Journal of Differential Geometry最新文献

Higher order Jacobian matrix theory and invariants of singularities 高阶雅可比矩阵理论与奇点不变量

IF 2.5 1区数学

Journal of Differential Geometry Pub Date : 2026-10-01 DOI: 10.4310/jdg/1789484463

Shuanghe Fan,Stephen S.-T. Yau,Huaiqing Zuo

{"title":"Higher order Jacobian matrix theory and invariants of singularities","authors":"Shuanghe Fan,Stephen S.-T. Yau,Huaiqing Zuo","doi":"10.4310/jdg/1789484463","DOIUrl":"https://doi.org/10.4310/jdg/1789484463","url":null,"abstract":"The higher order Nash blow-up of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. Inspired by the higher order Nash blow-up, we develop higher order Jacobian matrix theory, a generalization of the Jacobian matrix theory in the usual sense. The most significant contribution of this work is the higher order generalization of similarity transformation. It bridges the two long-standing, distinct research directions in the study of higher-order Jacobian matrices: the one from the higher order Nash blowup of a hypersurface, and the one representing a higher-order tangent map of a morphism. As an application, we use it to construct different contact and right invariants of isolated singularities, generalizing the Tjurina algebra and the Milnor algebra in singularity theory respectively. It is worth mentioning that the Tjurina algebra and the Milnor algebra were constructed using first-order derivatives, but now we are constructing invariants using arbitrary higher-order derivatives. The higher order Jacobian matrix theory also has potential applications and inspirations in multiple branches of mathematics: inspiring the establishment of higher order Hessian matrix theory and projective invariants, calculating explicit expression for the inverse of an automorphism of a power series ring, and performing explicit calculations in finite determinacy.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"154 1","pages":"217-268"},"PeriodicalIF":2.5,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021099","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}

引用次数: 0

Involutions and the Chern-Simons filtration in instanton Floer homology 瞬子花同调中的对合和chen - simons滤波

IF 2.5 1区数学

Journal of Differential Geometry Pub Date : 2026-10-01 DOI: 10.4310/jdg/1789483138

Antonio Alfieri,Irving Dai,Abhishek Mallick,Masaki Taniguchi

引用次数: 0

A new Poincaré type rigidity phenomenon with applications 一种新的庞卡罗式刚性现象及其应用

IF 2.5 1区数学

Journal of Differential Geometry Pub Date : 2026-09-01 DOI: 10.4310/jdg/1788448633

Ming Xiao

{"title":"A new Poincaré type rigidity phenomenon with applications","authors":"Ming Xiao","doi":"10.4310/jdg/1788448633","DOIUrl":"https://doi.org/10.4310/jdg/1788448633","url":null,"abstract":"We discover a new Poincaré type phenomenon by establishing an optimal rigidity theorem for local CR (Cauchy-Riemann) mappings between circle bundles that are defined in a canonical way over (possibly reducible) bounded symmetric domains. We prove such a local CR map, if nonconstant, must extend to a rational biholomorphism between the corresponding disk bundles. The result includes as a special case the classical Poincaré-Tanaka-Alexander theorem. Among other applications, we show, for two irreducible bounded symmetric domains with rank at least two, a local CR diffeomorphism between (open connected pieces of) their anti-canonical circle bundles extends to a norm-preserving holomorphic isomorphism between their anti-canonical bundles. The statement fails in the rank one case. As another application, we construct, for any $n geq 2$, a countably infinite family of compact locally homogeneous strongly pseudoconvex CR hypersurfaces (in complex manifolds) of real dimension $2 n+1$ with transverse symmetry such that they are all obstruction flat and Bergman logarithmically flat. Moreover, their local CR structures are mutually inequivalent. Such a family cannot exist in dimension three by known results: A Bergman logarithmically flat CR hypersurface must be spherical, and so is a compact obstruction flat CR hypersurface with transverse symmetry.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"379 1","pages":"97-135"},"PeriodicalIF":2.5,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148894744","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}

引用次数: 0

Realization of tropical curves in Abelian surfaces 热带曲线在阿贝尔曲面中的实现

IF 2.5 1区数学

Journal of Differential Geometry Pub Date : 2026-09-01 DOI: 10.4310/jdg/1788448114

