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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":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":"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
{"title":"Harmonic maps from $S^3$ into $S^2$ with low Morse index","authors":"Tristan Rivière","doi":"10.4310/jdg/1695236594","DOIUrl":"https://doi.org/10.4310/jdg/1695236594","url":null,"abstract":"We prove that any smooth harmonic map from $S^3$ into $S^2$ of Morse index less or equal than $4$ has to be an harmonic morphism, that is the successive composition of an isometry of $S^3$, the Hopf fibration and an holomorphic map from $mathbb{C}P^1$ into itself.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135388385","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: 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":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":"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
Creating Stein surfaces by topological isotopy 用拓扑同位素法创建斯坦因表面
1区 数学
Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236593
Robert E. Gompf
{"title":"Creating Stein surfaces by topological isotopy","authors":"Robert E. Gompf","doi":"10.4310/jdg/1695236593","DOIUrl":"https://doi.org/10.4310/jdg/1695236593","url":null,"abstract":"We combine Freedman’s topology with Eliashberg’s holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded $2$-complex $K$ in a complex surface, after $C^0$-small topological ambient isotopy, is the intersection of an uncountable nested family of Stein regular neighborhoods that are all topologically ambiently isotopic rel $K$, but frequently realize uncountably many diffeomorphism types. These arise from the Cantor set levels of a topological mapping cylinder. The boundaries of the neighborhoods are $3$-manifolds that are only topologically embedded, but still satisfy a notion of pseudoconvexity. Such $3$-manifolds share some basic properties of hypersurfaces that are strictly pseudoconvex in the usual smooth sense, but they are far more common. The complementary notion of topological pseudoconcavity is realized by uncountably many diffeomorphism types homeomorphic to $mathbb{R}^4$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135637914","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
Isoperimetry and volume preserving stability in real projective spaces 实射影空间中的等距性和保体积稳定性
1区 数学
Journal of Differential Geometry Pub Date : 2023-09-01 DOI: 10.4310/jdg/1695236595
Celso Viana
{"title":"Isoperimetry and volume preserving stability in real projective spaces","authors":"Celso Viana","doi":"10.4310/jdg/1695236595","DOIUrl":"https://doi.org/10.4310/jdg/1695236595","url":null,"abstract":"We classify the volume preserving stable hypersurfaces in the real projective space $mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $mathbb{RP}^k subset mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on $mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $mathbb{S}^n$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","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":"135685689","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: 3
Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $mathrm{SU}(m)$ or $mathrm{U}(m)$ 具有规范群$ mathm {SU}(m)$或$ mathm {U}(m)$的$G_2$-实例的模空间的正则取向
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2023-06-01 DOI: 10.4310/jdg/1686931600
D. joyce, M. Upmeier
{"title":"Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $mathrm{SU}(m)$ or $mathrm{U}(m)$","authors":"D. joyce, M. Upmeier","doi":"10.4310/jdg/1686931600","DOIUrl":"https://doi.org/10.4310/jdg/1686931600","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47224374","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
The renormalized volume of a $4$-dimensional Ricci-flat ALE space 4维ricci -平坦ALE空间的重整化体积
1区 数学
Journal of Differential Geometry Pub Date : 2023-03-01 DOI: 10.4310/jdg/1683307004
Olivier Biquard, Hans-Joachim Hein
{"title":"The renormalized volume of a $4$-dimensional Ricci-flat ALE space","authors":"Olivier Biquard, Hans-Joachim Hein","doi":"10.4310/jdg/1683307004","DOIUrl":"https://doi.org/10.4310/jdg/1683307004","url":null,"abstract":"We introduce a natural definition of the renormalized volume of a $4$-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of $4$-dimensional Ricci-flat ALE spaces are Kronheimer’s gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer’s period map.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136096819","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
On the topology and index of minimal surfaces II 关于极小曲面的拓扑和索引2
1区 数学
Journal of Differential Geometry Pub Date : 2023-03-01 DOI: 10.4310/jdg/1683307005
Otis Chodosh, Davi Maximo
{"title":"On the topology and index of minimal surfaces II","authors":"Otis Chodosh, Davi Maximo","doi":"10.4310/jdg/1683307005","DOIUrl":"https://doi.org/10.4310/jdg/1683307005","url":null,"abstract":"For an immersed minimal surface in $mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in $mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134996012","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
Morse homotopy for the $SU(2)$-Chern–Simons perturbation theory $SU（2）$-Chern-Simons微扰理论的Morse同胚性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2023-02-01 DOI: 10.4310/jdg/1680883580
Tatsuro Shimizu
{"title":"Morse homotopy for the $SU(2)$-Chern–Simons perturbation theory","authors":"Tatsuro Shimizu","doi":"10.4310/jdg/1680883580","DOIUrl":"https://doi.org/10.4310/jdg/1680883580","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42400811","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
Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles 具有充分正则或反正则线束的非奇异射影$n$-折叠上Chern类的不等式
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-11-01 DOI: 10.4310/jdg/1675712992
Rong Du, Hao Sun
{"title":"Inequalities of Chern classes on nonsingular projective $n$-folds with ample canonical or anti-canonical line bundles","authors":"Rong Du, Hao Sun","doi":"10.4310/jdg/1675712992","DOIUrl":"https://doi.org/10.4310/jdg/1675712992","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49150549","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
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