Maximal K-Destabilizers

• Maximal K-destabilizers are unique non-Archimedean metrics that identify the most destabilizing direction in polarized algebraic varieties.
• They emerge from a variational maximization over geodesic rays, connecting non-Archimedean pluripotential theory and classical test-configuration methods.
• Their quantization bridges maximal Chow-destabilizers with Kähler geometry, offering explicit analytic and computational tools, especially in toric cases.

A maximal K-destabilizer is a distinguished non-Archimedean metric or “direction” in the infinite-dimensional space of Kähler metrics, which most strongly evidences the K-unstability of a polarized algebraic variety. These objects play a central role in the paper of K-stability—a notion from algebraic and differential geometry characterizing the existence of canonical Kähler metrics—by quantifying and organizing the ways in which a variety fails to admit such metrics. Maximal K-destabilizers refine and deepen the classical theory of test-configurations and connect variational, non-Archimedean, and combinatorial approaches to stability. Recent developments show that maximal K-destabilizers also arise as the metric limits of maximal Chow-destabilizers, providing a quantized bridge between Geometric Invariant Theory (GIT) and differential geometry (Yao, 20 Nov 2025).

1. K-Stability, Test-Configurations, and Non-Archimedean K-Energy

Let $(X,L)$ denote an $n$-dimensional polarized projective manifold over $\C$. An ample test-configuration for $(X,L)$ consists of a flat family $\pi\colon\X\to\C$ equipped with a $\C^*$-action compatible with the standard action on the base, together with a relatively ample $\Q$-line bundle $\L$ restricting to $(X,L)$ over $t\neq0$. The associated central fiber $(\X_0, \L_0)$ encodes degenerations of $(X,L)$ which can be used to test for K-stability.

For a test-configuration, denote by $w_k$ the total weight of the $\C^*$-action on $H^0(\X_0, \L_0^k)$, and $N_k = \dim H^0(\X_0, \L_0^k)$. The expansion

$\frac{w_k}{kN_k} = \Ena(\X, \L) - \frac{1}{2k}\DF(\X, \L) + O(k^{-2})$

relates algebro-geometric and non-Archimedean invariants: $\Ena$ (non-Archimedean Monge–Ampère energy) and $\DF$ (Donaldson–Futaki invariant). K-semistability requires $\DF(\X, \L)\ge0$ for all test-configurations; otherwise, $(X,L)$ is K-unstable.

Boucksom–Jonsson non-Archimedean pluripotential theory associates to each ample test-configuration a continuous plurisubharmonic (psh) metric $\phi$ on the Berkovich analytification $L^{an}$, extending fundamental energies:

• Monge–Ampère: $\Ena \colon \PSH(L) \to \mathbb{R}\cup\{-\infty\}$
• K-energy: $\Mna \colon \E^1(L) \to (-\infty, \infty]$

For test-configuration metrics, one recovers

$\Mna(\phi_{\X, \L}) = \DF(\X, \L) - V^{-1}(\X_0 - \X_{0,\mathrm{red}})\cdot \L^n \leq \DF(\X, \L)$

giving a non-Archimedean perspective on destabilization (Yao, 20 Nov 2025).

2. Variational Characterization and Existence of Maximal K-Destabilizers

The maximal K-destabilizer is defined via a variational maximization over geodesic rays in the Hadamard space $\E^2(L)$ (the space of finite-energy metrics with the $d_2$ metric). For an $L^2$-geodesic ray $\ell = \{\ell_t\}_{t\ge 0}$, the radial K-energy is: $\Mrad(\ell) = \lim_{t\to \infty} \frac{1}{t} \M(\ell_t)$

A theorem of Xia [“Xia '21”] states: $\inf_{\phi\in \E^2(L)}\sqrt{\Cal(\phi)} = \sup_{\ell \in \R^2(L)} \frac{-\Mrad(\ell)}{\|\ell\|_2}$ When the supremum is negative (the geodesically K-unstable case), there exists, up to time scaling and asymptotic equivalence, a unique geodesic ray $\ell^{K}$ maximizing $-\Mrad(\ell)/\|\ell\|_2$. The corresponding non-Archimedean metric

$\phi^K = \limNA(\ell^K)\in \E^1(L)$

is the maximal K-destabilizer, unique up to positive scaling (Yao, 20 Nov 2025).

The maximal K-destabilizer is thus the metric, or filtration, which realizes the steepest descent for the non-Archimedean K-energy, encapsulating the “direction” in which K-unstability is most severe.

