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Item type:Publication, Berglund–Hübsch transpose and Sasaki–Einstein rational homology 7-spheres(Springer Science and Business Media Deutschland GmbH, 2024-08-01)We show that links of isolated hypersurface singularities defined by invertible polynomials coming from the Johnson and Kollár list of Kähler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-Hübsch transpose rule coming from classical mirror symmetry constructions. Actually, this rule produces twins, that is, links with same degree, Milnor number and homology H3, with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number vary. The Berglund-Hübsch transpose rule not only gives a framework to better understand the existence of Sasaki-Einstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7-sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the Berglund-Hübsch transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Sasaki-Einstein 7-Manifolds and Orlik’s Conjecture(Springer Science+Business Media, 2023-11-16)We study the homology groups of certain 2-connected 7-manifolds admitting quasiregular Sasaki-Einstein metrics, among them, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres, extending the list given by Boyer, Galicki and Nakamaye in [6]. As a consequence, we exhibit new families of positive Sasakian homotopy 9-spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer [12] showing new examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected sums of S3 × S4. Actually we show that manifolds of the form #k(S3 × S4) admit Sasaki-Einstein metrics for 22 different values of k. All these links arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities where Orlik's conjecture holds due to a recent result by Hertling and Mase [19]. Mathematics Subject Classification 53C25; 57R60. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Beglund-Hübsch Transpose and Sasaki-Einstein Rational Homology 7-Spheres(Cornell University, 2023-11-27)We show that links of invertible polynomials coming from the Johnson and Kollár list of Kähler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-Hübsch transpose rule coming from classical mirror symmetry constructions. Actually, this rule produces twins, that is, links with same degree, Milnor number and homology H_3, with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number may vary. The Berglund-Hübsch transpose rule not only gives a framework to better understand the existence of SasakiEinstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7 -sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the BH-transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, 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.1 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, 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.1