Kähler-Einstein Cone Metrics

• Kähler-Einstein metrics with cone singularities are Kähler metrics on complex varieties that satisfy modified Einstein equations incorporating divisor terms to impose prescribed conical angles.
• They utilize pluripotential theory, the complex Monge–Ampère formulation, and the Minimal Model Program to establish existence, uniqueness, and regularity on klt pairs.
• These metrics have significant applications in complex differential geometry and moduli theory, exemplified by models like the conical Fubini–Study metric on projective spaces.

A Kähler-Einstein metric with cone singularities is a Kähler metric on a complex algebraic variety, possibly singular, that satisfies the Einstein condition $\operatorname{Ric}(\omega) = \lambda \omega + [D]$ (for an effective $\mathbb{Q}$-divisor $D$ encoding the cone's location and angle) and is locally equivalent near the divisor to a model metric with prescribed cone angles. The modern theory, building on methods from pluripotential theory, complex differential geometry, and the minimal model program, has systematically extended existence and regularity results to encompass klt pairs, log-resolutions, and cases where $D$ has simple or even normal crossings.

1. Definitions, Local Models, and Singular Structures

Let $X$ be a normal complex projective variety with $\dim X = n$ and let $D = \sum a_i D_i$ be an effective $\mathbb{Q}$-divisor. The pair $(X, D)$ is called Kawamata log-terminal (klt) when $0 \leq a_i < 1$ for each $\mathbb{Q}$0, $\mathbb{Q}$1 is $\mathbb{Q}$2-Cartier, and for any (hence every) log-resolution $\mathbb{Q}$3,

$\mathbb{Q}$4

The log-smooth locus $\mathbb{Q}$5 is the set of $\mathbb{Q}$6 where $\mathbb{Q}$7 is smooth and $\mathbb{Q}$8 has simple normal crossings---locally, $\mathbb{Q}$9.

A Kähler-Einstein metric with cone singularities of angle $D$0 along each divisor $D$1 ($D$2) is, on $D$3, a Kähler metric $D$4 such that in coordinates where $D$5,

$D$6

in the sense that on compact subsets, $D$7 for some $D$8.

The ample locus $D$9 for a big $D$0-Cartier divisor $D$1 is the largest Zariski open subset where $D$2 admits a smooth Kähler representative.

2. Existence and Uniqueness Results

Let $D$3 be a klt pair with coefficients $D$4 and log-smooth locus $D$5. The main theorems are:

• If $D$6 is big (or ample), there exists a unique, negatively curved Kähler-Einstein metric $D$7 in $D$8 satisfying

$D$9

and, on $X$0, $X$1 has cone singularities of angle $X$2 ($X$3) along each $X$4.

• If $X$5 is ample, there exists a unique, positively curved Kähler-Einstein metric in $X$6 with

$X$7

and the same cone behavior on $X$8.

A similar Ricci-flat statement applies when $X$9 and a nef and big polarization is chosen.

Uniqueness holds in each Kähler class due to the comparison principle for Monge–Ampère equations. For the classical example of $\dim X = n$0, $\dim X = n$1 a union of hyperplanes with equal weights, one recovers the Fubini-Study metric with conical behavior along those hyperplanes (Guenancia, 2012).

3. Analytic Framework: Monge–Ampère Formulation and Regularization

The analytic foundation relies on reformulating the Kähler-Einstein metric search as a non-pluripolar complex Monge–Ampère equation. For a smooth representative $\dim X = n$2 of $\dim X = n$3 and defining sections $\dim X = n$4 for $\dim X = n$5, set

$\dim X = n$6

where $\dim X = n$7 and $\dim X = n$8 is a smooth volume form. This is distributionally equivalent to

$\dim X = n$9

Regularization involves replacing singular weights with smooth approximations, applying either the continuity path or variational minimization methods, and obtaining uniform $D = \sum a_i D_i$0 (via Kołodziej-type $D = \sum a_i D_i$1 estimates with $D = \sum a_i D_i$2 guaranteed by $D = \sum a_i D_i$3), Laplacian and higher regularity estimates (Evans-Krylov, Schauder) away from the divisors. Passage to the limit yields a metric that is smooth on $D = \sum a_i D_i$4 and shows precisely quasi-isometric behavior to the model cone metric near each divisor chart (Guenancia, 2012).

