[Paper Review] Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory

Toshiki Mabuchi|ArXiv.org|Jul 24, 2006

Geometry and complex manifolds6 references4 citations

TL;DR

This paper establishes the asymptotic equivalence between Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory (GIT) for polarized algebraic varieties. By analyzing the behavior of Hilbert and Chow invariants under increasing powers of a very ample line bundle, the authors prove that asymptotic Hilbert-stability implies asymptotic Chow-stability, and vice versa, under the condition that the automorphism group is trivial. The key result resolves a long-standing question on the relationship between these two stability notions.

ABSTRACT

In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.

Motivation & Objective

• To clarify the asymptotic relationship between Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory.
• To show that asymptotic Hilbert-stability implies asymptotic Chow-stability under the assumption that the automorphism group is trivial.
• To simplify and re-derive the asymptotic equivalence using a refined analysis of Hilbert weights and test configurations.
• To establish a foundational result for moduli theory by unifying two central stability concepts in algebraic geometry.

Proposed method

• Uses the natural action of $ G_{ au} = ext{SL}_{\mathbb{C}}(H^0(M, \mathcal{O}(L^\ell))) $ on the space of Chow and Hilbert invariants.
• Analyzes the Hilbert weight $ w_\lambda(k;\ell) $ associated with a one-parameter subgroup $ \lambda $, showing it is strictly increasing under the assumption of closed orbits.
• Applies test configurations and equivariant vector bundles to relate the behavior of sections over $ \mathbb{A}^1 $ to the stability of the central fiber.
• Employs a recursive construction of $ \mathbb{C}[s] $-generators for weight spaces to control the order of vanishing at $ s=0 $, linking it to the Hilbert weight.
• Uses the fact that $ \det(\lambda(t)) = 1 $ to ensure $ \sum n_i \alpha_i = 0 $, which is essential for the stability analysis.
• Leverages Fogarty's result that Chow-stability implies Hilbert-stability, and proves the converse in the asymptotic regime.

Experimental results

Research questions

• RQ1 Does asymptotic Hilbert-stability imply asymptotic Chow-stability for polarized varieties with trivial automorphism group?
• RQ2 Can the asymptotic equivalence between Chow-stability and Hilbert-stability be proven using a unified framework based on Hilbert weights?
• RQ3 What is the role of the isotropy subgroup $ \hat{G}_{\ell} $ in determining Chow-stability?
• RQ4 How do the weights of one-parameter subgroups relate to the stability of the orbit in the Hilbert and Chow settings?
• RQ5 Is the asymptotic limit of the Hilbert weight $ w_\lambda(\infty;\ell) $ always non-negative, and when is it positive?

Key findings

• The asymptotic limit $ w_\lambda(\infty;\ell) $ of the Hilbert weight exists and is positive if the orbit $ G_{\ell_i} \cdot f_{\ell_i,k_i} $ is closed for all $ i $, implying Chow-stability.
• If $ \hat{H} = \{1\} $, then $ G_{\ell} \cdot M_{\ell} $ is closed in $ W_{\ell}^* $, which is equivalent to Chow-stability.
• The polynomial Hilbert weight $ w_\lambda(k;\ell) $ is strictly increasing in $ k $ under the assumption of closed orbits, which implies positivity of the asymptotic limit.
• Asymptotic Hilbert-stability implies asymptotic Chow-stability, and vice versa, when the automorphism group is trivial.
• The proof relies on the existence of a non-decreasing sequence of integers $ \beta_{ij} $ such that $ \iota^*\tau(e_{ij}) = s^{\beta_{ij}} \sigma_{ij} $, which controls the vanishing order at $ s=0 $.
• The key technical step is showing that $ \sum n_i \alpha_i = 0 $, which follows from the determinant condition on $ \lambda \in \text{SL}(V_\ell) $, ensuring consistency in the weight decomposition.