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

Cuadros Valle J.; Gomez R.R.; Lope Vicente J. · Journal of Geometry and Physics (2025) · DOI: 10.1016/j.geomphys.2025.105531

Resumen

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.

Instituciones: Pontificia Universidad Católica del Perú