Takeo Nishinou

引用次数: 0

Conical Calabi–Yau metrics on toric affine varieties and convex cones 环仿射变异和凸锥上的圆锥Calabi-Yau度量

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432924

Robert J. Berman

引用次数: 6

The index formula for families of Dirac type operators on pseudomanifolds 伪流形上狄拉克型算子族的指数公式

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432923

Pierre Albin, Jesse Gell-Redman

引用次数: 13

Existence of multiple closed CMC hypersurfaces with small mean curvature 具有小平均曲率的多个闭合CMC超曲面的存在性

1区数学

Journal of Differential Geometry Pub Date : 2023-10-01 DOI: 10.4310/jdg/1696432925

Akashdeep Dey

{"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":"26 1","pages":"0"},"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}

引用次数: 9

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties 退化Calabi-Yau变型附近Maurer-Cartan方程的几何性质

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236591

Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma

{"title":"Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties","authors":"Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma","doi":"10.4310/jdg/1695236591","DOIUrl":"https://doi.org/10.4310/jdg/1695236591","url":null,"abstract":"Given a degenerate Calabi–Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin–Vilkovisky algebra $PV^{ast,ast}(X)$, producing a singular version of the extended Kodaira–Spencer differential graded Lie algebra in the Calabi–Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov–Tian–Todorov–type unobstructedness theorem for smoothing of singular Calabi–Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi–Yau varieties (as studied by Friedman [$href{https://doi.org/10.2307/2006955}{22}$] and Kawamata–Namikawa [$href{ https://doi.org/10.1007/BF01231538}{41}$]) and maximally degenerate Calabi–Yau varieties (as studied by Kontsevich–Soibelman $[href{ https://link.springer.com/chapter/10.1007/0-8176-4467-9_9}{45}$] and Gross–Siebert [$href{ http://doi.org/10.4007/annals.2011.174.3.1}{30}$]). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov–Kontsevich [$href{https://doi.org/10.1155/S1073792898000166}{2}$, $href{https://doi.org/10.48550/arXiv.math/9903124}{1}$].","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388033","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}

引用次数: 18

Harmonic maps from $S^3$ into $S^2$ with low Morse index 低摩尔斯指数从$S^3$到$S^2$的谐波映射

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236594

Tristan Rivière

引用次数: 1

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters 非关键参数平面环面上$SU(3)$ Toda系统解的数量和偶数性

1区数学

Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236592

Zhijie Chen, Chang-Shou Lin

{"title":"On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters","authors":"Zhijie Chen, Chang-Shou Lin","doi":"10.4310/jdg/1695236592","DOIUrl":"https://doi.org/10.4310/jdg/1695236592","url":null,"abstract":"We study the $SU(3)$ Toda system with singular sources begin{equation*} begin{cases} Delta u+2e^{u}-e^{v}=4pi sum _{k=0}^{m} n_{1,k}delta _{p_{k}} quad text{ on }; E_{tau}, Delta v+2e^{v}-e^{u}=4pi sum _{k=0}^{m} n_{2,k}delta _{p_{k}} quad text{ on }; E_{tau}, end{cases} end{equation*} where $E_{tau}:=mathbb{C}/(mathbb{Z}+mathbb{Z}tau )$ with $operatorname{Im}tau gt 0$ is a flat torus, $delta _{p_{k}}$ is the Dirac measure at $p_{k}$, and $n_{i,k}in mathbb{Z}_{geq 0}$ satisfy $sum _{k}n_{1,k}not equiv sum _{k} n_{2,k} mod 3$. This is known as the non-critical case and it follows from a general existence result of [$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$] that solutions always exist. In this paper we prove that (i) The system has at most begin{equation*} frac{1}{3times 2^{m+1}}prod _{k=0}^{m}(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2) in mathbb{N} end{equation*} solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_{0}=0$, the system has even solutions if and only if at least one of ${n_{1,0}, n_{2,0}}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}lt n_{2,0}$, then except for finitely many $tau $’s modulo $SL(2,mathbb{Z})$ action, the system has exactly $frac{n_{1,0}+1}{2}$ even solutions. Differently from [$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388235","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}

引用次数: 2