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

Jaime Cuadros Valle, Ralph R. Gomez, Joe Lope Vicente

TL;DR

This work investigates the existence of extremal Sasaki metrics on links arising from invertible polynomials by leveraging the Berglund-Hübsch transpose and a relative K-stability framework. By combining the generalized Lichnerowicz obstruction with BH duality, the authors construct explicit families of links whose Sasaki-Reeb cones are large yet entirely obstructed to extremal metrics, including homotopy and rational homology spheres. They develop chain-cycle and Thom-Sebastiani sum constructions to produce perturbations on BH duals that enlarge the cone while maintaining obstruction, yielding numerous new examples beyond Brieskorn-Pham polynomials. The results illuminate the stability (or lack thereof) of canonical Sasaki metrics under mirror-symmetry-type dualities and broaden the understanding of extremal metrics in Sasakian geometry with connections to the topology of links and mirror symmetry.

Abstract

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.

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

TL;DR

This work investigates the existence of extremal Sasaki metrics on links arising from invertible polynomials by leveraging the Berglund-Hübsch transpose and a relative K-stability framework. By combining the generalized Lichnerowicz obstruction with BH duality, the authors construct explicit families of links whose Sasaki-Reeb cones are large yet entirely obstructed to extremal metrics, including homotopy and rational homology spheres. They develop chain-cycle and Thom-Sebastiani sum constructions to produce perturbations on BH duals that enlarge the cone while maintaining obstruction, yielding numerous new examples beyond Brieskorn-Pham polynomials. The results illuminate the stability (or lack thereof) of canonical Sasaki metrics under mirror-symmetry-type dualities and broaden the understanding of extremal metrics in Sasakian geometry with connections to the topology of links and mirror symmetry.

Abstract

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.
Paper Structure (10 sections, 14 theorems, 192 equations)

This paper contains 10 sections, 14 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sasaki-Reeb cone and extremal Sasaki metrics
  4. Links of hypersurface singularities and Sasakian geometry
  5. Topology of links
  6. Berglund-Hübsch and the Sasaki-Reeb cone
  7. Extremal Sasaki metrics and Berglund-Hübsch transpose
  8. Examples derived from chain-cycle type singularities
  9. Examples derived from Thom-Sebastiani sums of chain type singularities
  10. Examples derived from Thom-Sebastiani sums of more general invertible polynomials

Key Result

Theorem 2.1

BGS Let $L_f(\mathbf{w}, d)$ be either a negative or null link of an isolated hypersurface singularity with underlying CR-structure $(\mathcal{D}, J)$. Then its Sasaki-Reeb cone $\mathfrak{t}^{+}(\mathcal{D}, J)$ is one-dimensional.

Theorems & Definitions (22)

  • Theorem 2.1
  • Lemma 2.2
  • Conjecture 2.3: Orlik
  • Example 1
  • Theorem 4.1: BvC
  • Lemma 4.2
  • Theorem 4.3
  • Example 2
  • Theorem 4.4
  • Lemma 4.5
  • ...and 12 more