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

Abstract

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.

We rewrote certain parts of the article, corrected some misprints and errors and add a new section on the applications of our findings. Accepted for publications in Annals of Global Analysis and Geometry

References in corpus (6)

• Canonical Sasakian Metrics
• The integral monodromy of isolated quasihomogeneous singularities
• Sasaki-Einstein 7-manifolds and Orlik's conjecture
• Beglund-Hübsch transpose and Sasaki-Einstein rational homology 7-spheres
• Compact moduli of Calabi-Yau cones and Sasaki-Einstein spaces
• Exceptional Fano 3-folds from rational curves