Pub Date : 2024-08-06DOI: 10.1515/advgeom-2024-0014
Charu Goel, Sarah Hess, Salma Kuhlmann
For n, d ∈ ℕ, the cone 𝓟n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σn+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σn+1,2d and 𝓟n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties Vk−n ⊊ … ⊊ Vn ⊊ … ⊊ V0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V0, …, Vn (and Vn+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σn+1,2d. We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.
对于 n, d∈ ℕ，度数为 2d 的 n + 1 变数中正半定实数形式的锥𝓟 n+1,2d 包含可表示为实数形式有限平方和的子锥Σ n+1,2d 。希尔伯特[11] 证明了这些锥体在希尔伯特情形 (n + 1, 2d) 中完全重合，即 n + 1 = 2 或 2d = 2 或 (n + 1, 2d) = (3, 4)。在本文中，我们通过[4]中的格拉姆矩阵方法，在不可还原的投影变种 V k-n ⊊ ... ⊊ Vn ⊊ ... ⊊ V 0 的滤波上，诱导出介于 Σ n+1,2d 和 𝓟 n+1,2d 之间的中间锥的滤波，其中包含维罗纳变种。通过证明 V 0，...，V n（以及当 n = 2 时的 V n+1）是最小度的变项，我们证明了相应的中间锥与Σ n+1,2d 重合。此外，我们还证明，在 n ≥ 3 的 (n + 1)-ary 四元数和 n ≥ 2 的 (n + 1)-ary 六元数的非希尔伯特情况下，所有剩余的圆锥内含都是严格的。
{"title":"Cones between the cones of positive semidefinite forms and sums of squares","authors":"Charu Goel, Sarah Hess, Salma Kuhlmann","doi":"10.1515/advgeom-2024-0014","DOIUrl":"https://doi.org/10.1515/advgeom-2024-0014","url":null,"abstract":"For <jats:italic>n</jats:italic>, <jats:italic>d</jats:italic> ∈ ℕ, the cone 𝓟<jats:sub> <jats:italic>n</jats:italic>+1,2<jats:italic>d</jats:italic> </jats:sub> of positive semidefinite real forms in <jats:italic>n</jats:italic> + 1 variables of degree 2<jats:italic>d</jats:italic> contains the subcone <jats:italic>Σ</jats:italic> <jats:sub> <jats:italic>n</jats:italic>+1,2<jats:italic>d</jats:italic> </jats:sub> of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the <jats:italic>Hilbert cases</jats:italic> (<jats:italic>n</jats:italic> + 1, 2<jats:italic>d</jats:italic>) with <jats:italic>n</jats:italic> + 1 = 2 or 2<jats:italic>d</jats:italic> = 2 or (<jats:italic>n</jats:italic> + 1, 2<jats:italic>d</jats:italic>) = (3, 4). In this paper, we induce a filtration of intermediate cones between <jats:italic>Σ</jats:italic> <jats:sub> <jats:italic>n</jats:italic>+1,2<jats:italic>d</jats:italic> </jats:sub> and 𝓟<jats:sub> <jats:italic>n</jats:italic>+1,2<jats:italic>d</jats:italic> </jats:sub> via the Gram matrix approach in [4] on a filtration of irreducible projective varieties <jats:italic>V</jats:italic> <jats:sub> <jats:italic>k</jats:italic>−<jats:italic>n</jats:italic> </jats:sub> ⊊ … ⊊ <jats:italic>V<jats:sub>n</jats:sub> </jats:italic> ⊊ … ⊊ <jats:italic>V</jats:italic> <jats:sub>0</jats:sub> containing the Veronese variety. Here, <jats:italic>k</jats:italic> is the dimension of the vector space of real forms in <jats:italic>n</jats:italic> + 1 variables of degree <jats:italic>d</jats:italic>. By showing that <jats:italic>V</jats:italic> <jats:sub>0</jats:sub>, …, <jats:italic>V</jats:italic> <jats:sub> <jats:italic>n</jats:italic> </jats:sub> (and <jats:italic>V</jats:italic> <jats:sub> <jats:italic>n</jats:italic>+1</jats:sub> when <jats:italic>n</jats:italic> = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with <jats:italic>Σ</jats:italic> <jats:sub> <jats:italic>n</jats:italic>+1,2<jats:italic>d</jats:italic> </jats:sub>. We moreover prove that, in the non-Hilbert cases of (<jats:italic>n</jats:italic> + 1)-ary quartics for <jats:italic>n</jats:italic> ≥ 3 and (<jats:italic>n</jats:italic> + 1)-ary sextics for <jats:italic>n</jats:italic> ≥ 2, all the remaining cone inclusions are strict.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943427","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}
We consider an anisotropic area-preserving nonlocal flow for closed convex plane curves, which is a generalization of the model introduced by Pan and Yang (J. Differential Equations 266 (2019), 3764–3786) when τ = 1. Under this flow, the evolving curve maintains its convexity and converges to a homothety of a smooth symmetric strictly convex plane curve in the C∞ sense. The analysis of the asymptotic behavior of this flow implies the possibility of deforming one curve into another within the framework of Minkowski geometry.
