[Paper Review] On the existence of conic Kahler-Einstein metrics

Gang Tian, Feng Wang|arXiv (Cornell University)|Mar 29, 2019

Geometry and complex manifolds18 references13 citations

TL;DR

This paper establishes the existence of conic Kähler-Einstein metrics on log Fano manifolds under log K-polystability, using a variational approximation method to overcome the lack of semi-ampleness in the conical divisors. The key result confirms the converse of the log K-stability condition for conic Kähler-Einstein metrics, extending the Yau-Tian-Donaldson conjecture to the conical setting with multiple divisors and general cone angles.

ABSTRACT

In this paper, we prove the conic version of YTD conjecture on log Fano manifolds.

Motivation & Objective

• To prove the existence of conic Kähler-Einstein metrics on log Fano manifolds with multiple divisors and general cone angles.
• To establish the converse of the log K-polystability condition for conic Kähler-Einstein metrics, generalizing the Yau-Tian-Donaldson conjecture to the conical case.
• To overcome the technical challenge of non-semi-ample divisors in conical metric approximation by using a variational method instead of the continuity method.
• To extend the partial $C^0$ estimate and apply Cheeger-Colding theory to the singular limit of approximating metrics.
• To show that log Futaki invariants vanish under log K-polystability, ensuring the existence of a solution to the conical Kähler-Einstein equation.

Proposed method

• Approximate the conic Kähler-Einstein metric using a family of smooth metrics with Ricci curvature bounded from below, avoiding reliance on semi-ampleness of divisors.
• Use a variational method to construct a sequence of metrics satisfying the conical Kähler-Einstein equation, replacing the continuity method used in prior works.
• Apply Cheeger-Colding and Cheeger-Colding-Tian theory to analyze the Gromov-Hausdorff limit of the approximating metrics.
• Establish a partial $C^0$ estimate in the conical setting, adapting techniques from Tian's earlier work despite non-uniform cone angles approaching $2eta_i$ rather than $2eta_i \to 2\pi$.
• Use a one-parameter family of metrics $\omega_s$ and analyze the evolution of the functional $\int_M X(f_s) \omega_s^n$ to show it is constant, implying the existence of a solution.
• Prove that the log Futaki invariant vanishes under log K-polystability, using the constancy of the functional and the vanishing of the limit at $s=1$.

Experimental results

Research questions

• RQ1 Does log K-polystability imply the existence of a conic Kähler-Einstein metric with cone angles $2\pi\beta^i$ along each divisor $D_i$ in a log Fano manifold?
• RQ2 Can the continuity method be replaced by a variational approach when the conical divisors are not semi-ample?
• RQ3 How can the partial $C^0$ estimate be adapted in the conical setting when cone angles do not approach $2\pi$?
• RQ4 What is the role of the log Futaki invariant in the existence of conic Kähler-Einstein metrics, and how does it vanish under log K-polystability?
• RQ5 Can the Gromov-Hausdorff limit of approximating metrics recover a weakly conic Kähler-Einstein metric, and what does this imply for the existence result?

Key findings

• The main result confirms that if $(M, \sum_{i=1}^k (1 - \beta^i) D_i)$ is log K-polystable, then there exists a conic Kähler-Einstein metric with angle $2\pi\beta^i$ along each $D_i$.
• The variational method successfully replaces the continuity method in the absence of semi-ampleness, enabling the construction of approximating metrics.
• The partial $C^0$ estimate is established in the conical setting, despite the challenge that cone angles do not tend to $2\pi$.
• The functional $\int_M X(f_s) \omega_s^n$ is shown to be constant in $s$, implying the existence of a solution to the conical Kähler-Einstein equation.
• The log Futaki invariant vanishes under log K-polystability, which is a necessary condition for the existence of a conic Kähler-Einstein metric, and is shown to be sufficient via the variational approach.
• The solution is obtained by proving that the set $I$ of $T \in [0,1]$ for which the metric exists is both open and closed, hence $I = [0,1]$, implying existence at $T=1$.