[Paper Review] Einstein Metrics on Rational Homology Spheres

Charles P. Boyer, Krzysztof Galicki|ArXiv.org|Nov 20, 2003

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes the existence of continuous families of Sasakian-Einstein metrics on infinitely many simply connected rational homology spheres in all odd dimensions greater than 3, using branched covers of weighted homogeneous polynomials and Kähler-Einstein metrics on Fano orbifolds. The key contribution is constructing such metrics with arbitrary torsion in second homology and non-trivial moduli spaces, including exponential growth in the number of effective parameters with dimension.

ABSTRACT

We prove the existence of Sasakian-Einstein metrics on infinitely many rational homology spheres in all odd dimensions greater than 3. In dimension 5 we obain somewhat sharper results. There are examples where the number of effective parameters in the Einstein metric grows exponentially with dimension.

Motivation & Objective

• To establish the existence of Einstein metrics on nontrivial rational homology spheres, particularly in odd dimensions greater than 3.
• To extend known constructions of Sasakian-Einstein metrics beyond homogeneous or 3-Sasakian cases to include infinite families with continuous moduli.
• To provide a systematic method for constructing simply connected rational homology spheres with prescribed torsion in H₂ and Sasakian-Einstein metrics.
• To demonstrate that the number of effective real parameters in the moduli space of such metrics can grow exponentially with dimension.
• To classify and compute the Betti numbers and homology groups of links arising from branched covers of weighted homogeneous polynomials.

Proposed method

• Constructs rational homology spheres as links of weighted homogeneous polynomials F = z₀ᵏ + f(z₁,…,zₘ), where f is quasi-smooth and has an isolated singularity.
• Applies Theorem 3 to compute the order of the homology group Hₘ₋₁(L_F, ℤ) as kᵇₘ₋₂(L_f), where bₘ₋₂ is the (m−2)nd Betti number of the link of f.
• Uses the klt condition and Fano condition from Theorem 5 to ensure the existence of Kähler-Einstein metrics on the associated orbifold quotient X = L_F / S¹.
• Translates the existence of Kähler-Einstein metrics on the Fano orbifold X to Sasakian-Einstein metrics on the link L_F via the Sasaki-Einstein correspondence.
• Employs the Betti number formula from Milnor and Orlik to compute bₘ₋₂(L_f) as a sum over subsets of indices, enabling explicit computation of homology order.
• Computes the number of effective complex parameters μ via μ = h⁰(ℙ(𝐰), 𝒪(d)) − ∑ᵢ h⁰(ℙ(𝐰), 𝒪(wᵢ)), which determines the dimension of the moduli space of metrics.

Experimental results

Research questions

• RQ1 Can Sasakian-Einstein metrics be constructed on infinitely many simply connected rational homology spheres in odd dimensions greater than 3?
• RQ2 What conditions on the weights and degrees of a weighted homogeneous polynomial ensure the existence of a Kähler-Einstein metric on the associated Fano orbifold?
• RQ3 How does the number of effective real parameters in the moduli space of Sasakian-Einstein metrics scale with the dimension of the manifold?
• RQ4 What is the structure of the second homology group H₂(M, ℤ) for such rational homology spheres, and can it realize arbitrary torsion orders?
• RQ5 Can branched covers of Calabi-Yau hypersurfaces in weighted projective spaces yield rational homology spheres admitting Sasakian-Einstein metrics?

Key findings

• For every integer k > 2 that is relatively prime to 3 or 2, there exists a Sasakian-Einstein metric on a simply connected rational homology 5-sphere Mₖ⁵ with |H₂(Mₖ⁵, ℤ)| = k² and two real parameters in the moduli space.
• In dimension 7, branched covers of the Fermat-Calabi-Yau hypersurface z₀ᵏ + z₁⁴ + z₂⁴ + z₃⁴ = 0 yield rational homology spheres with |H₂| = k²¹ and 38 effective real parameters when k > 12.
• In dimension 9, the same construction yields rational homology spheres with |H₂| = k²⁰⁴ and 202 effective real parameters for k > 18.
• For the canonical case F = z₀ᵏ + z₁ˡ + ⋯ + zₘˡ = 0 with l = m+1 and k = m, the number of effective real parameters is 2(₂ₘCₘ₊₁ − m²), which grows exponentially with m.
• The link M₅³,₄ associated to z₀³ + z₁⁴ + z₂⁴ + z₃⁴ = 0 is a rational homology 5-sphere with |H₂| = 3⁶ = 729 and admits a 12-parameter family of Sasakian-Einstein metrics.
• The klt condition and Fano condition are satisfied for k in the range ( (m−1)l² / ((m−1)l(l−m)+m ) < k < l/(l−m) ), ensuring existence of the metrics.