[Paper Review] Nonexistence for complete Kähler Einstein metrics on some noncompact manifolds

Peng Gao, Shing‐Tung Yau+1 more|arXiv (Cornell University)|Mar 30, 2016

Geometry and complex manifolds11 references3 citations

TL;DR

This paper proves that complete Kähler–Einstein metrics do not exist on the complement of a subvariety of codimension ≥2 in a compact Kähler manifold, nor on the complement of an exceptional divisor that contracts to a canonical or terminal singularity. The result extends to klt pairs and relies on curvature and singularity analysis, resolving a long-standing question about canonical metrics on noncompact Kähler manifolds.

ABSTRACT

Let $M$ be a compact Kähler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ cannot admit any complete Kähler--Einstein metric if blow-down of $E$ is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.

Motivation & Objective

• To resolve the open problem of whether complete Kähler–Einstein metrics can exist on noncompact Kähler manifolds obtained by removing a subvariety of codimension ≥2 from a compact Kähler manifold.
• To investigate the existence of complete Kähler–Einstein metrics on the complement of an exceptional divisor in a birational morphism where the target has canonical or terminal singularities.
• To extend the nonexistence result to the setting of log pairs (X,D) with Kawamata log terminal (klt) singularities.
• To provide a geometric obstruction to the existence of such metrics based on curvature and singularity type, using analytic and algebraic geometry techniques.
• To apply the results to compactifications of locally symmetric spaces, showing that singular boundary components (cusps) are necessary for the existence of complete Kähler–Einstein metrics.

Proposed method

• Uses the Bonnet–Myers compactness theorem to rule out positive Ricci curvature Kähler–Einstein metrics on noncompact manifolds.
• Applies curvature estimates and asymptotic analysis of metrics with Poincaré–Mok–Yau (PMY) type behavior to show nonexistence of complete Kähler–Einstein metrics with nonpositive Ricci curvature.
• Employs resolution of singularities and log resolution techniques to reduce the problem to analyzing the singular locus of codimension 2.
• Leverages the notion of discrepancy and the definition of klt (Kawamata log terminal) singularities to characterize local singularity types.
• Applies Hartog’s extension theorem and Zariski’s main theorem to ensure that singularities of codimension ≥2 do not affect the canonical bundle structure.
• Uses the fact that klt pairs have quotient singularities in codimension 2 (via Lemma 4.3) to reduce the problem to a local model where nonexistence is established.

Experimental results

Research questions

• RQ1 Can a complete Kähler–Einstein metric exist on the complement of a subvariety of codimension ≥2 in a compact Kähler manifold?
• RQ2 Does the blow-up of a compact Kähler manifold along a subvariety of codimension ≥2 admit a complete Kähler–Einstein metric on the complement of the exceptional divisor?
• RQ3 Under what conditions on the singularities of the target variety can a complete Kähler–Einstein metric exist on the complement of an exceptional divisor in a birational morphism?
• RQ4 Does the klt (Kawamata log terminal) condition on a pair (X,D) obstruct the existence of complete Kähler–Einstein metrics on the complement of the exceptional divisor?
• RQ5 Can the nonexistence result be extended to the case of open manifolds arising from desingularization of Satake compactifications of locally symmetric spaces?

Key findings

• There are no complete Kähler–Einstein metrics on $M - N$ when $M$ is a compact Kähler manifold and $N$ is a subvariety of codimension ≥2.
• If $f: M o X$ is a birational morphism with $X$ having only canonical or terminal singularities and the singular locus of $X$ has codimension 2, then $M - E$ admits no complete Kähler–Einstein metric.
• For a complex surface $M$ with exceptional divisor $E$ contracting to an A-D-E singularity, $M - E$ admits no complete Kähler–Einstein metric.
• The nonexistence result extends to klt pairs: if $(X, D)$ is klt, then $M - E$ admits no complete Kähler–Einstein metric, where $E$ is the exceptional divisor of a resolution.
• The result implies that Satake compactifications of locally symmetric spaces must include singular boundary components (cusps), as otherwise complete Kähler–Einstein metrics would exist on the complement of the boundary, contradicting the nonexistence theorem.
• The curvature condition $- ho ext{Id} o ext{Ric} o 0$ with $ ho eq 0$ is incompatible with the existence of complete Kähler–Einstein metrics on $M - N$ for codimension ≥2 subvarieties $N$.