Pub Date : 2024-07-15DOI: 10.1215/00127094-2023-0048
Tudor Pădurariu, Yukinobu Toda
{"title":"Categorical and K-theoretic Donaldson–Thomas theory of C3 (part I)","authors":"Tudor Pădurariu, Yukinobu Toda","doi":"10.1215/00127094-2023-0048","DOIUrl":"https://doi.org/10.1215/00127094-2023-0048","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141832766","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 : 2024-07-15DOI: 10.1215/00127094-2023-0051
R. Friedman, R. Laza
. We prove that the higher direct images R q f ∗ Ω p Y /S of the sheaves of relative K¨ahler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have k -Du Bois local complete intersection singularities, for p ≤ k and all q ≥ 0, generalizing a result of Du Bois (the case k = 0). We then propose a definition of k -rational singularities extending the definition of rational singularities, and show that, if X is a k -rational variety with either isolated or local complete intersection singularities, then X is k -Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties. In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the k - rationality definition proposed here is equivalent to a previously given numerical definition for k - rational singularities. As an immediate consequence, it follows that for hypersurface singularities, k -Du Bois singularities are ( k − 1)-rational. Independently, we have proved that the latter statement also holds for isolated local complete intersection singularities, and conjecture that it holds more generally for all local complete intersection singularities.
.我们证明，在 p ≤ k 和所有 q ≥ 0 的情况下，相对 K ¨ahler 二项性的剪切的高直映像 R q f ∗ Ω p Y /S 是局部自由的，并且与任意基数变化的 flat 适当族相容，这些族的fibers 具有 k -Du Bois 局部完全交集奇点，这推广了 Du Bois 的一个结果（k = 0 的情况）。然后，我们提出了扩展有理奇点定义的 k 有理奇点定义，并证明如果 X 是具有孤立奇点或局部完全交点奇点的 k 有理综，那么 X 就是 k 杜波依斯。作为应用，我们讨论了族中霍奇数的行为和奇异卡拉比优（Calabi-Yau）变体的无碍性。在附录中，Morihiko Saito 证明了在超曲面奇点的情况下，这里提出的 k - 理性定义等同于之前给出的 k - 理性奇点的数值定义。因此，对于超曲面奇点，k -杜波依斯奇点是 ( k - 1)- 理性的。另外，我们还证明了后一种说法对于孤立的局部完全交点奇点也是成立的，并猜想这种说法对于所有局部完全交点奇点都是普遍成立的。
{"title":"Higher Du Bois and higher rational singularities","authors":"R. Friedman, R. Laza","doi":"10.1215/00127094-2023-0051","DOIUrl":"https://doi.org/10.1215/00127094-2023-0051","url":null,"abstract":". We prove that the higher direct images R q f ∗ Ω p Y /S of the sheaves of relative K¨ahler differentials are locally free and compatible with arbitrary base change for flat proper families whose fibers have k -Du Bois local complete intersection singularities, for p ≤ k and all q ≥ 0, generalizing a result of Du Bois (the case k = 0). We then propose a definition of k -rational singularities extending the definition of rational singularities, and show that, if X is a k -rational variety with either isolated or local complete intersection singularities, then X is k -Du Bois. As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties. In an appendix, Morihiko Saito proves that, in the case of hypersurface singularities, the k - rationality definition proposed here is equivalent to a previously given numerical definition for k - rational singularities. As an immediate consequence, it follows that for hypersurface singularities, k -Du Bois singularities are ( k − 1)-rational. Independently, we have proved that the latter statement also holds for isolated local complete intersection singularities, and conjecture that it holds more generally for all local complete intersection singularities.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141833597","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 : 2024-06-01DOI: 10.1215/00127094-2023-0038
Tao Li
{"title":"Taut foliations of 3-manifolds with Heegaard genus 2","authors":"Tao Li","doi":"10.1215/00127094-2023-0038","DOIUrl":"https://doi.org/10.1215/00127094-2023-0038","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141402413","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 : 2023-10-01DOI: 10.1215/00127094-2022-0090
Jonas Luhrmann, Wilhelm Schlag
We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable nonresonance property exhibited by the quadratic nonlinearity in the Klein–Gordon equation for the perturbation, and a variable coefficient quadratic normal form. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known nonintegrable models, for instance, to the asymptotic stability problem for the kink of the ϕ4 model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein–Gordon equations in one space dimension.
