Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sasaki-Einstein 7-manifolds and Orlik's conjecture

Jaime Cuadros, Joe Lope

Abstract

We calculate the homology groups of certain 2-connected 7-manifolds admitting quasi-regular Sasaki-Einstein metrics. These manifolds are links that arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities. Among these links, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres and 124 new examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected sums of $S^{3} \times S^{4}.$ Furthermore, we found that manifolds of the form $k \#\left(S^{3} \times S^{4}\right)$ admit Sasaki-Einstein metrics for 22 different values of $k.$ We also describe the diffeomorphism type of certain families of homotopy 9-spheres admitting positive Ricci curvature. These manifolds are branched covers of $S^{11}$ branched over Sasaki-Einstein rational homology 7-spheres.

Sasaki-Einstein 7-manifolds and Orlik's conjecture

Abstract

We calculate the homology groups of certain 2-connected 7-manifolds admitting quasi-regular Sasaki-Einstein metrics. These manifolds are links that arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities. Among these links, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres and 124 new examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected sums of Furthermore, we found that manifolds of the form admit Sasaki-Einstein metrics for 22 different values of We also describe the diffeomorphism type of certain families of homotopy 9-spheres admitting positive Ricci curvature. These manifolds are branched covers of branched over Sasaki-Einstein rational homology 7-spheres.
Paper Structure (9 sections, 7 theorems, 37 equations)

This paper contains 9 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Sasaki-Einstein metrics on links and Orlik's conjecture
  3. Links and Sasaki-Einstein metrics
  4. The Topology of Links and Orlik's Conjecture
  5. Results
  6. Positive Ricci curvature on homotopy spheres
  7. Rational homology 7-spheres admitting Sasaki-Einstein structures from Johnson-Kollár list
  8. Sasaki-Einstein 7-manifolds of the form $\#k(S^3\times S^4)$ from the Johnson-Kollár's list and Cheltsov's list
  9. appendix

Key Result

Theorem 2.1

The link $L_{f}(\mathbf{w}, d)$ admits a Sasaki-Einstein structure if and only if the Fano orbifold $\mathcal{Z}_{f}$ admits a Kähler-Einstein orbifold metric of scalar curvature $4 n(n+1).$

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Conjecture 2.3: Orlik
  • Lemma 2.4
  • proof
  • Theorem 3.1: Boyer et al. BGN2
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Theorem 3.3
  • ...and 2 more