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].
{"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":"117 1","pages":""},"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}
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.
{"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":"19 4","pages":""},"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}
Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, R. Mazzeo, M. Mulase, A. Neitzke
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.
{"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":"117 1","pages":"223-253"},"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}
In this paper, we prove that the open Gromov–Witten invariants defined in [20] on K3 surfaces satisfy the Kontsevich–Soibelman wall-crossing formula. One hand, this gives a geometric interpretation of the slab functions in Gross–Siebert program. On the other hands, the open Gromov–Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties [26][27] but on compact Calabi–Yau surfaces.
{"title":"Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces","authors":"Yu-Shen Lin","doi":"10.4310/jdg/1609902017","DOIUrl":"https://doi.org/10.4310/jdg/1609902017","url":null,"abstract":"In this paper, we prove that the open Gromov–Witten invariants defined in [20] on K3 surfaces satisfy the Kontsevich–Soibelman wall-crossing formula. One hand, this gives a geometric interpretation of the slab functions in Gross–Siebert program. On the other hands, the open Gromov–Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties [26][27] but on compact Calabi–Yau surfaces.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"20 4","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503390","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}
The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $ggeqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne-Mumford compactification $overline{mathcal M}_g$ of the moduli $mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.
{"title":"Limit of Weierstrass measure on stable curves","authors":"Ngai-fung Ng, Sai-Kee Yeung","doi":"10.4310/jdg/1668186789","DOIUrl":"https://doi.org/10.4310/jdg/1668186789","url":null,"abstract":"The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $ggeqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne-Mumford compactification $overline{mathcal M}_g$ of the moduli $mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42515865","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}
Source: Journal of Differential Geometry, Volume 116, Number 3
资料来源:《Journal of Differential Geometry》第116卷第3期
{"title":"Index to Volume 126","authors":"","doi":"10.4310/jdg/1606964419","DOIUrl":"https://doi.org/10.4310/jdg/1606964419","url":null,"abstract":"<p><strong>Source: </strong>Journal of Differential Geometry, Volume 116, Number 3</p>","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"21 3","pages":""},"PeriodicalIF":2.5,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503389","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":"New proof for the regularity of Monge–Ampère type equations","authors":"Xu-jia Wang, Yating Wu","doi":"10.4310/jdg/1606964417","DOIUrl":"https://doi.org/10.4310/jdg/1606964417","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"116 1","pages":"543-553"},"PeriodicalIF":2.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42466754","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":"The $L_p$ Minkowski problem for the electrostatic $mathfrak{p}$-capacity","authors":"Zou Du, Xiong Ge","doi":"10.4310/jdg/1606964418","DOIUrl":"https://doi.org/10.4310/jdg/1606964418","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"116 1","pages":"555-596"},"PeriodicalIF":2.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46870650","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":"Space of Ricci flows (II)—Part B: Weak compactness of the flows","authors":"Xiuxiong Chen, Bing Wang","doi":"10.4310/jdg/1599271253","DOIUrl":"https://doi.org/10.4310/jdg/1599271253","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"116 1","pages":"1-123"},"PeriodicalIF":2.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48727375","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":"Riemann–Hilbert problems for the resolved conifold and non-perturbative partition functions","authors":"T. Bridgeland","doi":"10.4310/jdg/1594260015","DOIUrl":"https://doi.org/10.4310/jdg/1594260015","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"115 1","pages":"395-435"},"PeriodicalIF":2.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48491802","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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