Non-existence of extremal Sasaki metrics and the Berglund-Hübsch transpose

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2025-05-13 DOI:10.1016/j.geomphys.2025.105531

Jaime Cuadros Valle , Ralph R. Gomez , Joe Lope Vicente

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引用次数: 0

Abstract

We use the Berglund-Hübsch transpose rule from classical mirror symmetry in the context of Sasakian geometry [11] and results on relative K-stability in the Sasaki setting developed by Boyer and van Coevering in [6] to exhibit examples of Sasaki manifolds with big Sasaki cones that have no extremal Sasaki metrics at all. Previously, examples with this feature were produced in [6] for Brieskorn-Pham polynomials or their deformations. Our examples are based on the more general framework of invertible polynomials. In particular, we construct families of links that preserve the emptiness of the extremal Sasaki-Reeb cone via the Berglund-Hübsch rule: if the link does not admit extremal Sasaki metrics then its Berglund-Hübsch dual preserves this property and moreover this dual admits a representative in its local moduli with a larger Sasaki-Reeb cone which remains obstructed to admitting extremal Sasaki metrics. Some of the examples exhibited here have the homotopy type of a sphere or are rational homology spheres.

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极值Sasaki度量的不存在性和berglund - h bsch转置

我们使用Sasaki几何[11]背景下经典镜像对称的berglund - h bsch转置规则，以及Boyer和van Coevering在[6]中开发的Sasaki环境中相对k -稳定性的结果来展示具有大Sasaki锥的Sasaki流形的例子，这些流形根本没有极值Sasaki度量。以前，具有此特征的例子是在b[6]中为Brieskorn-Pham多项式或其变形生成的。我们的例子是基于更一般的可逆多项式框架。特别地，我们通过berglund - h bsch规则构造了保持极值Sasaki- reeb锥空性的环系：如果该环不允许极值Sasaki度量，则其berglund - h bsch对偶保留了这一性质，并且该对偶在其局部模中允许一个具有更大的Sasaki- reeb锥的代表，该锥仍然阻碍允许极值Sasaki度量。这里展示的一些例子具有球的同伦类型或者是有理同伦球。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity