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The stable cohomology of moduli spaces of sheaves on surfaces 曲面上轮轴模空间的稳定上同调
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-06-01 DOI: 10.4310/jdg/1659987893
Izzet Coskun, Matthew Woolf
{"title":"The stable cohomology of moduli spaces of sheaves on surfaces","authors":"Izzet Coskun, Matthew Woolf","doi":"10.4310/jdg/1659987893","DOIUrl":"https://doi.org/10.4310/jdg/1659987893","url":null,"abstract":"Let X be a smooth, irreducible, complex projective surface, H a polarization on X. Let γ = (r, c,∆) be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves MX,H(γ). When the rank r = 1, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank r ≥ 1 and the first Chern class c, then the Betti numbers (and more generally the Hodge numbers) of MX,H(r, c,∆) stabilize as the discriminant ∆ tends to infinity and that the stable Betti numbers are independent of r and c. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when X is a rational surface and KX · H < 0, then the classes [MX,H(γ)] stabilize in an appropriate completion of the Grothendieck ring of varieties as ∆ tends to ∞. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when X is a rational surface, H · KX < 0 and there are no strictly semistable objects in MX,H(γ).","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44619126","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: 10
Visible actions of compact Lie groups on complex spherical varieties 复球面上紧李群的可见作用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2022-02-01 DOI: 10.4310/jdg/1645207534
Yu-ichi Tanaka
{"title":"Visible actions of compact Lie groups on complex spherical varieties","authors":"Yu-ichi Tanaka","doi":"10.4310/jdg/1645207534","DOIUrl":"https://doi.org/10.4310/jdg/1645207534","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45735296","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
Collapsing ancient solutions of mean curvature flow 塌缩平均曲率流的古老解
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-10-01 DOI: 10.4310/jdg/1632506300
T. Bourni, Mathew T. Langford, G. Tinaglia
{"title":"Collapsing ancient solutions of mean curvature flow","authors":"T. Bourni, Mathew T. Langford, G. Tinaglia","doi":"10.4310/jdg/1632506300","DOIUrl":"https://doi.org/10.4310/jdg/1632506300","url":null,"abstract":"We construct a compact, convex ancient solution of mean curvature flow in $mathbb{R}^{n+1}$ with $O(1) times O(n)$ symmetry that lies in a slab of width $pi$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $pi$ and in no smaller slab.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665165","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: 33
Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85. 对“非负截面曲率和非唯一灵魂的模空间”的修正，J.Diff.Geom。89（2011），第1号，49–85。
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-09-01 DOI: 10.4310/jdg/1631124246
I. Belegradek, S. Kwasik, R. Schultz
{"title":"Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85.","authors":"I. Belegradek, S. Kwasik, R. Schultz","doi":"10.4310/jdg/1631124246","DOIUrl":"https://doi.org/10.4310/jdg/1631124246","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42694039","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
Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori 矩形环面上多重格林函数临界点的精确个数与非退化性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-07-01 DOI: 10.4310/JDG/1625860623
Zhijie Chen, Changshou Lin
{"title":"Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori","authors":"Zhijie Chen, Changshou Lin","doi":"10.4310/JDG/1625860623","DOIUrl":"https://doi.org/10.4310/JDG/1625860623","url":null,"abstract":"Let $E_{tau}:= mathbb{C}/(mathbb{Z}+ mathbb{Z} tau)$ be a flat torus and $G(z; tau)$ be the Green function on $E_{tau}$. Consider the multiple Green function $G_{n}$ on$(E_{tau})^{n}$: [ G_n (z_1, cdots ,z_n ; tau) := sum_{i lt j} G(z_i - z_j ; tau) - n sum_{i=1}^n G(z_i ; tau). ] We prove that for $ tau in i mathbb{R}_{gt 0}$, i.e. $E_tau$ is a rectangular torus, $G_n$ has exactly $2n + 1$ critical points modulo the permutation group $S_n$ and all critical points are non-degenerate. More precisely, there are exactly $n$ (resp. $n+1$) critical points $ boldsymbol{a}$’s with the Hessian satisfying $(-1)^n det D^2 G_n (boldsymbol{a} ; tau) lt 0$ (resp. $gt 0$). This confirms a conjecture in [4]. Our proof is based on the connection between $G_n$ and the classical Lame equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of $G_{n}$ in terms of the monodromy data of the Lame equation. As an application, we show that the mean field equation [ Delta_u + e^u = rho delta_0 textrm{ on } E_tau ] has exactly $n$ solutions for $8 pi n - rho gt 0$ small, and exactly $n+1$ solutions for $rho - 8 pi n gt 0$ small.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47882033","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: 7