我们考虑了封闭凸平面曲线的各向异性面积保留非局部流，它是潘、杨（《微分方程学报》，266 (2019)，3764-3786）在τ = 1时引入的模型的广义化。在此流动条件下，演化曲线保持其凸性，并收敛于 C ∞ 意义上的光滑对称严格凸平面曲线的同调。对这种流的渐近行为的分析意味着在闵科夫斯基几何框架内将一条曲线变形为另一条曲线的可能性。
{"title":"Anisotropic area-preserving nonlocal flow for closed convex plane curves","authors":"Tianyu Zhao, Yunlong Yang, Yueyue Mao, Jianbo Fang","doi":"10.1515/advgeom-2023-0025","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0025","url":null,"abstract":"We consider an anisotropic area-preserving nonlocal flow for closed convex plane curves, which is a generalization of the model introduced by Pan and Yang (J. Differential Equations 266 (2019), 3764–3786) when <jats:italic>τ</jats:italic> = 1. Under this flow, the evolving curve maintains its convexity and converges to a homothety of a smooth symmetric strictly convex plane curve in the <jats:italic>C</jats:italic> <jats:sup>∞</jats:sup> sense. The analysis of the asymptotic behavior of this flow implies the possibility of deforming one curve into another within the framework of Minkowski geometry.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"3 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559343","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0026
Genki Omori
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several cases of closed surfaces.
{"title":"The balanced superelliptic mapping class groups are generated by three elements","authors":"Genki Omori","doi":"10.1515/advgeom-2023-0026","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0026","url":null,"abstract":"The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several cases of closed surfaces.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559008","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0022
Taro Hayashi
Let X be a K3 surface and let g be a non-symplectic involution of X such that the fixed points set contains a curve of genus 8 or more. In this paper, we show that the quotient space X/〈g〉 is determined by the fixed points set and the action of g on rational curves on X.
设 X 是一个 K3 曲面，设 g 是 X 的一个非交错内卷，使得定点集包含一条属 8 或以上的曲线。在本文中，我们将证明商空间 X/〈g〉是由定点集和 g 对 X 上有理曲线的作用决定的。
{"title":"Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more","authors":"Taro Hayashi","doi":"10.1515/advgeom-2023-0022","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0022","url":null,"abstract":"Let <jats:italic>X</jats:italic> be a <jats:italic>K</jats:italic>3 surface and let <jats:italic>g</jats:italic> be a non-symplectic involution of <jats:italic>X</jats:italic> such that the fixed points set contains a curve of genus 8 or more. In this paper, we show that the quotient space <jats:italic>X</jats:italic>/〈<jats:italic>g</jats:italic>〉 is determined by the fixed points set and the action of <jats:italic>g</jats:italic> on rational curves on <jats:italic>X</jats:italic>.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"31 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558847","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0031
Annamaria Iezzi, Motoko Qiu Kawakita, Marco Timpanella
We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a generalization of Wiman’s sextics [38] and Edge’s sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in the table www.manypoints.org, see [37].