4. Extensions, Examples, and Variants

• The regularity and existence theory for cone singularities extends to normal crossing and self-intersecting divisors. For such $D = \sum a_i D_i$5, after reduction via étale neighborhoods to a setting with simple normal crossings, one invokes the regularity theory for klt pairs to conclude the existence of genuine cone singularities (Guenancia, 2015).
• Threshold for coefficients: The restriction $D = \sum a_i D_i$6 (i.e., $D = \sum a_i D_i$7) is technical, ensuring $D = \sum a_i D_i$8-integrability of the right-hand side so that global $D = \sum a_i D_i$9 estimates can be established; potential for reduction of this bound via sharper PDE techniques is noted in the literature (Guenancia, 2012).
• Limits and special geometries: In the case $\mathbb{Q}$0, Ricci-flat cone singular Kähler metrics exist in each Kähler class. On $\mathbb{Q}$1 with $\mathbb{Q}$2 a union of hyperplanes, the classical Fubini-Study metric exhibits such cone behavior and serves as a model (Guenancia, 2012).
• Relationship to vanishing/parallelism of holomorphic tensors: Under these metric backgrounds, Bochner techniques demonstrate vanishing and extension properties of holomorphic tensor fields, generalizing theorems of Lichnerowicz and Kobayashi (Campana et al., 2011).

5. Historical Context and Relation to Broader Theory

• For smooth divisors $\mathbb{Q}$3 and $\mathbb{Q}$4 ample, the cone singularity construction was established by Brendle, Jeffres–Mazzeo–Rubinstein, and Campana–Guenancia–Păun under $\mathbb{Q}$5; Mazzeo–Rubinstein (unpublished/announced) extended to all $\mathbb{Q}$6.
• The approach adopted here generalizes to Kähler-Einstein metrics on varieties with log-terminal singularities and allows for ampleness to be replaced by bigness, leveraging techniques developed in the Minimal Model Program (MMP) and non-pluripolar complex Monge–Ampère equations as in BEGZ.
• These structures underpin much of the recent progress in the field, from moduli theory to the study of K-stability and the Yau–Tian–Donaldson conjecture.

6. Proof Strategy and Functional Analytic Approach

• Reduction to smooth models: When $\mathbb{Q}$7 is merely big, one uses the MMP or Zariski decomposition to reduce to a setup with a projective manifold $\mathbb{Q}$8, a semipositive and big class, and a simple normal crossing boundary. Existence and uniqueness then follows from full Monge–Ampère mass theory (Guenancia, 2012).
• Continuity path or variational method: The singular data and cohomology class are approximated by smooth positive data. Solutions to the regularized Monge–Ampère equation are obtained by the continuity method or variational minimization.
• A priori estimates: Kołodziej-type $\mathbb{Q}$9 bounds lead to uniform $(X, D)$0 estimates; Laplacian estimates follow from control of bisectional curvature (for $(X, D)$1). Evans–Krylov and Schauder estimates provide higher regularity.
• Passage to the limit: Compactness and local smoothness away from divisors yield convergence to a legitimate Kähler–Einstein metric with the prescribed cone singularity structure.

7. Context, Limitations, and Future Directions

• Ampleness and angle bounds: The theory as proven requires the ample (or big) class and the restriction on coefficients ($(X, D)$2), both technical necessities for the analytic approach. Open problems remain in further lowering the angle bound, treating the full log-canonical setting for $(X, D)$3, and in specifying finer boundary regularity in the presence of higher-codimension or more general singular sets.
• Connections to moduli and metric compactness: The metric structures furnished here are central in the investigation of the compactness properties of moduli spaces of Kähler-Einstein varieties with prescribed degenerations, as well as the fine structure of metric tangent cones and Gromov-Hausdorff limits.

References:

Guenancia, H., "Kähler–Einstein metrics with cone singularities on klt pairs" (Guenancia, 2012); Campana, F., Guenancia, H., Păun, M., "Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields" (Campana et al., 2011); Guenancia, H., "Kähler–Einstein metrics with conic singularities along self-intersecting divisors" (Guenancia, 2015).