Pub Date : 2022-04-18DOI: 10.1515/advgeom-2022-0005
A. De Schepper
Abstract The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔6(𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over 𝕂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other “degenerate composition algebras” as the algebras used to construct the square.
{"title":"Geometric characterisation of subvarieties of 𝓔6(𝕂) related to the ternions and sextonions","authors":"A. De Schepper","doi":"10.1515/advgeom-2022-0005","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0005","url":null,"abstract":"Abstract The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔6(𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over 𝕂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other “degenerate composition algebras” as the algebras used to construct the square.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"69 - 106"},"PeriodicalIF":0.5,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41844586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1515/advgeom-2022-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/advgeom-2022-frontmatter2","DOIUrl":"https://doi.org/10.1515/advgeom-2022-frontmatter2","url":null,"abstract":"","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49044369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1515/advgeom-2022-0002
Antonio G. Ache, M. Warren
Abstract For an embedded submanifold Σ ⊂ ℝN, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].
{"title":"Approximating coarse Ricci curvature on submanifolds of Euclidean space","authors":"Antonio G. Ache, M. Warren","doi":"10.1515/advgeom-2022-0002","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0002","url":null,"abstract":"Abstract For an embedded submanifold Σ ⊂ ℝN, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"215 - 243"},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47775597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-27DOI: 10.1515/advgeom-2023-0016
Andres Fernandez Herrero
Abstract We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar arguments apply to Gieseker semistable G-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G-bundles admitting a quasiprojective good moduli space can be written naturally as a G-linearized global quotient Y/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G-Higgs bundles on a family of curves is smooth over any base in characteristic 0.
{"title":"On automorphisms of semistable G-bundles with decorations","authors":"Andres Fernandez Herrero","doi":"10.1515/advgeom-2023-0016","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0016","url":null,"abstract":"Abstract We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar arguments apply to Gieseker semistable G-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G-bundles admitting a quasiprojective good moduli space can be written naturally as a G-linearized global quotient Y/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G-Higgs bundles on a family of curves is smooth over any base in characteristic 0.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"17 2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41310758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1515/advgeom-2021-0030
M. Abbassi, Ibrahim Lakrini
Abstract We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.
{"title":"Harmonic sections of vector bundles with spherically symmetric metrics","authors":"M. Abbassi, Ibrahim Lakrini","doi":"10.1515/advgeom-2021-0030","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0030","url":null,"abstract":"Abstract We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"135 - 150"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47307732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1515/advgeom-2021-0008
M. Misawa, Nobumitsu Nakauchi
Abstract The radial map w(x) = x ‖x‖−1 is a well-known example of harmonic maps and p-harmonic ones into spheres with a point singularity. In this paper we give two examples of harmonic maps and p-harmonic ones into spheres of higher dimensions with the singularity xixj ‖x‖−2 or the singularity xixjxk ‖x‖−3.
{"title":"Two examples of harmonic maps into spheres","authors":"M. Misawa, Nobumitsu Nakauchi","doi":"10.1515/advgeom-2021-0008","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0008","url":null,"abstract":"Abstract The radial map w(x) = x ‖x‖−1 is a well-known example of harmonic maps and p-harmonic ones into spheres with a point singularity. In this paper we give two examples of harmonic maps and p-harmonic ones into spheres of higher dimensions with the singularity xixj ‖x‖−2 or the singularity xixjxk ‖x‖−3.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"23 - 31"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42968268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1515/advgeom-2021-0035
B. Tosun
Abstract We give an overview of the research related to the topological characterization of Stein domains in complex two-dimensional space, and an instance of their many important connections to smooth manifold topology in dimension four. One goal is to motivate and explain the following remarkable conjecture of Gompf: no Brieskorn integral homology sphere (other than S3) admits a pseudoconvex embedding in ℂ2, with either orientation. We include some new examples and results that consider the conjecture for families of rational homology spheres which are Seifert fibered, and integral homology spheres which are hyperbolic.
{"title":"Stein domains in ℂ2 with prescribed boundary","authors":"B. Tosun","doi":"10.1515/advgeom-2021-0035","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0035","url":null,"abstract":"Abstract We give an overview of the research related to the topological characterization of Stein domains in complex two-dimensional space, and an instance of their many important connections to smooth manifold topology in dimension four. One goal is to motivate and explain the following remarkable conjecture of Gompf: no Brieskorn integral homology sphere (other than S3) admits a pseudoconvex embedding in ℂ2, with either orientation. We include some new examples and results that consider the conjecture for families of rational homology spheres which are Seifert fibered, and integral homology spheres which are hyperbolic.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"9 - 22"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49074488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1515/advgeom-2022-frontmatter1
{"title":"Frontmatter","authors":"","doi":"10.1515/advgeom-2022-frontmatter1","DOIUrl":"https://doi.org/10.1515/advgeom-2022-frontmatter1","url":null,"abstract":"","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47132666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.1515/advgeom-2021-0034
Ramesh Sharma, Sharief Deshmukh
Abstract A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.
{"title":"Ricci almost solitons with associated projective vector field","authors":"Ramesh Sharma, Sharief Deshmukh","doi":"10.1515/advgeom-2021-0034","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0034","url":null,"abstract":"Abstract A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"22 1","pages":"1 - 8"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46105538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-07DOI: 10.1515/advgeom-2023-0017
E. Aksnes
Abstract The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.
{"title":"Tropical Poincaré duality spaces","authors":"E. Aksnes","doi":"10.1515/advgeom-2023-0017","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0017","url":null,"abstract":"Abstract The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"23 1","pages":"345 - 370"},"PeriodicalIF":0.5,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42017833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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