{"title":"New sextics of genus 6 and 10 attaining the Serre bound","authors":"Annamaria Iezzi, Motoko Qiu Kawakita, Marco Timpanella","doi":"10.1515/advgeom-2023-0031","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0031","url":null,"abstract":"We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a generalization of Wiman’s sextics [38] and Edge’s sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in the table <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"uri\" xlink:href=\"http://www.manypoints.org\">www.manypoints.org</jats:ext-link>, see [37].","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559014","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0028
Debaditya Raychaudhury
We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, OX(1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative for many pairs (X, OX(1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously k-regular for some positive integer k ≤ dim X, then F is a GV−(k−1) sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (X, OX(1)), and we show that this function can be extended to a continuous function on N1(X)ℝ. We also provide syzygetic consequences of our results for Oℙ(E)(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties. In particular, we show that Oℙ(E)(1) satisfies the Np property if the base-point freeness threshold of the class of OX(1) in N1(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito.
我们研究极化不规则光滑投影变项 (X, O X (1)) 上无扭相干剪切的连续 CM 正则性及其与泛型消失理论的关系。这种卡斯特诺沃-芒福德正则性的连续变体是由穆斯托帕引入的，他提出了这样一个问题：连续 1-regular 的剪切 F 是否是 GV？在这里，我们对许多对（X, O X (1)）给出了肯定的回答，其中包括任何极化无性杂交的情况。此外，对于这些对子，我们证明了如果 F 对于某个正整数 k ≤ dim X 是连续 k-regular 的，那么 F 就是 GV-(k-1) sheaf。此外，我们将连续 CM-regularity 的概念扩展到极化无性变体 (X, O X (1)) 上的ℚ扭曲束上的实值函数，并证明该函数可以扩展为 N 1(X)ℝ 上的连续函数。我们还提供了与极化无常变体上的 0 规则束ɛ相关联的ℙ(ɛ) 上 Oℙ(E)(1) 的协同结果。我们特别指出，如果 N 1(X) 中 O X (1) 类的基点自由阈值小于 1/(p + 2)，则 Oℙ(E)(1) 满足 Np 特性。这一结果是通过伊藤敦撰写的附录 A 中的一个定理得到的。
{"title":"Continuous CM-regularity and generic vanishing","authors":"Debaditya Raychaudhury","doi":"10.1515/advgeom-2023-0028","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0028","url":null,"abstract":"We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (<jats:italic>X</jats:italic>, O<jats:italic> <jats:sub>X</jats:sub> </jats:italic>(1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative for many pairs (<jats:italic>X</jats:italic>, O<jats:italic> <jats:sub>X</jats:sub> </jats:italic>(1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously <jats:italic>k</jats:italic>-regular for some positive integer <jats:italic>k</jats:italic> ≤ dim <jats:italic>X</jats:italic>, then F is a GV<jats:sub>−(<jats:italic>k</jats:italic>−1)</jats:sub> sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (<jats:italic>X</jats:italic>, O<jats:italic> <jats:sub>X</jats:sub> </jats:italic>(1)), and we show that this function can be extended to a continuous function on <jats:italic>N</jats:italic> <jats:sup>1</jats:sup>(<jats:italic>X</jats:italic>)<jats:sub>ℝ</jats:sub>. We also provide syzygetic consequences of our results for O<jats:sub>ℙ(E)</jats:sub>(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties. In particular, we show that O<jats:sub>ℙ(E)</jats:sub>(1) satisfies the <jats:italic>N<jats:sub>p</jats:sub> </jats:italic> property if the base-point freeness threshold of the class of O<jats:italic> <jats:sub>X</jats:sub> </jats:italic>(1) in <jats:italic>N</jats:italic> <jats:sup>1</jats:sup>(<jats:italic>X</jats:italic>) is less than 1/(<jats:italic>p</jats:italic> + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559419","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0029
Hiromichi Takagi
In our previous paper [31], we show that all primeℚ-Fano 3-folds X with only 1/2(1, 1, 1)-singularities in certain 5 classes can be embedded as linear sections into bigger dimensionalℚ-Fano varieties called key varieties; each key variety is constructed from data of the Sarkisov link starting from the blow-up at one 1/2(1, 1, 1)-singularity of X. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link for each X as a linear section of the dual variety. In a natural context describing the variety dual to the key variety of X of genus 5 with one 1/2(1, 1, 1)-singularity, we also characterize a general canonical curve of genus 9 with a g72.$g_{7}^{2}.$
在我们之前的论文[31]中，我们证明了所有素ℚ-法诺 3 折叠 X 在某些 5 类中只有 1/2(1, 1, 1)奇异性，都可以作为线性部分嵌入到更大维度的ℚ-法诺变种中，称为关键变种；每个关键变种都是从 X 的一个 1/2(1, 1, 1)奇异性处的炸开开始的萨基索夫链的数据构造的。作为一种应用，我们将每个 X 的萨基索夫链的基本部分解释为对偶变种的线性部分。在描述与具有一个 1/2(1, 1, 1)奇异性的 X 属 5 的关键变种对偶的自然背景下，我们还描述了具有一个 g 7 2 的属 9 的一般典型曲线。
{"title":"Duality related with key varieties of ℚ-Fano threefolds constructed from projective bundles","authors":"Hiromichi Takagi","doi":"10.1515/advgeom-2023-0029","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0029","url":null,"abstract":"In our previous paper [31], we show that all primeℚ-Fano 3-folds <jats:italic>X</jats:italic> with only 1/2(1, 1, 1)-singularities in certain 5 classes can be embedded as linear sections into bigger dimensionalℚ-Fano varieties called key varieties; each key variety is constructed from data of the Sarkisov link starting from the blow-up at one 1/2(1, 1, 1)-singularity of <jats:italic>X</jats:italic>. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link for each <jats:italic>X</jats:italic> as a linear section of the dual variety. In a natural context describing the variety dual to the key variety of <jats:italic>X</jats:italic> of genus 5 with one 1/2(1, 1, 1)-singularity, we also characterize a general canonical curve of genus 9 with a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_advgeom-2023-0029_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>g</m:mi> <m:mrow> <m:mn>7</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>.</m:mo> </m:math> <jats:tex-math>$g_{7}^{2}.$</jats:tex-math> </jats:alternatives> </jats:inline-formula>","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559010","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0032
Adam Thompson
The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the L2-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger–Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a complete description of the Bach flow on simply connected nilmanifolds.
巴赫流是定义在四曲面上的四阶几何流。对于紧凑流形，它是韦尔曲率 L 2-norm 的负梯度流。在本文中，我们研究了四维简单连接零曼形上的巴赫流，这些零曼形的李代数是不可分解的。我们的研究表明，从任意左不变度量开始的巴赫流在所有正时间内都存在，并且在重定标后会在尖的切格-格罗莫夫意义上收敛到一个非梯度的膨胀巴赫孤子。将我们的结果与海利韦尔之前的结果结合起来，就能完整地描述简单相连无芒物上的巴赫流。
{"title":"Bach flow of simply connected nilmanifolds","authors":"Adam Thompson","doi":"10.1515/advgeom-2023-0032","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0032","url":null,"abstract":"The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the <jats:italic>L</jats:italic> <jats:sup>2</jats:sup>-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger–Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a complete description of the Bach flow on simply connected nilmanifolds.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558837","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 : 2024-01-24DOI: 10.1515/advgeom-2023-0030
Zhining Liu
We study the classification problem for polarized varieties with high nef value. We give a complete list of isomorphism classes for normal polarized varieties with high nef value. This generalizes classical work on the smooth case by Fujita, Beltrametti and Sommese. As a consequence we obtain that polarized varieties with slc singularities and high nef value are birationally equivalent to projective bundles over nodal curves.
{"title":"A note on polarized varieties with high nef value","authors":"Zhining Liu","doi":"10.1515/advgeom-2023-0030","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0030","url":null,"abstract":"We study the classification problem for polarized varieties with high nef value. We give a complete list of isomorphism classes for normal polarized varieties with high nef value. This generalizes classical work on the smooth case by Fujita, Beltrametti and Sommese. As a consequence we obtain that polarized varieties with slc singularities and high nef value are birationally equivalent to projective bundles over nodal curves.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559013","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 : 2023-10-01DOI: 10.1515/advgeom-2023-0019
Ethan Cotterill, Cristhian Garay, Johana Luviano
Abstract The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry , and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p -adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.
{"title":"Exploring tropical differential equations","authors":"Ethan Cotterill, Cristhian Garay, Johana Luviano","doi":"10.1515/advgeom-2023-0019","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0019","url":null,"abstract":"Abstract The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry , and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p -adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119562","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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