通过 Berglund-Hübsch 转置证明极值 Sasaki 度量的不存在性
Non-existence of extremal Sasaki metrics via the Berglund-Hübsch transpose
AI总结：
本文利用 Berglund-Hübsch 转置与 Sasaki 相对 K-稳定性，构造并证明可逆多项式链的极值 Sasaki 度量不存在性可由其对偶保持。
摘要：
我们将经典镜像对称中的 Berglund-Hübsch 转置规则应用于 Sasaki 几何背景，并结合 Boyer 和 van Coevering 发展的 Sasaki 情形下相对 K-稳定性结果，以给出具有大 Sasaki 锥、且完全不容许任何极值 Sasaki 度量的 Sasaki 流形例子。此前，Boyerer 和 van Coevering 已针对 Brieskorn-Pham 多项式或其形变构造出具有这一特征的例子。我们的例子建立在更一般的可逆多项式框架之上。特别地，我们通过 Berglund-Hübsch 规则构造了保持极值 Sasaki-Reeb 锥为空性的链族：若该链不容许极值 Sasaki 度量，则其 Berglund-Hübsch 对偶也保持这一性质；此外，该对偶在其局部模空间中存在一个代表，具有更大的 Sasaki-Reeb 锥，而该锥在容许极值 Sasaki 度量方面仍受到阻碍。这里给出的一些例子具有球面的同伦型，或是有理同调球面。
英文摘要：
We use the Berglund-Hübsch transpose rule from classical mirror symmetry in the context of Sasakian geometry and results on relative K-stability in the Sasaki setting developed by Boyer and van Coevering to exhibit examples of Sasaki manifolds of big Sasaki cones that do not admit any extremal Sasaki metrics at all. Previously, examples with this feature were produced by Boyer and van Coevering 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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