[Paper Review] Kaehler-Einstein metrics on orbifolds and Einstein metrics on spheres

Alessandro Ghigi, Janós Kollár|ArXiv.org|Jul 14, 2005

Geometry and complex manifolds12 references4 citations

TL;DR

This paper generalizes a method for constructing Kähler-Einstein metrics on orbifolds via Galois coverings to identity maps with trivial Galois groups, enabling new Einstein metrics on odd-dimensional spheres and Kähler-Einstein metrics on degree 2 Del Pezzo surfaces with $A_1$ or $A_2$ singularities. The key advance is a factor-of-$n$ improvement in the existence bounds over prior work, yielding over $10^6$ new Einstein metrics on $S^9$.

ABSTRACT

A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metrics on degree 2 Del Pezzo surfaces with A_1 or A_2-singularities.

Motivation & Objective

• To generalize the Arezzo-Ghigi-Pirola method for constructing Kähler-Einstein metrics on manifolds via Galois coverings to the case of orbifolds with trivial Galois group.
• To apply this generalized method to identity maps $\pi: (\mathbb{P}^n, \Delta) \to \mathbb{P}^n$ that are nontrivial in the orbifold category, enabling new constructions of Kähler-Einstein metrics.
• To leverage the Kobayashi lifting theorem to obtain new Einstein metrics on odd-dimensional spheres from Kähler-Einstein orbifold metrics on $\mathbb{P}^n$ with $\mathbb{Q}$-divisors.
• To improve the existence bounds for orbifold Kähler-Einstein metrics on $\mathbb{P}^n$ with $n+2$ hyperplanes and $\mathbb{Q}$-divisors, achieving a factor-of-$n$ improvement over Boyer-Galicki-Kollár.
• To construct new examples of Kähler-Einstein metrics on degree 2 Del Pezzo surfaces with $A_1$ or $A_2$ singularities using the same method.

Proposed method

• Generalize the Galois covering method from manifolds to orbifolds, focusing on identity maps $\pi: (\mathbb{P}^n, \Delta) \to \mathbb{P}^n$ that are trivial set-theoretically but nontrivial as orbifold coverings.
• Use the orbifold structure defined by a $\mathbb{Q}$-divisor $\Delta = \sum_{i=0}^{n+1} (1 - \frac{1}{m_i}) D_i$, where $D_i$ are hyperplanes in general position in $\mathbb{P}^n$.
• Apply the existence criterion for orbifold Kähler-Einstein metrics based on the condition $1 < \sum_{i=0}^{n+1} \frac{1}{m_i} < 1 + (n+1) \min_i \frac{1}{m_i}$, which improves the bound from Boyer-Galicki-Kollár by a factor of $n$.
• Leverage the isomorphism between the weighted projective space $\mathbb{P}(w_0, \dots, w_{n+1})$ and $\mathbb{P}^{n+1}$ via $z_i \mapsto x_i = z_i^{m_i}$ to identify the orbifold $(X, \Delta_X)$ with $ (\mathbb{P}^n, \Delta) $.
• Use the Kobayashi lifting theorem to lift the orbifold Kähler-Einstein metric on $ (\mathbb{P}^n, \Delta) $ to an Einstein metric on the link $L(m_0, \dots, m_{n+1}) \subset \mathbb{C}^{n+2}$, which is diffeomorphic to $S^{2n+1}$.
• Apply the method to degree 2 Del Pezzo surfaces with $A_{n-1}$ singularities by analyzing the double cover $\pi: S \to \mathbb{P}^2$ ramified over a quartic curve and verifying integrability of $\eta^{-\lambda}$ for $\lambda \leq \frac{1}{2}$.

Experimental results

Research questions

• RQ1 Can the Arezzo-Ghigi-Pirola method for constructing Kähler-Einstein metrics via Galois coverings be extended to orbifolds with trivial Galois group?
• RQ2 Does the improved bound $1 < \sum \frac{1}{m_i} < 1 + (n+1) \min \frac{1}{m_i}$ guarantee the existence of Kähler-Einstein metrics on $ (\mathbb{P}^n, \sum (1 - \frac{1}{m_i}) D_i) $ with $n+2$ hyperplanes?
• RQ3 Can the resulting orbifold Kähler-Einstein metric on $ (\mathbb{P}^n, \Delta) $ be lifted to an Einstein metric on the sphere $S^{2n+1}$ via the link construction?
• RQ4 Do the improved bounds yield new examples of Einstein metrics on odd-dimensional spheres beyond those known from Boyer-Galicki-Kollár?
• RQ5 Can the method be applied to construct Kähler-Einstein metrics on singular Del Pezzo surfaces, such as degree 2 surfaces with $A_1$ or $A_2$ singularities?

Key findings

• The paper establishes a new existence criterion for orbifold Kähler-Einstein metrics on $ (\mathbb{P}^n, \sum_{i=0}^{n+1} (1 - \frac{1}{m_i}) D_i) $, requiring $1 < \sum_{i=0}^{n+1} \frac{1}{m_i} < 1 + (n+1) \min_i \frac{1}{m_i}$, which improves the bound from Boyer-Galicki-Kollár by a factor of $n$.
• For $m_0=2, m_1=3, m_2=5$, and $m_3 \in \{17,19,23,29,31,37,41,43,47,49,53,59\}$, the conditions yield 12 new Einstein metrics on $S^5$.
• The method produces at least $10^3$ new Einstein metrics on $S^7$, $10^6$ on $S^9$, and so on, for higher odd dimensions.
• The construction applies to degree 2 Del Pezzo surfaces with $A_1$ or $A_2$ singularities, proving they admit orbifold Kähler-Einstein metrics via the integrability of $\eta^{-\lambda}$ for $\lambda \leq \frac{1}{2}$.
• The method confirms the existence of orbifold Kähler-Einstein metrics on diagonalizable quartic Del Pezzo surfaces (degree 4), as previously shown by Mabuchi and Mukai, via a new proof using the orbifold Kähler-Einstein criterion.
• The isomorphism between the Brieskorn–Pham singularity link and $S^{2n+1}$, combined with the orbifold metric, ensures that the Einstein metric on the sphere is determined up to isometry by the $m_i$ values, except in rare cases with holomorphic contact structures when $n$ is odd.