arXiv subjects

Joe Lope Vicente

Publications and source records attributed to Joe Lope Vicente.

5 recordsLinked to original sources

arXiv2026

Sasaki-Einstein rational homology spheres, rational varieties and the Berglund-Hübsch rule

We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hypersurfaces defined by cycle polynomials, on the estimate given by Johnson and Kollár to determine Kähler-Einstein orbifold metrics. We also construct weighted hypersurfaces that contain the rational varieties described above as codimension two subvarieties and, due to the refined estimate for cyclic polynomials, we find conditions on the weights and degrees of these hypersurfaces so their corresponding smooth links admit Sasaki-Einstein metrics. Finally we study the effect of the Berglund-Hübsch transpose rule on the topology and on the existence of Sasaki-Einstein metrics on the links studied and generalize all the results given in arXiv:2311.15998 for rational homology 7-spheres to rational homology $(4n-1)$-spheres, that is, we show invariance of these two features under the transpose rule.

math.DG↗

arXiv2026

The local moduli of Sasaki-Einstein rational homology 7-spheres and invertible polynomials

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, 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.

math.DG↗

arXiv2026

Exceptional Fano 3-folds from rational curves

We show exceptionality of certain families of non-quasismooth weighted hypersurfaces. In particular these admit Kähler-Einstein metrics. Our examples are produced by the monomials generating the complex deformations of orbifolds whose corresponding $S^1$-Seifert bundles are smooth rational homology 7-spheres admitting Sasaki-Einstein metrics. From our construction, it follows that these exceptional Fano hypersurfaces describe elements in the boundary of the K-moduli of $\mathbb{Q}$-Fano 3-folds.

math.AG↗

arXiv2025

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

We use the Berglund-Hübsch 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 big Sasaki cones that do not admit any extremal Sasaki metrics at all. 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 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.

math.DG↗

arXiv2024

Beglund-Hübsch transpose and Sasaki-Einstein rational homology 7-spheres

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.

math.DG↗