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

Jaime Cuadros Valle, Joe Lope Vicente

TL;DR

This work studies the local moduli of Sasaki-Einstein metrics on links L_f that arise from invertible polynomials associated with Johnson–Kollar’s anticanonically embedded Fano 3-folds of index 1, focusing on links that are rational homology 7-spheres. By translating the problem to deformations of the transverse holomorphic structure and counting monomials of degree d under weight constraints, the authors derive explicit dimension formulas for the local moduli across three polynomial types: cycle-type, Thom-Sebastiani sums (chain–cycle), and Berglund–Hübsch duals of chain–cycle polynomials. They show that cycle-type links have zero-dimensional local moduli, provide a closed-form expression for the Thom-Sebastiani case μ = m_2/(v_0 v_1) − 1/v_0 − 1/v_1 − 1, and identify a small set of Berglund–Hübsch duals with nontrivial real moduli, including five exceptional instances. Overall, the results constrain the landscape of inequivalent Sasaki-Einstein structures on these rational homology 7-spheres and connect deformation theory to the arithmetic of invertible polynomials and their weights.

Abstract

We compute the dimension of the local moduli space of Sasaki-Einstein metrics for links of invertible polynomials coming from the list of Johnson and Kollár of anticanonically embedded Fano 3-folds of index 1 that produce rational homology 7-spheres, that is, 7-manifolds whose rational-homology equals that of the 7-sphere. In order to do so, we propose additional conditions to the Diophantine equations associated to this problem. We also find solutions for the problem associated to the moduli for the Berglund-Hübsch duals of links arising from Thom-Sebastiani sums of chain and cycle polynomials. In particular we give an explicit description of local moduli of the complex structure for the corresponding Kähler-Einstein orbifold which in this setting is a rational homology projective 3 -spaces with quotient singularities.

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

TL;DR

This work studies the local moduli of Sasaki-Einstein metrics on links L_f that arise from invertible polynomials associated with Johnson–Kollar’s anticanonically embedded Fano 3-folds of index 1, focusing on links that are rational homology 7-spheres. By translating the problem to deformations of the transverse holomorphic structure and counting monomials of degree d under weight constraints, the authors derive explicit dimension formulas for the local moduli across three polynomial types: cycle-type, Thom-Sebastiani sums (chain–cycle), and Berglund–Hübsch duals of chain–cycle polynomials. They show that cycle-type links have zero-dimensional local moduli, provide a closed-form expression for the Thom-Sebastiani case μ = m_2/(v_0 v_1) − 1/v_0 − 1/v_1 − 1, and identify a small set of Berglund–Hübsch duals with nontrivial real moduli, including five exceptional instances. Overall, the results constrain the landscape of inequivalent Sasaki-Einstein structures on these rational homology 7-spheres and connect deformation theory to the arithmetic of invertible polynomials and their weights.

Abstract

We compute the dimension of the local moduli space of Sasaki-Einstein metrics for links of invertible polynomials coming from the list of Johnson and Kollár of anticanonically embedded Fano 3-folds of index 1 that produce rational homology 7-spheres, that is, 7-manifolds whose rational-homology equals that of the 7-sphere. In order to do so, we propose additional conditions to the Diophantine equations associated to this problem. We also find solutions for the problem associated to the moduli for the Berglund-Hübsch duals of links arising from Thom-Sebastiani sums of chain and cycle polynomials. In particular we give an explicit description of local moduli of the complex structure for the corresponding Kähler-Einstein orbifold which in this setting is a rational homology projective 3 -spaces with quotient singularities.

Paper Structure

This paper contains 8 sections, 19 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sasakian structures on links and the Berglund-Hübsch rule
  4. Deformations of transverse holomorphic Sasakian structures
  5. Local moduli of Sasaki-Einstein metrics on $\mathbb{Q}$-homology 7-spheres
  6. $\mathbb{Q}$-homology 7-spheres: the cycle type
  7. $\mathbb{Q}$-homology 7-spheres: Thom-Sebastiani sums of invertible polynomials
  8. $\mathbb{Q}$-homology sphere: the Berglund-Hübsch dual of chain-cycle polynomials

Key Result

Theorem 2.2.1

Let $Z_f$ be a quasi-smooth weighted hypersurface in $\mathbb{C P}(\mathbf{w})$ corresponding to the weighted homogenous polynomial $f$ of degree $d$ and weight vector $\mathbf{w}=\left(w_0, \ldots, w_n\right)$. Let $\mathfrak{A u t}\left(Z_f\right)$ denote the group of automorphisms of $Z_f$. Assum Furthermore, if the index $I=|\mathbf{w}|-d>0$ and $Z_f$ admits a Kähler-Einstein metric for a gene

Theorems & Definitions (43)

  • Theorem 2.2.1
  • Theorem 2.2.2
  • Lemma 3.1.1
  • Lemma 3.1.2
  • proof
  • Lemma 3.1.3
  • proof
  • Remark 3.1.1
  • Theorem 3.1.1
  • proof
  • ...and 33 more