[Paper Review] Cheeger-Colding-Tian theory for conic Kahler-Einstein metrics

Gang Tian, Feng Wang|arXiv (Cornell University)|Jul 19, 2018

Geometry and complex manifolds7 references4 citations

TL;DR

This paper extends the Cheeger-Colding-Tian theory of tangent cones and singular strata to conical Kähler-Einstein metrics on Fano varieties with normal crossing divisors. By adapting Cheeger-Colding's Gromov-Hausdorff limit arguments to conical singularities, it establishes the existence of tangent cones and stratification of singularities, showing that the singular set has Hausdorff dimension at most $2n-2$, with cone angles determined by the limit of cone angles along the divisor components.

ABSTRACT

In this paper is to extend the Cheeger-Colding Theory to the class of conic Kahler-Einstein metrics. This extension provides a technical tool for [LTW] in which we prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with certain singularity.

Motivation & Objective

• To extend the Cheeger-Colding-Tian theory of tangent cones and singular strata to the setting of conical Kähler-Einstein metrics on Fano varieties with normal crossing divisors.
• To establish the existence of tangent cones at every point of the Gromov-Hausdorff limit space of such metrics, even in the presence of conical singularities.
• To prove that the singular set in the limit space is stratified with dimension at most $2n-2$, generalizing the classical result for Ricci-bounded Riemannian manifolds.
• To characterize the cone angles in the tangent cones in terms of the limit of the original cone angles along the divisor components.
• To provide a technical foundation for the Yau-Tian-Donaldson conjecture in the singular Fano case, particularly for varieties with controlled singularities.

Proposed method

• Adapts the Cheeger-Colding approach to Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below to the conical Kähler-Einstein setting.
• Uses a class of Riemannian manifolds with singularities defined by a set $\mathcal{S}$ of zero $n$-dimensional Hausdorff measure, where the metric is smooth on the regular part $\mathcal{R}$.
• Employs cut-off functions $\gamma_\epsilon$ with controlled $L^2$-norm of the gradient to handle the singular set in the approximation process.
• Applies the Bochner formula and harmonic approximation techniques in annular regions to control the gradient of distance-like functions.
• Uses the volume comparison and curvature estimates from conical Kähler-Einstein metrics to control the limit geometry.
• Applies Stokes' theorem and holonomy arguments on circle bundles over Riemann surfaces to relate the Euler characteristic and cone angles in the limit.

Experimental results

Research questions

• RQ1 Do tangent cones exist at every point of the Gromov-Hausdorff limit space of conical Kähler-Einstein metrics with Ricci curvature bounded below?
• RQ2 What is the stratification of the singular set in the limit space, and what is the maximal possible Hausdorff dimension of the singular strata?
• RQ3 How are the cone angles of the tangent cones related to the original cone angles of the conical Kähler-Einstein metrics?
• RQ4 Can the Cheeger-Colding-Tian theory of tangent cones and singular strata be extended to the conical Kähler-Einstein setting?
• RQ5 What is the topological and geometric structure of the limit space under bounded Ricci curvature and volume, with conical singularities?

Key findings

• For any sequence of conical Kähler-Einstein metrics in $\mathcal{M}(n,k,\delta,V)$, the Gromov-Hausdorff limit space $X$ admits tangent cones at every point, which are metric cones.
• The singular set $\mathcal{S}$ in the limit space satisfies $\mathcal{S} = \mathcal{S}_{2n-2}$, and each stratum $\mathcal{S}_k$ has Hausdorff dimension at most $k$.
• The cone angle $1 - \bar{\beta}$ of a tangent cone $T_xX \cong \mathbb{C}_{\bar{\beta}} \times \mathbb{R}^{2n-2}$ is given by $1 - \bar{\beta} = \sum m_i (1 - \beta_\infty^i)$, where $\beta_\infty^i$ are the limits of the original cone angles.
• The limit space $X$ of such metrics lies in the class $\mathcal{M}(V, \pi\sqrt{(2n-1)/\delta}, 2n)$, ensuring uniform geometric control.
• The existence of tangent cones is established via a harmonic approximation argument in annular regions, using cut-off functions with controlled gradient energy.
• The cone angle formula is derived from a topological argument involving the Euler characteristic of Riemann surfaces and the curvature of line bundles over singular fibers.