therein) , and then studied firstly in mathematics by C. Hertling cite{Het1}. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^*$ geometry and obtain the following result. Let $finmathbb{C}[z_0, dots, z_{n+2}]$ be a nondegenerate homogeneous polynomialof degree $n+2$, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity $(mathbb{C}^{n+2}, f)$, both can be written as a $tt^*$ structure. We proved that there exists a $tt^*$ substructure on Landau-Ginzburg side, which should correspond to the $tt^*$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $tt^*$ geometries between these two models except the isomorphism between real structures.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129659370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abeer Al Ahmadieh, Mario Kummer, Miruna-Stefana Sorea
{"title":"A generalization of the space of complete quadrics","authors":"Abeer Al Ahmadieh, Mario Kummer, Miruna-Stefana Sorea","doi":"10.4418/2021.76.2.9","DOIUrl":"https://doi.org/10.4418/2021.76.2.9","url":null,"abstract":"To any homogeneous polynomial $h$ we naturally associate a variety $Omega_h$ which maps birationally onto the graph $Gamma_h$ of the gradient map $nabla h$ and which agrees with the space of complete quadrics when $h$ is the determinant of the generic symmetric matrix. We give a sufficient criterion for $Omega_h$ being smooth which applies for example when $h$ is an elementary symmetric polynomial. In this case $Omega_h$ is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when $Omega_h$ is not smooth.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128833055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mirror Symmetry for a Cusp Polynomial Landau–Ginzburg Orbifold","authors":"A. Basalaev, A. Takahashi","doi":"10.1093/IMRN/RNAB145","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB145","url":null,"abstract":"For any triple of positive integers $A' = (a_1',a_2',a_3')$ and $c in mathbb{C}^*$, cusp polynomial $f_{A'} = x_1^{a_1'}+x_2^{a_2'}+x_3^{a_3'}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle-Lenzing orbifold projective line $mathbb{P}^1_{a_1',a_2',a_3'}$. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial $f_{A'}$, turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of $mathbb{P}^1_{a_1',a_2',a_3'}$. \u0000In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$ - a symmetry group of a cusp polynomial $f_{A'}$, we introduce the Frobenius manifold of a pair $(f_{A'},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov-Witten theory of Geigle-Lenzing weighted projective line $mathbb{P}^1_{A,Lambda}$, indexed by another set $A$ and $Lambda$, distinct points on $mathbb{C}setminus{0,1}$. \u0000For some special values of $A'$ with the special choice of $G$ it happens that $mathbb{P}^1_{A'} cong mathbb{P}^1_{A,Lambda}$. Combining our mirror symmetry isomorphism for the pair $(A,Lambda)$, together with the ``usual'' one for $A'$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta-function.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"52 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125776548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hodge-Newton filtration for $p$-divisible groups with ramified endomorphism structure.","authors":"Andrea Marrama","doi":"10.25537/dm.2022v27.1805-1863","DOIUrl":"https://doi.org/10.25537/dm.2022v27.1805-1863","url":null,"abstract":"Let $mathcal{O}_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with perfect residue field. We prove the existence of the Hodge-Newton filtration for $p$-divisible groups over $mathcal{O}_K$ with additional endomorphism structure for the ring of integers of a finite, possibly ramified field extension of $mathbb{Q}_p$. The argument is based on the Harder-Narasimhan theory for finite flat group schemes over $mathcal{O}_K$. In particular, we describe a sufficient condition for the existence of a filtration of $p$-divisible groups over $mathcal{O}_K$ associated to a break point of the Harder-Narasimhan polygon.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130740627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Virtual classes for hypersurfaces via\u0000 two-periodic complexes","authors":"M. Shoemaker","doi":"10.1090/CONM/763/15326","DOIUrl":"https://doi.org/10.1090/CONM/763/15326","url":null,"abstract":"These expository notes are based on a series of lectures given at the May 2018 Snowbird workshop, Crossing the Walls in Enumerative Geometry. We give an introductory treatment of the notion of a virtual fundamental class in algebraic geometry, and describe a new construction of the virtual fundamental class for Gromov-Witten theory of a hypersurface. The results presented here are based on joint work with I. Ciocan-Fontanine, D. Favero, J. Gu'er'e, and B. Kim.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128990196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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