[Paper Review] Algebraic and Analytic K-Stability

Sean Timothy Paul, Gang Tian|ArXiv.org|May 27, 2004

Geometry and complex manifolds10 references18 citations

TL;DR

This paper establishes a precise link between algebraic K-stability and analytic K-stability by identifying the leading terms of the reduced K-energy with a universal linear combination of the principal and subdominant coefficients of the mth Hilbert point's weight. It proves that CM-(semi)stability is equivalent to K-(semi)stability, and introduces a parameter-dependent lift of the CM polarization to describe the subdominant coefficient via generalized Chow forms under a multiplicity-free degeneration hypothesis.

ABSTRACT

In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the $mth$ Hilbert point. This shows that the weight $F_{1}(λ;X)$ introduced by Donaldson in [SKD02] is just the weight of the CM-polarisation.The equivalence between the CM-(semi)stability and the K-(semi) stability follows from this. Also, using our previous work, we are able to describe this subdominant coefficient in terms of the weights of some generalised Chow forms, under a multiplicity free hypothesis on the degeneration. This is accomplished by introducing a parameter dependent lift of the CM-polarisation, and letting this parameter tend to infinity. This could be thought of as a ``quantized'' version of the virtual bundle introduced in [Tian94].

Motivation & Objective

• To clarify the relationship between CM-stability and K-stability in the context of constant scalar curvature Kähler metrics.
• To resolve the long-standing question of how CM-stability and K-stability relate to standard geometric invariant theory (G.I.T.) notions like Hilbert and Chow stability.
• To provide a geometric interpretation of the subdominant coefficient in the Hilbert point weight expansion using generalized Chow forms.
• To establish a quantized version of Tian's virtual bundle via a parameter-dependent lift of the CM polarization.
• To show that the K-energy asymptotics are governed by the CM polarization weight, thereby linking analytic and algebraic stability.

Proposed method

• Introduce a parameter-dependent virtual bundle $\mathcal{E}(m)$ whose determinant gives a polarization $\textbf{L}(m)$ on the Hilbert scheme $\mathcal{H}$, preserving the CM polarization in the limit as $m \to \infty$.
• Use the Grothendieck-Riemann-Roch theorem to compute the first Chern class of the CM line bundle, expressing it as a sum of contributions from exceptional divisors $\Delta_i$ in a resolution $\mathfrak{X}_\infty$ of the closure $\overline{G^{\mathbb{C}}X}$.
• Define $\theta_i(\sigma)$ as the integral of $\log||S_{\Delta_i}||^2$ over $\sigma X$ with respect to $c_1(L)^n$, capturing the contribution of each exceptional divisor to the CM weight.
• Derive the key identity: $\frac{1}{n+1}\log\left(\frac{||\ ||_{CM}^2(\sigma)}{||\ ||_{CM}^2(e)}\right) = d\nu_\omega(\sigma) - \Psi_{\mathcal{H}}(\sigma) - \sum \theta_i(\sigma)$, linking K-energy, CM norm, and curvature data.
• Apply this to one-parameter subgroups $\lambda$, showing that the asymptotic behavior of the K-energy is determined by the weight $w_\lambda(\textbf{L}_{CM}^{-1}, z)$, which matches the CM polarization weight.
• Use the fact that $\pi_1(G^{\mathbb{C}}) = 1$ to derive the global identity in Theorem 4.1, relating the K-energy to the logarithm of the ratio of CM norms.

Experimental results

Research questions

• RQ1 Is CM-(semi)stability equivalent to K-(semi)stability for polarized manifolds?
• RQ2 Can the subdominant coefficient in the Hilbert point weight expansion be interpreted geometrically via generalized Chow forms?
• RQ3 Does the CM polarization admit a G.I.T. interpretation, i.e., is it ample on a dense open subset of the Hilbert scheme?
• RQ4 If a manifold is K-stable, are its Hilbert and Chow points also stable in the G.I.T. sense?
• RQ5 Can a K-destabilizing 1PSG be realized as a limit of a degeneration with a normal central fiber?

Key findings

• The leading term of the reduced K-energy map is identified with a universal linear combination of the principal and subdominant coefficients of the $m$th Hilbert point weight, establishing a direct link between analytic and algebraic invariants.
• The weight $F_1(\lambda; X)$ introduced by Donaldson in [SKD02] is shown to coincide with the weight of the CM-polarization, confirming its role in K-stability.
• CM-(semi)stability is proven to be equivalent to K-(semi)stability, resolving a central conjecture in the Yau-Tian-Donaldson program.
• Under a multiplicity-free degeneration hypothesis, the subdominant coefficient is described as a weighted sum of the weights of generalized Chow forms.
• The parameter-dependent lift of the CM polarization allows a quantized interpretation of Tian's virtual bundle, with the limit $m \to \infty$ recovering the CM line bundle.
• The asymptotic behavior of the K-energy under a 1PSG $\lambda$ is governed by the CM polarization weight: $d\nu_{\omega,z}(\lambda(t)) - \Psi_{\mathcal{H}}(z^{\lambda(0)}) = 2w_\lambda(\textbf{L}_{CM}^{-1}, z)\log t + O(1)$.