Donaldson-Thomas (abbreviated as DT) theory is a sheaf theoretic technique of enumerating curves on a Calabi-Yau threefold. Classical DT invariants give a virtual count of Gieseker stable sheaves provided that no strictly semistable sheaves exist. This assumption was later lifted by the work of Joyce and Song who defined generalized DT invariants using Hall algebras and the Behrend function, their method being motivic in nature. In this talk, we will present a new approach towards generalized DT theory, obtaining an invariant as the degree of a virtual cycle inside a Deligne-Mumford stack. The main components are an adaptation of Kirwans partial desingularization procedure and recent results on the structure of moduli of sheaves on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem and Jun Li. Special Note: Pre-talk at 1:30P. Host: James McKernan Friday, September 28, 2018 2:00 PM AP&M 5829 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
唐纳森-托马斯（简称 DT）理论是一种枚举 Calabi-Yau 三折上曲线的剪子理论技术。经典的 DT 变量给出了 Gieseker 稳定剪切的虚拟计数，前提是不存在严格半稳态的剪切。乔伊斯和宋后来利用霍尔代数和贝伦德函数定义了广义的 DT 变量，从本质上讲，他们的方法是动机式的。在本讲座中，我们将介绍一种实现广义 DT 理论的新方法，即通过德利尼-芒福德堆栈内部虚拟循环的度数获得不变式。其主要组成部分是对 Kirwans 部分去奇化过程的改编，以及关于 Calabi-Yau 三折上剪子的模结构的最新成果。基于与 Young-Hoon Kiem 和 Jun Li 的合作成果。特别提示：下午1:30预讲。主持人： James McKernan
{"title":"Generalized Donaldson–Thomas invariants via Kirwan blowups","authors":"Jun Li, Y. Kiem, M. Savvas","doi":"10.4310/jdg/1721071499","DOIUrl":"https://doi.org/10.4310/jdg/1721071499","url":null,"abstract":"Donaldson-Thomas (abbreviated as DT) theory is a sheaf theoretic technique of enumerating curves on a Calabi-Yau threefold. Classical DT invariants give a virtual count of Gieseker stable sheaves provided that no strictly semistable sheaves exist. This assumption was later lifted by the work of Joyce and Song who defined generalized DT invariants using Hall algebras and the Behrend function, their method being motivic in nature. In this talk, we will present a new approach towards generalized DT theory, obtaining an invariant as the degree of a virtual cycle inside a Deligne-Mumford stack. The main components are an adaptation of Kirwans partial desingularization procedure and recent results on the structure of moduli of sheaves on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem and Jun Li. Special Note: Pre-talk at 1:30P. Host: James McKernan Friday, September 28, 2018 2:00 PM AP&M 5829 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141713159","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":"Green's functions and complex Monge–Ampère equations","authors":"Bin Guo, Duong H. Phong, Jacob Sturm","doi":"10.4310/jdg/1721071497","DOIUrl":"https://doi.org/10.4310/jdg/1721071497","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141705546","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":"Sharp existence, symmetry and asymptotics results for the singular $SU(3)$ Toda system with critical parameters","authors":"Zhijie Chen, Chang-Shou Lin","doi":"10.4310/jdg/1721071493","DOIUrl":"https://doi.org/10.4310/jdg/1721071493","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141716251","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":"From Seiberg-Witten to Gromov: MCE and singular symplectic forms","authors":"Yi-Jen Lee","doi":"10.4310/jdg/1717772424","DOIUrl":"https://doi.org/10.4310/jdg/1717772424","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"141410231","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":"Intersection de Rham complexes in positive characteristic","authors":"Mao Sheng, Zebao Zhang","doi":"10.4310/jdg/1717772421","DOIUrl":"https://doi.org/10.4310/jdg/1717772421","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"141395650","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}
Sagun Chanillo, A. Logunov, E. Malinnikova, D. Mangoubi
{"title":"Local version of Courant’s nodal domain theorem","authors":"Sagun Chanillo, A. Logunov, E. Malinnikova, D. Mangoubi","doi":"10.4310/jdg/1707767334","DOIUrl":"https://doi.org/10.4310/jdg/1707767334","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140525585","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}
It is shown that any affine toric variety $Y$, which is $mathbb{Q}$-Gorenstein, admits a conical Ricci flat Kähler metric, which is smooth on the regular locus of $Y$. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of $Y$. The case when the vertex point of $Y$ is an isolated singularity was previously shown by Futaki–Ono–Wang. The proof is based on an existence result for the inhomogeneous Monge–Ampère equation in $mathbb{R}^m$ with exponential right hand side and with prescribed target given by a proper convex cone, combined with transversal a priori estimates on $Y$.