3. Boucksom–Jonsson Non-Archimedean Framework and Convex Analysis

Within the Boucksom–Jonsson framework:

• $\PSH(L)$ is the non-Archimedean space of psh metrics, materially the “envelope” of rational test-configuration metrics.
• The subset $\E^1(L)$, where $\Ena(\phi) > -\infty$, is a complete metric space $(\E^1(L), d_1)$, with $\E^2(L)$ a $CAT(0)$ subspace.
• The non-Archimedean K-energy $\Mna$ extends as a continuous, convex, and homogeneous functional on $\E^1(L)$, convex along NA geodesics.
• Steepest descent directions for $\Mna$ correspond to maximal destabilizers, supported by joint work of Berman–Boucksom–Darvas–Lempert–Jonsson which ensures that the radial K-energy of geodesic rays coincides with the NA K-energy on their limits (Yao, 20 Nov 2025).

The maximal K-destabilizer is thus uniquely characterized as minimizing the penalized energy functional

$\breve\Mna(\phi) = \Mna(\phi) + \tfrac12 \|\phi\|_2^2 \quad(\phi \in \E^2(L))$

attained for the “optimal” (most destabilizing) metric.

4. Quantization and Relation to Maximal Chow-Destabilizers

In the field of GIT, destabilization is characterized by maximally destabilizing filtrations (or norms) on spaces of global sections, known as maximal Chow-destabilizers. When $(X,L)$ is K-unstable, its projective embedding $X \hookrightarrow \mathbb{P}H^0(X, kL)^*$ becomes Chow-unstable for large $k$. Kempf–Rousseau theory produces a unique (up to scaling) maximal Chow-destabilizer $\chi_k^*\in \mathcal{N}(H^0(X,kL))$ minimizing a renormalized Chow weight $M_{X,k}(\chi)$, subject to norm normalization.

Boucksom–Jonsson established tools for relating these finite-level (quantized) destabilizers to non-Archimedean metrics via the NA Fubini–Study and sup-norm maps:

• $\mathrm{FS}_k\colon\mathcal{N}(H^0(X, kL)) \to \CPSH(L)$
• $\mathrm{SN}_k\colon\CPSH(L) \to \mathcal{N}(H^0(X, kL))$ providing an intertwining structure.

The main quantization theorem states that, under natural continuity hypotheses,

$$\phi_k\,=\,\mathrm{FS}_k(\chi_k^*)\longrightarrow \phi^{\mathrm{K}}$$

in the strong NA topology as $k\to\infty$ (Yao, 20 Nov 2025). Thus, the maximal K-destabilizer is the infinite-level limit of maximal Chow-destabilizers, cementing a fundamental bridge between algebro-geometric and pluripotential analytic approaches.

5. Central Proof Techniques and Estimates

The existence and characterization of the maximal K-destabilizer proceed from:

• Reformulating the problem as minimization of the penalized K-energy functional $\breve\Mna$ over $\E^2(L)$.
• Constructing quantized K-energy functionals

$\mathcal{Q}_k(\phi) = 2k(\Ena(\phi) - E_k \circ \mathrm{SN}_k(\phi)) \approx \Mna(\phi),$

where $E_k(\cdot)$ is the average of jumping numbers.

• Demonstrating that minimization of

$$\mathcal{Q}_k(\phi) + \frac{1}{2k^2}\|\mathrm{SN}_k(\phi)\|_2^2$$

over $\CPSH(L)$ pulls back to the Chow weight problem in $\mathcal{N}(H^0(X, kL))$, uniquely solved by $\chi_k^*$.

• Passing to the $k\to\infty$ limit, one recovers the unique maximizer of $-\Mrad(\ell)/\|\ell\|_2$.

This program rigorously connects finite-level (quantized) algebraic stability criteria to infinite-dimensional pluripotential optimality (Yao, 20 Nov 2025).

6. The Toric Case and Explicit Constructions

In the toric case $(X, L)$, with moment polytope $P\subset M_\mathbb{R}$:

• Attention can be restricted to $T_\C$-invariant norms and metrics.
• Maximal Chow-destabilizers correspond to convex piecewise–linear functions on $P$ minimizing discrete secondary-polytope weights.
• The maximal K-destabilizer is the optimum of Székelyhidi’s variational problem: among convex $f\colon P\to\mathbb{R}$,

$$\frac{\mathcal{L}(f)}{\|f\|_{L^2(P)}},\qquad \mathcal{L}(f) = \int_P f\,dx - \int_{\partial P}f\,d\sigma.$$

• The quantization result recovers the classical optimal toric test-configuration as $k\to\infty$.

A plausible implication is that in the toric regime, maximal K-destabilizers admit explicit convex analytic representation, rendering them amenable to combinatorial and computational methods (Yao, 20 Nov 2025).

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In summary, maximal K-destabilizers are uniquely determined non-Archimedean metrics signifying the “steepest descent” toward K-unstability, and their deep relationship with maximal Chow-destabilizers highlights the unity of GIT, variational, and metric approaches in modern stability theory. The quantization principle ensures that foundational stability problems in Kähler geometry can be studied at both algebraic and analytic levels, linked via the structure of maximal destabilization.