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Non-existence of extremal Sasaki metrics via the Berglund-H\'ubsch transpose
Non-existence of extremal Sasaki metrics via the Berglund-H\'ubsch transpose
- Publication Year :
- 2024
-
Abstract
- We use the Berglund-H\"ubsch 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 complexity 3 or complexity 4 that do not admit any extremal Sasaki metrics in its whole Sasaki-Reeb cone which is of Gorenstein type. 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 examples of links with the following property: if the link satisfies the Lichnerowicz obstruction of Gauntlett, Martelli, Sparks and Yau, then its Berglund-H\"ubsch dual preserves this obstruction and moreover this dual admits a perturbation in its local moduli which is obstructed to admitting extremal Sasaki metrics in its whole Sasaki-Reeb cone. In particular, we exhibit examples that have the homotopy type of a sphere or are rational homology spheres.<br />Comment: The previous version of this article has been corrected (minor inaccuracies and typos) and improved. We also have added subsection 4.3 with more examples for more general invertible polynomials
- Subjects :
- Mathematics - Differential Geometry
53C25, 57R60
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2409.09720
- Document Type :
- Working Paper