Publicación

Berglund–Hübsch transpose and Sasaki–Einstein rational homology 7-spheres

Jaime Cuadros Valle · Cuadros Valle J. · Ralph R. Gomez · Gomez R.R. · Joe Lope Vicente · Lope Vicente J.
2024 Communications in Mathematical Physics DOI: 10.1007/s00220-024-05093-5

Resumen

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.

Autores y colaboradores

Authors

Ralph R. Gomez
Gomez R.R.

Palabras clave

Metrics Geometry

Proyectos

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Item type:Research Project,
Holomorphic and Symplectic fillings on positive and null Sasaki 5-manifolds
P_HSF-S5
Please see file Anexo for a detailed explanation of certain definitions. It is known that many contact manifolds admit symplectic fillings, that is, they can be realized as boundary of compact symplectic manifolds. It is also known that one can extend this definition at the level of complex geometry. In this situation the analogue is given holomorphic fillings:a compact complex manifold (N,J) is a holomorphic filling of the contact manifold (M, D) if N (M,J) is strictly pseudo-convex and $D=TMcap JTM$. Due to a result of Marinescu, G., Yeganefer, N (see [MY]) Sasakian manifolds, which are of contact type, admit holomorphic fillings and thus Kählerian fillings. There is classification of Sasakian manifolds in terms of the basic Chern class (one obtains positive, negative and null structures). We are interested in simply connected 5 manifolds admitting both positive and null structures. Positive Sasakian manifolds usually can be realized as S^1-Seifert bundles over a log Del Pezzo surfaces. In the null case, they are S^1-Seifert bundles over K3 orbifolds. We think, for certain cases: it is possible to determine the (strong) symplectic filling of associated to the null case and the holomorphic filling of the positive case. The reason to choose these two structures is trying to understand an interesting connection discovered by Alexeev and Nikulin in the classification of certain Del Pezzo surfaces using K3 lattice theory. We expect to reinterpret this connection at the level of fillings.
P_HSF-S5
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