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    The local moduli of Sasaki-Einstein rational homology 7-spheres and invertible polynomials
    (Springer Science+Business Media, 2026-03-01)
    We study the local moduli space of Sasaki-Einstein metrics on links of invertible polynomials defining rational homology 7-spheres. All these polynomials are either of cycle type or are given as Thom Sebastiani sums of a cycle block and another atomic block. We found that for polynomials of cycle type, the local moduli spaces of Sasaki-Einstein metrics are zero dimensional. For the Thom-Sebastiani sums of an atomic block and a cycle polynomial, the dimensions of the local moduli spaces of Sasaki-Einstein metrics are positive in general. Since all the links under study in this article remain Sasaki-Einstein rational homology 7-spheres under the Berglund-Hübsch rule from classical mirror symmetry (Berglund and Hübsch, Nucl Phys B 393:377–391 (1993), Cuadros et al., Commun Math Phys 405:199 (2024)), we are able to find solutions for the problem associated to the moduli for the Berglund-Hübsch transpose duals of this type of links. For the purpose of doing this, we give specific description of the moduli spaces of complex structures on the weighted quasismooth hypersurfaces cut out by the corresponding invertible polynomials and, in particular, from this description, we can produce families of quasismooth weighted hypersurfaces that degenerate to non-quasismooth with at worst klt singularities.
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    Non-existence of extremal Sasaki metrics and the Berglund-Hübsch transpose
    (Elsevier BV, 2025-08-01)
    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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