Pub Date : 2020-09-11DOI: 10.1215/00127094-2022-0059
Jeongseok Oh, Richard P. Thomas
Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.
{"title":"Counting sheaves on Calabi–Yau 4-folds, I","authors":"Jeongseok Oh, Richard P. Thomas","doi":"10.1215/00127094-2022-0059","DOIUrl":"https://doi.org/10.1215/00127094-2022-0059","url":null,"abstract":"Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46681636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-21DOI: 10.1215/00127094-2022-0042
Lin Chen
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(!(t)!)$-equivariant and $U^-(!(t)!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and arXiv:1607.00586. We also generalize our results to the Beilinson-Drinfeld Grassmannian $Gr_{G,X^I}$ and to the affine flag variety $Fl_G$.
{"title":"Nearby cycles on Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and long intertwining functor","authors":"Lin Chen","doi":"10.1215/00127094-2022-0042","DOIUrl":"https://doi.org/10.1215/00127094-2022-0042","url":null,"abstract":"Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(!(t)!)$-equivariant and $U^-(!(t)!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and arXiv:1607.00586. We also generalize our results to the Beilinson-Drinfeld Grassmannian $Gr_{G,X^I}$ and to the affine flag variety $Fl_G$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47423413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-19DOI: 10.1215/00127094-2023-0007
Corey Bregman, Ruth Charney, K. Vogtmann
For any right-angled Artin group $A_{Gamma}$ we construct a finite-dimensional space $mathcal{O}_{Gamma}$ on which the group $text{Out}(A_{Gamma})$ of outer automorphisms of $A_{Gamma}$ acts properly. We prove that $mathcal{O}_{Gamma}$ is contractible, so that the quotient is a rational classifying space for $text{Out}(A_{Gamma})$. The space $mathcal{O}_{Gamma}$ blends features of the symmetric space of lattices in $mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $mathcal{O}_{Gamma}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{Gamma}$.
{"title":"Outer space for RAAGs","authors":"Corey Bregman, Ruth Charney, K. Vogtmann","doi":"10.1215/00127094-2023-0007","DOIUrl":"https://doi.org/10.1215/00127094-2023-0007","url":null,"abstract":"For any right-angled Artin group $A_{Gamma}$ we construct a finite-dimensional space $mathcal{O}_{Gamma}$ on which the group $text{Out}(A_{Gamma})$ of outer automorphisms of $A_{Gamma}$ acts properly. We prove that $mathcal{O}_{Gamma}$ is contractible, so that the quotient is a rational classifying space for $text{Out}(A_{Gamma})$. The space $mathcal{O}_{Gamma}$ blends features of the symmetric space of lattices in $mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $mathcal{O}_{Gamma}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{Gamma}$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43758456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-15DOI: 10.1215/00127094-2019-0083
M. Drmota, C. Mauduit, J. Rivat
If q1 and q2 are two coprime bases, f (resp. g) a strongly q1-multiplicative (resp. strongly q2-multiplicative) function of modulus 1 and θ a real number, we estimate the sums ∑ n≤x Λ(n)f(n)g(n) exp(2iπθn) (and ∑ n≤x μ(n)f(n)g(n) exp(2iπθn)), where Λ denotes the von Mangoldt function (and μ the Möbius function). The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation and combinatorial arguments. We deduce from these estimates a Prime Number Theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.