{"title":"Conical Calabi–Yau metrics on toric affine varieties and convex cones","authors":"Robert J. Berman","doi":"10.4310/jdg/1696432924","DOIUrl":"https://doi.org/10.4310/jdg/1696432924","url":null,"abstract":"It is shown that any affine toric variety $Y$, which is $mathbb{Q}$-Gorenstein, admits a conical Ricci flat Kähler metric, which is smooth on the regular locus of $Y$. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of $Y$. The case when the vertex point of $Y$ is an isolated singularity was previously shown by Futaki–Ono–Wang. The proof is based on an existence result for the inhomogeneous Monge–Ampère equation in $mathbb{R}^m$ with exponential right hand side and with prescribed target given by a proper convex cone, combined with transversal a priori estimates on $Y$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135948526","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}
We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We establish a formula for the Chern character of their index.
{"title":"The index formula for families of Dirac type operators on pseudomanifolds","authors":"Pierre Albin, Jesse Gell-Redman","doi":"10.4310/jdg/1696432923","DOIUrl":"https://doi.org/10.4310/jdg/1696432923","url":null,"abstract":"We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We establish a formula for the Chern character of their index.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135948774","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}
Min-max theory for constant mean curvature (CMC) hypersurfaces has been developed by Zhou–Zhu $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$ and Zhou $[href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$. In particular, in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$, Zhou and Zhu proved that for any $c gt 0$, every closed Riemannian manifold $(M^{n+1}, g), 3 leq n + 1 leq 7$, contains a closed $c$-CMC hypersurface. In this article we will show that the min-max theory for CMC hypersurfaces in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}, href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$ can be extended in higher dimensions using the regularity theory of stable CMC hypersurfaces, developed by Bellettini–Wickramasekera $[href{https://doi.org/10.48550/arXiv.1802.00377}{4}, href{https://doi.org/10.48550/arXiv.1902.09669}{5}]$ and Bellettini–Chodosh–Wickramasekera $[href{https://doi.org/10.1016/j.aim.2019.05.023}{3}]$. Furthermore, we will prove that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g), n+1 geq 3$, tends to infinity as $c to 0^+$. More quantitatively, there exists a constant $gamma_0$, depending on $g$, such that for all $c gt 0$, there exist at least $gamma_0 c^{-frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$.
{"title":"Existence of multiple closed CMC hypersurfaces with small mean curvature","authors":"Akashdeep Dey","doi":"10.4310/jdg/1696432925","DOIUrl":"https://doi.org/10.4310/jdg/1696432925","url":null,"abstract":"Min-max theory for constant mean curvature (CMC) hypersurfaces has been developed by Zhou–Zhu $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$ and Zhou $[href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$. In particular, in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$, Zhou and Zhu proved that for any $c gt 0$, every closed Riemannian manifold $(M^{n+1}, g), 3 leq n + 1 leq 7$, contains a closed $c$-CMC hypersurface. In this article we will show that the min-max theory for CMC hypersurfaces in $[href{https://doi.org/10.1007/s00222-019-00886-1}{39}, href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$ can be extended in higher dimensions using the regularity theory of stable CMC hypersurfaces, developed by Bellettini–Wickramasekera $[href{https://doi.org/10.48550/arXiv.1802.00377}{4}, href{https://doi.org/10.48550/arXiv.1902.09669}{5}]$ and Bellettini–Chodosh–Wickramasekera $[href{https://doi.org/10.1016/j.aim.2019.05.023}{3}]$. Furthermore, we will prove that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g), n+1 geq 3$, tends to infinity as $c to 0^+$. More quantitatively, there exists a constant $gamma_0$, depending on $g$, such that for all $c gt 0$, there exist at least $gamma_0 c^{-frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135948783","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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