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.

💡 Research Summary

The paper investigates the local deformation theory of Sasaki‑Einstein (SE) metrics on 7‑dimensional rational homology spheres that arise as links of isolated hypersurface singularities defined by invertible (also called “non‑degenerate”) weighted homogeneous polynomials. By the Kreuzer‑Skarke classification, every invertible polynomial can be written, up to a permutation of variables, as a Thom‑Sebastiani sum of three atomic blocks: Fermat, chain, and loop (cycle). The authors focus on the two families that actually produce rational homology 7‑spheres admitting SE metrics: (i) pure loop‑type polynomials and (ii) Thom‑Sebastiani sums of a loop block with a second atomic block (either a chain or a Fermat block).

Main results for the two families

1. Pure loop (cycle) type – For a loop polynomial
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