{"title":"Prime numbers in two bases","authors":"M. Drmota, C. Mauduit, J. Rivat","doi":"10.1215/00127094-2019-0083","DOIUrl":"https://doi.org/10.1215/00127094-2019-0083","url":null,"abstract":"If q1 and q2 are two coprime bases, f (resp. g) a strongly q1-multiplicative (resp. strongly q2-multiplicative) function of modulus 1 and θ a real number, we estimate the sums ∑ n≤x Λ(n)f(n)g(n) exp(2iπθn) (and ∑ n≤x μ(n)f(n)g(n) exp(2iπθn)), where Λ denotes the von Mangoldt function (and μ the Möbius function). The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation and combinatorial arguments. We deduce from these estimates a Prime Number Theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":"169 1","pages":"1809-1876"},"PeriodicalIF":2.5,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47289272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-15DOI: 10.1215/00127094-2022-0011
S. Lynch
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
{"title":"Convexity estimates for hypersurfaces moving by concave curvature functions","authors":"S. Lynch","doi":"10.1215/00127094-2022-0011","DOIUrl":"https://doi.org/10.1215/00127094-2022-0011","url":null,"abstract":"We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42043778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-18DOI: 10.1215/00127094-2022-0076
Tsao-Hsien Chen, D. Nadler
We construct a stratified homeomorphism between the space of $ntimes n$ real matrices with real eigenvalues and the space of $ntimes n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $GL_n(mathbb R)$-adjoint orbits and $O_n(mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kahler quotients of linear spaces. We discuss applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.
{"title":"Real and symmetric matrices","authors":"Tsao-Hsien Chen, D. Nadler","doi":"10.1215/00127094-2022-0076","DOIUrl":"https://doi.org/10.1215/00127094-2022-0076","url":null,"abstract":"We construct a stratified homeomorphism between the space of $ntimes n$ real matrices with real eigenvalues and the space of $ntimes n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $GL_n(mathbb R)$-adjoint orbits and $O_n(mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kahler quotients of linear spaces. We discuss applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46569461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-15DOI: 10.1215/00127094-2019-0075
V. Pilloni
— We investigate the p-adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on the Siegel threefolds. 2000 Mathematics Subject Classification: 11F33, 11G18, 14G35
{"title":"Higher coherent cohomology and p -adic modular forms of singular weights","authors":"V. Pilloni","doi":"10.1215/00127094-2019-0075","DOIUrl":"https://doi.org/10.1215/00127094-2019-0075","url":null,"abstract":"— We investigate the p-adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on the Siegel threefolds. 2000 Mathematics Subject Classification: 11F33, 11G18, 14G35","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48500603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.1215/00127094-2022-0036
R. Kenyon, I. Prause
We prove a new integrability principle for gradient variational problems in $mathbb{R}^2$, showing that solutions are explicitly parameterized by $kappa$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $kappa$, where $kappa$ is the square root of the Hessian determinant of the surface tension.
{"title":"Gradient variational problems in R2","authors":"R. Kenyon, I. Prause","doi":"10.1215/00127094-2022-0036","DOIUrl":"https://doi.org/10.1215/00127094-2022-0036","url":null,"abstract":"We prove a new integrability principle for gradient variational problems in $mathbb{R}^2$, showing that solutions are explicitly parameterized by $kappa$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $kappa$, where $kappa$ is the square root of the Hessian determinant of the surface tension.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47543479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-28DOI: 10.1215/00127094-2022-0046
R. Oliver, F. Thorne
Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (log n)^2})$ for an explicit constant $c$.
{"title":"Upper bounds on number fields of given degree and bounded discriminant","authors":"R. Oliver, F. Thorne","doi":"10.1215/00127094-2022-0046","DOIUrl":"https://doi.org/10.1215/00127094-2022-0046","url":null,"abstract":"Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (log n)^2})$ for an explicit constant $c$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46491984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-19DOI: 10.1215/00127094-2022-0048
A. Sah
We improve the upper bound for diagonal Ramsey numbers to [R(k+1,k+1)leexp(-c(log k)^2)binom{2k}{k}] for $kge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.
{"title":"Diagonal Ramsey via effective quasirandomness","authors":"A. Sah","doi":"10.1215/00127094-2022-0048","DOIUrl":"https://doi.org/10.1215/00127094-2022-0048","url":null,"abstract":"We improve the upper bound for diagonal Ramsey numbers to [R(k+1,k+1)leexp(-c(log k)^2)binom{2k}{k}] for $kge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal \"effective quasirandomness\" results about convergence of graphs. This optimality represents a natural barrier to improvement.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47237326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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