[Paper Review] Extremal Kähler metrics

Gábor Székelyhidi|arXiv (Cornell University)|May 19, 2014

Geometry and complex manifolds39 references21 citations

TL;DR

This paper surveys recent progress on Calabi's extremal Kähler metrics, establishing a conjectural link between the existence of such metrics and algebro-geometric stability (K- and $̂{K}$-stability). It proves equality in the Calabi functional's infimum formula for ruled surfaces via test-configurations, showing that minimizing sequences decompose into complete extremal metrics or collapsing fibrations, with a limiting filtration achieving maximal destabilization.

ABSTRACT

This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.

Motivation & Objective

• To clarify the relationship between the existence of extremal Kähler metrics and algebro-geometric stability conditions such as K-stability and $̂{K}$-stability.
• To investigate the behavior of minimizing sequences for the Calabi functional when no extremal metric exists.
• To understand the geometric decomposition of Kähler manifolds under instability, analogous to 3-manifold geometrization.
• To establish equality in the infimum formula for the Calabi functional via test-configurations and their limits.

Proposed method

• Uses the Calabi functional $\mathrm{Cal}(\omega) = \int_M (S(\none) - \underline{S})^2 \omega^n$ to measure deviation from constant scalar curvature.
• Applies the momentum construction to explicitly construct minimizing sequences of metrics on ruled surfaces $M = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$ over genus 2 curves.
• Constructs sequences of test-configurations $\chi_i$ with increasing exponents to realize the infimum of the Calabi functional.
• Analyzes pointed limits of metrics to identify complete extremal metrics on $M \setminus S_0$ and $M \setminus S_\infty$, or collapsing circle fibrations.
• Introduces a limiting filtration $\chi$ as the limit of test-configurations $\chi_i$, which achieves the supremum in the stability inequality.
• Relies on the formula $\lim_{i\to\infty} \|S(\omega_i) - \underline{S}\|_{L^2} = \lim_{i\to\infty} -c_n \frac{\mathrm{Fut}(\chi_i)}{\|\chi_i\|}$ to prove equality in the stability conjecture.

Experimental results

Research questions

• RQ1 Under what algebro-geometric stability conditions does a Kähler class admit an extremal metric?
• RQ2 What happens to minimizing sequences of the Calabi functional when no extremal metric exists?
• RQ3 Can the infimum of the Calabi functional be realized via test-configurations, and does equality hold in the stability conjecture?
• RQ4 How do geometric decompositions or collapsing phenomena emerge in unstable cases?
• RQ5 Does a maximally destabilizing filtration exist in the limit of test-configurations, and what is its role in the stability conjecture?

Key findings

• For ruled surfaces $M = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$ over genus 2 curves, equality holds in the infimum formula (23) for all polarizations $L$.
• When $m < k_1$, minimizing sequences converge to an extremal metric in the Kähler class $\Omega_m$.
• For $k_1 \leq m \leq k_2$, pointed limits yield complete extremal metrics on $M \setminus S_0$ and $M \setminus S_\infty$, with volume additivity.
• When $m > k_2$, limits include collapsing circle fibrations and complete extremal metrics, with total volume strictly less than the original.
• The test-configurations $\chi_i$ have exponents tending to infinity, indicating increasingly complex degenerations into chains of copies of $M$.
• A limiting filtration $\chi$ exists as the limit of $\chi_i$, achieving the supremum in the stability inequality and playing a role analogous to the Harder-Narasimhan filtration.