Harmonic quasi-isometric maps into Gromov hyperbolic $operatorname{CAT}(0)$-spaces Gromov双曲$operatorname{CAT}(0)$-空间中的调和拟等距映射
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-07-01 DOI: 10.4310/JDG/1625860625
H. Sidler, S. Wenger
{"title":"Harmonic quasi-isometric maps into Gromov hyperbolic $operatorname{CAT}(0)$-spaces","authors":"H. Sidler, S. Wenger","doi":"10.4310/JDG/1625860625","DOIUrl":"https://doi.org/10.4310/JDG/1625860625","url":null,"abstract":"We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a proper, Gromov hyperbolic, $operatorname{CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist–Hulin.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47343349","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
Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds 紧流形上Ginzburg-Landau方程最小-极大解的存在性及极限行为
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-06-01 DOI: 10.4310/JDG/1622743143
Daniel Stern
{"title":"Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds","authors":"Daniel Stern","doi":"10.4310/JDG/1622743143","DOIUrl":"https://doi.org/10.4310/JDG/1622743143","url":null,"abstract":"We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $geq 2$. We investigate the limiting behavior of these critical points as $varepsilon to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $varepsilon to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43631625","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: 22
On the existence of harmonic $Z_2$ spinors 关于调和Z_2旋量的存在性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-03-01 DOI: 10.4310/JDG/1615487003
Aleksander Doan, Thomas Walpuski
{"title":"On the existence of harmonic $Z_2$ spinors","authors":"Aleksander Doan, Thomas Walpuski","doi":"10.4310/JDG/1615487003","DOIUrl":"https://doi.org/10.4310/JDG/1615487003","url":null,"abstract":"We prove the existence of singular harmonic Z2 spinors on 3–manifolds with b1 > 1. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic Z2 spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce [Joy17] regarding Donaldson and Segal’s proposal for counting G2–instantons [DS11].","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42370502","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: 4
Fukaya $A_infty$-structures associated to Lefschetz fibrations. III Fukaya $A_infty$ -与Lefschetz纤维相关的结构。3。
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-03-01 DOI: 10.4310/jdg/1615487005
Paul Seidel
{"title":"Fukaya $A_infty$-structures associated to Lefschetz fibrations. III","authors":"Paul Seidel","doi":"10.4310/jdg/1615487005","DOIUrl":"https://doi.org/10.4310/jdg/1615487005","url":null,"abstract":"Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503391","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
From the Hitchin section to opers through nonabelian Hodge 从希钦部分到非贝利式霍奇的歌剧
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2021-02-01 DOI: 10.4310/JDG/1612975016
Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, R. Mazzeo, M. Mulase, A. Neitzke
{"title":"From the Hitchin section to opers through nonabelian Hodge","authors":"Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, R. Mazzeo, M. Mulase, A. Neitzke","doi":"10.4310/JDG/1612975016","DOIUrl":"https://doi.org/10.4310/JDG/1612975016","url":null,"abstract":"For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point $mathbf{u}$ of the base of Hitchin’s integrable system for $(G,C)$. One family $nabla_{hbar ,mathbf{u}}$ consists of $G$-opers, and depends on $hbar in mathbb{C}^times$. The other family $nabla_{R, zeta,mathbf{u}}$ is built from solutions of Hitchin’s equations, and depends on $zeta in mathbb{C}^times , R in mathbb{R}^+$. We show that in the scaling limit $R to 0, zeta = hbar R$, we have $nabla_{R,zeta,mathbf{u}} to nabla_{hbar,mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45535930","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: 13
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