Pub Date : 2022-03-02DOI: 10.1515/crelle-2022-0092
F. Xie
Abstract We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. • Case 1: The quadric surface bundle has a smooth section. • Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Finally, we apply the results to certain complete intersections of quadrics.
{"title":"Residual categories of quadric surface bundles","authors":"F. Xie","doi":"10.1515/crelle-2022-0092","DOIUrl":"https://doi.org/10.1515/crelle-2022-0092","url":null,"abstract":"Abstract We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. • Case 1: The quadric surface bundle has a smooth section. • Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Finally, we apply the results to certain complete intersections of quadrics.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76927341","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 : 2022-03-01DOI: 10.1515/crelle-2022-frontmatter784
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2022-frontmatter784","DOIUrl":"https://doi.org/10.1515/crelle-2022-frontmatter784","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89237972","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 : 2022-03-01DOI: 10.1515/crelle-2022-0037
Marcin Bilski, J. Bochnak, W. Kucharz
Abstract Let f : X → ℝ {f:Xrightarrowmathbb{R}} be a function defined on a connected nonsingular real algebraic set X in ℝ n {mathbb{R}^{n}} . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim X ≥ 2 {operatorname{dim}Xgeq 2} and k is a positive integer, then f is a regular function whenever the restriction f | C {f|_{C}} is a regular function for every algebraic curve C in X that is a 𝒞 k {mathcal{C}^{k}} submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q {x^{p}=y^{q}} for some primes p < q {p
摘要设f:X→{f:Xrightarrowmathbb{R}}是定义在连通的非奇异实代数集合X上的函数，该函数定义在一个连通的非奇异实代数集合X上，并定义在∈n {mathbb{R} ^{n}}上。我们证明了f的正则性可以通过控制f对X中的代数曲线或代数曲面的限制来检测。如果dim (X)≥2 {operatorname{dim} X geq 2}且k是正整数，则只要限制f| C f|{＿C{是X中与单位圆同态的 k }}{mathcal{C} ^{k}}子流形的每一个代数曲线C的正则函数，且该曲线非奇异或恰好有一个奇异，f就是正则函数。在后一种情况下，对于{某些素数p
{"title":"Hartogs-type theorems in real algebraic geometry, I","authors":"Marcin Bilski, J. Bochnak, W. Kucharz","doi":"10.1515/crelle-2022-0037","DOIUrl":"https://doi.org/10.1515/crelle-2022-0037","url":null,"abstract":"Abstract Let f : X → ℝ {f:Xrightarrowmathbb{R}} be a function defined on a connected nonsingular real algebraic set X in ℝ n {mathbb{R}^{n}} . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim X ≥ 2 {operatorname{dim}Xgeq 2} and k is a positive integer, then f is a regular function whenever the restriction f | C {f|_{C}} is a regular function for every algebraic curve C in X that is a 𝒞 k {mathcal{C}^{k}} submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q {x^{p}=y^{q}} for some primes p < q {p<q} . If dim X ≥ 3 {operatorname{dim}Xgeq 3} , then f is a regular function whenever the restriction f | S {f|_{S}} is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89549024","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 : 2022-02-25DOI: 10.1515/crelle-2022-0007
Lan-Hsuan Huang, Dan A. Lee, C. Sormani
Abstract There is an error in the proof of Theorem 1.3 of the original article. Despite the problem, it is rigorously proved in joint work of the first two authors and Perales that Theorem 1.3 is true, using recent results of Allen and Perales that extend the work of Allen, Perales, and Sormani.
{"title":"Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)","authors":"Lan-Hsuan Huang, Dan A. Lee, C. Sormani","doi":"10.1515/crelle-2022-0007","DOIUrl":"https://doi.org/10.1515/crelle-2022-0007","url":null,"abstract":"Abstract There is an error in the proof of Theorem 1.3 of the original article. Despite the problem, it is rigorously proved in joint work of the first two authors and Perales that Theorem 1.3 is true, using recent results of Allen and Perales that extend the work of Allen, Perales, and Sormani.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83807559","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 : 2022-02-17DOI: 10.1515/crelle-2022-0073
John Man-shun Ma, A. Muhammad, Niels Moller
Abstract In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {mathbb{R}^{n+1}} . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.
{"title":"Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry","authors":"John Man-shun Ma, A. Muhammad, Niels Moller","doi":"10.1515/crelle-2022-0073","DOIUrl":"https://doi.org/10.1515/crelle-2022-0073","url":null,"abstract":"Abstract In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {mathbb{R}^{n+1}} . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91271601","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 : 2022-02-15DOI: 10.1515/crelle-2021-0089
Robert Cass
Abstract For a reductive group over an algebraically closed field of characteristic p>0{p>0} we construct the abelian category of perverse 𝔽p{mathbb{F}_{p}}-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F-regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽p{mathbb{F}_{p}}-sheaves.
{"title":"Perverse 𝔽p-sheaves on the affine Grassmannian","authors":"Robert Cass","doi":"10.1515/crelle-2021-0089","DOIUrl":"https://doi.org/10.1515/crelle-2021-0089","url":null,"abstract":"Abstract For a reductive group over an algebraically closed field of characteristic p>0{p>0} we construct the abelian category of perverse 𝔽p{mathbb{F}_{p}}-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F-regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽p{mathbb{F}_{p}}-sheaves.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76907017","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 : 2022-02-08DOI: 10.1515/crelle-2023-0021
Arend Bayer, Alexander Perry
Abstract We settle the last open case of Kuznetsov’s conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel–Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel–Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.
{"title":"Kuznetsov’s Fano threefold conjecture via K3 categories and enhanced group actions","authors":"Arend Bayer, Alexander Perry","doi":"10.1515/crelle-2023-0021","DOIUrl":"https://doi.org/10.1515/crelle-2023-0021","url":null,"abstract":"Abstract We settle the last open case of Kuznetsov’s conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel–Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories. Along the way, we obtain a categorical description of the periods of Gushel–Mukai varieties, which we use to resolve a conjecture of Kuznetsov and the second author on the birational categorical Torelli problem, as well as to give a simple proof of a theorem of Debarre and Kuznetsov on the fibers of the period map. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89755379","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 : 2022-02-01DOI: 10.1515/crelle-2023-0010
C. Scarpa
Abstract We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.
{"title":"Scalar curvature and deformations of complex structures","authors":"C. Scarpa","doi":"10.1515/crelle-2023-0010","DOIUrl":"https://doi.org/10.1515/crelle-2023-0010","url":null,"abstract":"Abstract We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kähler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kähler reduction, which is a hyperkähler reduction for a particular choice of the spectral function. The main tool for studying the system is a flat connection on the space of first-order deformations of the complex structure, that allows to obtain a formal complexification of the moment map equations. Using this connection, we describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90765136","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 : 2022-02-01DOI: 10.1515/crelle-2022-frontmatter783
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2022-frontmatter783","DOIUrl":"https://doi.org/10.1515/crelle-2022-frontmatter783","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81732109","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 : 2022-01-30DOI: 10.1515/crelle-2023-0033
G. Dospinescu, Vytautas Paškūnas, Benjamin Schraen
Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.
{"title":"Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence","authors":"G. Dospinescu, Vytautas Paškūnas, Benjamin Schraen","doi":"10.1515/crelle-2023-0033","DOIUrl":"https://doi.org/10.1515/crelle-2023-0033","url":null,"abstract":"Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75767282","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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