Kwokwai Chan, Naichung Conan Leung, Ziming Nikolas Ma
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 [$href{https://doi.org/10.1155/S1073792898000166}{2}$, $href{https://doi.org/10.48550/arXiv.math/9903124}{1}$].
{"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}
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.
{"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}
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.
我们研究了具有奇异源的$SU(3)$ Toda系统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*}，其中$E_{tau}:=mathbb{C}/(mathbb{Z}+mathbb{Z}tau )$和$operatorname{Im}tau gt 0$是一个平面环面，$delta _{p_{k}}$是在$p_{k}$处的狄拉克测度，$n_{i,k}in mathbb{Z}_{geq 0}$满足$sum _{k}n_{1,k}not equiv sum _{k} n_{2,k} mod 3$。这就是所谓的非临界情况，它由[$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$]的一般存在性结果得出，解总是存在的。本文证明了(i)系统最多有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*}个解。我们有几个例子表明这个上界应该是尖锐的。我们的证明很好地结合了分析中的先验估计和代数几何中的经典bsamzout定理。(ii)对于$m=0$和$p_{0}=0$，系统有偶解当且仅当${n_{1,0}, n_{2,0}}$中至少有一个是偶解。进一步，如果$n_{1,0}$为奇数，$n_{2,0}$为偶数，$n_{1,0}lt n_{2,0}$，则除了$tau $的模$SL(2,mathbb{Z})$作用有限个外，系统恰好有$frac{n_{1,0}+1}{2}$个偶解。与[$href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$]不同的是，我们的证明是基于Toda系统的可积性，并且还隐含了Toda系统的四个奇异源偶解的一般不存在性结果。
{"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}
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 $mathbb{R}^4$.
{"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}
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 $mathbb{S}^n$.
{"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}
{"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}
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.
{"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}
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.
{"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}
{"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}
{"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}
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