[Paper Review] Kähler-Einstein metrics with cone singularities on klt pairs

Henri Guenancia|arXiv (Cornell University)|Dec 6, 2012

Geometry and complex manifolds13 references4 citations

TL;DR

This paper establishes the existence of Kähler-Einstein metrics with cone singularities along the divisor $D$ on klt pairs $(X,D)$ when $K_X + D$ is big or $-(K_X + D)$ is ample. Using regularization and Monge-Ampère theory in big cohomology classes, it proves that the Kähler-Einstein metric exhibits cone singularities along $D_{\geq 1/2}$ over the log-smooth locus intersected with the ample locus, extending known results to the general klt setting with coefficient constraints.

ABSTRACT

Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the negatively curved case).

Motivation & Objective

• To extend the existence of Kähler-Einstein metrics with cone singularities to general klt pairs beyond log-smooth or ample cases.
• To analyze the singular behavior of Kähler-Einstein metrics near the support of $D$ on the log-smooth locus of $X$.
• To establish cone singularity behavior in the case where $K_X + D$ is big or $-(K_X + D)$ is ample, under coefficient constraints in $[1/2, 1)$.
• To generalize prior results on edge metrics to klt pairs using big cohomology classes and pluripotential theory.
• To provide a unified framework for cone singularities in Kähler-Einstein metrics on singular varieties via regularization and approximation techniques.

Proposed method

• Utilizes the theory of Monge-Ampère equations in big cohomology classes, particularly the non-pluripolar product and finite energy currents.
• Applies regularization techniques to approximate singular metrics and control their Laplacian growth.
• Employs a Zariski decomposition via the minimal model program to reduce the problem to the log-canonical model where the metric is better understood.
• Uses approximation of the cone metric by smooth metrics in big cohomology classes to pass to the limit.
• Relies on the work of Boucksom on the ample locus and the finite generation of the canonical ring for klt pairs.
• Applies results from pluripotential theory, including the uniqueness and existence theorems for solutions to degenerate Monge-Ampère equations.

Experimental results

Research questions

• RQ1 Under what conditions do Kähler-Einstein metrics on klt pairs exhibit cone singularities along the divisor $D$?
• RQ2 How does the singular behavior of the Kähler-Einstein metric near $\mathrm{Supp}(D)$ depend on the coefficients of $D$?
• RQ3 Can the cone singularity result be extended from log-smooth pairs to general klt pairs when $K_X + D$ is big or $-(K_X + D)$ is ample?
• RQ4 What role does the ample locus $\mathrm{Amp}(K_X + D)$ play in the regularity and singular behavior of the Kähler-Einstein metric?
• RQ5 How does the regularization and approximation process in big cohomology classes ensure convergence to a solution with cone singularities?

Key findings

• The Kähler-Einstein metric on a klt pair $(X,D)$ with $K_X + D$ big has cone singularities along $D_{\geq 1/2}$ over $\mathrm{LS}(X,D_{\geq 1/2}) \cap \mathrm{Amp}(K_X + D) \setminus \mathrm{Supp}(D_{<1/2})$.
• When $-(K_X + D)$ is ample, any Kähler-Einstein metric has cone singularities along $D_{\geq 1/2}$ on $\mathrm{LS}(X,D_{\geq 1/2}) \setminus \mathrm{Supp}(D_{<1/2})$.
• The result generalizes previous work on edge metrics by Brendle, Campana-Guenancia-Păun, and Jeffres-Mazzeo-Rubinstein to the klt setting with coefficient constraints.
• The solution to the Monge-Ampère equation $\mathrm{MA}(\varphi) = e^{\varphi}\mu$ in big cohomology classes is shown to have finite energy and induce a metric with cone singularities.
• The regularization process ensures that the limit metric satisfies the Kähler-Einstein condition and inherits the cone structure from the approximating sequence.
• The proof relies on the Zariski decomposition of $\mu^* (K_X + D)$ via the minimal model program, reducing the problem to the log-canonical model where the metric is smooth outside the singular locus and $\mathrm{Supp}(D_{\mathrm{can}})$.