Back to Search Start Over

Non-existence of extremal Sasaki metrics via the Berglund-H\'ubsch transpose

Non-existence of extremal Sasaki metrics via the Berglund-H\'ubsch transpose

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

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2409.09720
Document Type :
Working Paper