[Paper Review] Stability of anti-canonically balanced metrics

Shunsuke Saito, Ryosuke Takahashi|arXiv (Cornell University)|Jul 19, 2016

Geometry and complex manifolds16 references3 citations

TL;DR

This paper establishes a slope formula for the quantized Ding functional along Bergman geodesic rays on Fano manifolds, linking its asymptotic behavior to Donaldson-Futaki invariants and Chow weights. It introduces a new algebro-geometric stability—F-stability—that is implied by the existence of anti-canonically balanced metrics, and proves a lower bound estimate for the $L^q$-norm of the deviation from Kähler-Einstein metrics.

ABSTRACT

We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existence of anti-canonically balanced metrics implies our stability. The relation between our stability and others is also discussed. As another application of the slope formula, we get the lower bound estimate on the Calabi like functionals on Fano manifolds.

Motivation & Objective

• To understand the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays on Fano manifolds.
• To define a new algebro-geometric stability (F-stability) based on the slope formula and its relation to balanced metrics.
• To establish a lower bound estimate for the $L^q$-norm of the function measuring deviation from Kähler-Einstein metrics.
• To clarify the relationship between F-stability, Chow stability, and other known stabilities such as K-stability.

Proposed method

• Derives the slope formula for the quantized Ding functional along Bergman geodesic rays using test configurations and quantized Futaki invariants.
• Introduces the quantized Futaki invariant $Fut_k(/mathcal{X},/mathcal{L})$ as a key invariant combining Donaldson-Futaki invariants and Chow weights.
• Defines F-polystability at level $k$ via the non-negativity of the slope formula, generalizing Chow stability.
• Applies Hölder's inequality and asymptotic analysis to estimate the $L^q$-norm of the function $B(\phi)$, measuring deviation from Kähler-Einstein metrics.
• Uses the $\mathbb{C}^*$-action on $H^0(X, -kK_X)$ to relate the infinitesimal generator $A$ to the slope of the functional.
• Employs the $p$-norm of test configurations and asymptotic expansion in $m$ to derive the lower bound on $||B(\phi)||_{L^q}$.

Experimental results

Research questions

• RQ1 How does the slope at infinity of the quantized Ding functional along Bergman geodesic rays relate to algebro-geometric invariants?
• RQ2 What is the precise relationship between the existence of anti-canonically $k$-balanced metrics and F-stability?
• RQ3 How does F-stability compare to Chow stability and other stability notions like K-stability?
• RQ4 Can the $L^q$-norm of the function $B(\phi)$ be bounded from below in terms of geometric invariants?
• RQ5 What is the role of the central fiber's singularities and $\mathbb{Q}$-Gorenstein condition in the slope formula?

Key findings

• The slope at infinity of the quantized Ding functional is given by $\frac{Fut_k(\mathcal{X},\mathcal{L})}{kN_k} - q$, where $q$ is a non-negative rational number depending on the central fiber.
• The quantity $q$ vanishes if and only if the test configuration is $\mathbb{Q}$-Gorenstein, $\mathcal{L} \cong -kK_{\mathcal{X}/\mathbb{C}}$, and the central fiber has at worst log terminal singularities.
• The existence of an anti-canonically $k$-balanced metric implies F-polystability at level $k$, establishing a link between analytic and algebro-geometric stability.
• Asymptotic Chow polystability implies asymptotic F-polystability, showing F-stability is a weaker condition than Chow stability.
• A lower bound on the $L^q$-norm of $B(\phi)$ is established: $||B(\phi)||_{L^q(\omega_\phi^n/n!)} \geq -DF(\mathcal{X},\mathcal{L}) + O(m^{-1})$ as $m \to \infty$, with $DF$ the Donaldson-Futaki invariant.
• The bound is derived via asymptotic analysis of the $p$-norm of test configurations and the $q$-norm of the matrix $\underline{M}(H)$, leading to a uniform lower bound in terms of the Donaldson-Futaki invariant.