{"title":"Asymptotic stability of the sine-Gordon kink under odd perturbations","authors":"Jonas Luhrmann, Wilhelm Schlag","doi":"10.1215/00127094-2022-0090","DOIUrl":"https://doi.org/10.1215/00127094-2022-0090","url":null,"abstract":"We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable nonresonance property exhibited by the quadratic nonlinearity in the Klein–Gordon equation for the perturbation, and a variable coefficient quadratic normal form. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known nonintegrable models, for instance, to the asymptotic stability problem for the kink of the ϕ4 model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein–Gordon equations in one space dimension.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119571","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 : 2023-09-01DOI: 10.1215/00127094-2022-0080
Jessica Fintzen, Tasho Kaletha, Loren Spice
We give a modification of Yu’s construction of supercuspidal representations of a connected reductive group G over a non-Archimedean local field F. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proved to be false for the original construction. This modification is also an essential ingredient in the construction of supercuspidal L-packets in a preprint by the second author. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field F. In particular, for regular supercuspidal parameters, we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a recent conjecture by the second author, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the conditions on F). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.
{"title":"A twisted Yu construction, Harish-Chandra characters, and endoscopy","authors":"Jessica Fintzen, Tasho Kaletha, Loren Spice","doi":"10.1215/00127094-2022-0080","DOIUrl":"https://doi.org/10.1215/00127094-2022-0080","url":null,"abstract":"We give a modification of Yu’s construction of supercuspidal representations of a connected reductive group G over a non-Archimedean local field F. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proved to be false for the original construction. This modification is also an essential ingredient in the construction of supercuspidal L-packets in a preprint by the second author. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field F. In particular, for regular supercuspidal parameters, we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a recent conjecture by the second author, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the conditions on F). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135637920","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 : 2023-09-01DOI: 10.1215/00127094-2022-0082
Irving Dai, Jennifer Hom, Matthew Stoffregen, Linh Truong
We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a Z∞-subgroup and a Z-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of Hendricks, Manolescu, and Zemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.
{"title":"An infinite-rank summand of the homology cobordism group","authors":"Irving Dai, Jennifer Hom, Matthew Stoffregen, Linh Truong","doi":"10.1215/00127094-2022-0082","DOIUrl":"https://doi.org/10.1215/00127094-2022-0082","url":null,"abstract":"We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a Z∞-subgroup and a Z-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of Hendricks, Manolescu, and Zemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387806","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 : 2023-09-01DOI: 10.1215/00127094-2022-0081
Jens Marklof, Matthew Welsh
We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence μ2≡Dmodm, with D>0 square-free and D≢1mod4. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.
{"title":"Fine-scale distribution of roots of quadratic congruences","authors":"Jens Marklof, Matthew Welsh","doi":"10.1215/00127094-2022-0081","DOIUrl":"https://doi.org/10.1215/00127094-2022-0081","url":null,"abstract":"We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence μ2≡Dmodm, with D>0 square-free and D≢1mod4. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894822","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}
{"title":"Bessel models for real unitary groups: The tempered case","authors":"Hang Xue","doi":"10.1215/00127094-2022-0018","DOIUrl":"https://doi.org/10.1215/00127094-2022-0018","url":null,"abstract":". We prove the local Gan–Gross–Prasad conjecture for tempered L -packets of real unitary groups. The proof is based on theta lifts and is very simple.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42615128","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 : 2023-01-01DOI: 10.1215/00127094-2022-0088
Didier Bresch, Pierre-Emmanuel Jabin, Zhenfu Wang
This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the s{'e}minaire Laurent Schwartz proceeding [https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full proof of results announced by the authors in [C. R. Acad. Sciences Section Maths (2019)].
{"title":"Mean field limit and quantitative estimates with singular attractive kernels","authors":"Didier Bresch, Pierre-Emmanuel Jabin, Zhenfu Wang","doi":"10.1215/00127094-2022-0088","DOIUrl":"https://doi.org/10.1215/00127094-2022-0088","url":null,"abstract":"This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the s{'e}minaire Laurent Schwartz proceeding [https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full proof of results announced by the authors in [C. R. Acad. Sciences Section Maths (2019)].","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008444","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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