K-Stability Theory in Algebraic Geometry

• K-Stability theory is a rigorous framework in algebraic geometry that uses test configurations and invariants to characterize canonical metrics on Fano and polarized varieties.
• It employs valuation-theoretic and non-Archimedean invariants such as the β- and δ-invariants to establish quantitative stability criteria and uniform K-stability.
• Its wall-crossing and semi-algebraic chamber decomposition approach facilitates moduli space construction and connects to Geometric Invariant Theory for birational transformations.

K-stability theory is a rigorous framework in algebraic geometry that characterizes the existence and moduli of canonical metrics—primarily Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics—on Fano and polarized varieties through algebro-geometric stability conditions. It has become central to the modern approach to the Yau–Tian–Donaldson conjecture and the construction of K-moduli spaces. The theory uses test configurations, non-Archimedean geometry, and valuative invariants to encode the obstruction to canonical metrics and to stratify moduli spaces via wall-crossing and chamber decomposition.

1. Fundamental Objects and Criteria

K-stability is defined for a triple $(X,\Delta;L)$, where $X$ is a normal projective variety, $\Delta$ is an effective Weil $\mathbb Q$-divisor (allowing log pairs and boundaries), and $L$ is an ample (often anticanonical) line bundle. For log Fano pairs $(X,\Delta)$, the central stability criterion is that $-K_X-\Delta$ is ample and $(X,\Delta)$ is klt.

A test configuration is a flat, $C^*$-equivariant family $(\mathcal{X},\mathcal{L})$ over $\mathbb C$ (or $\AA^1$), with generic fiber $(X,L)$ and a central fiber equipped with a $C^*$-action. The Donaldson–Futaki (DF) invariant is constructed via the asymptotic expansion of Hilbert and weight polynomials: $$h(k) = a_0 k^{n} + a_1 k^{n-1} + O(k^{n-2}), \quad w(k) = b_0 k^{n+1} + b_1 k^{n} + O(k^{n-1}),$$ yielding

$$\mathrm{DF}(\mathcal{X}, \mathcal{L}) = \frac{b_0 a_1 - b_1 a_0}{a_0}.$$

K-semistability demands $\mathrm{DF} \geq 0$ for all test configurations; K-stability requires strict positivity except for trivial configurations; K-polystability requires that only product-type test configurations have vanishing DF.

Uniform K-stability strengthens the condition by requiring a uniform gap relative to a norm (e.g., the minimum norm or the $J$-invariant).

2. Valuative and Non-Archimedean Invariants

Recent advances have reformulated K-stability in terms of valuation-theoretic invariants, especially the $\beta$- and $\delta$-invariants. For a prime divisor $E$ over $X$, define

$$\beta_{X,\Delta}(E) = A_{X,\Delta}(E) - S_{X,\Delta}(E),$$

where $A_{X,\Delta}(E)$ is the log discrepancy and

$$S_{X,\Delta}(E) = \frac{1}{(-K_X-\Delta)^d} \int_0^\infty \mathrm{vol}(-K_X-\Delta - tE) \,dt.$$

The pair $(X, \Delta)$ is K-semistable iff $\beta_{X,\Delta}(E) \geq 0$ for all divisors $E$, and uniformly K-stable iff there exists $\varepsilon > 0$ with $\beta_{X,\Delta}(E) \geq \varepsilon$ for all $E$ (Zhou et al., 2019, Boucksom et al., 2022). The $\delta$-invariant is

$$\delta(X, \Delta) := \inf_E \frac{A_{X,\Delta}(E)}{S_{X,\Delta}(E)}.$$

Uniform K-stability is equivalent to $\delta(X, \Delta) > 1$ (Boucksom et al., 2022, Liu et al., 2024). These invariants appear as minimizers of the non-Archimedean Mabuchi and Ding functionals in the Berkovich analytification of $X$, unifying test configuration and filtration approaches.

3. Wall-Crossing, Semi-Algebraic Chamber Decomposition, and K-Moduli Spaces

For log Fano pairs or "couples" $(X, \sum D_j)$, the space of coefficients $(x_1, \ldots, x_k)\in \mathbb R_{\geq 0}^k$ parameterizes log boundaries. The log Fano domain consists of tuples $(x_1, ..., x_k)$ such that $(X, \sum x_j D_j)$ is klt and $-K_X - \sum x_j D_j$ is ample.

For a fixed compact polytope $P$, the K-semistable locus $(X, \sum D_j)_P$ is a semi-algebraic subset of $P$ defined by polynomial inequalities. One obtains a finite semi-algebraic chamber decomposition

$P = \bigsqcup_i A_i$

so that K-(semi/poly)stability is constant in each chamber $A_i$ (Liu et al., 2024). In the case of a single divisor ($k=1$), the chamber decomposition reduces to a finite interval decomposition with walls at algebraic thresholds.

This structure underpins the wall-crossing theory of K-moduli: changing boundary coefficients across a wall induces a birational transformation of the moduli stack, generalizing classical GIT wall-crossing phenomena (Liu et al., 2024).

4. Relationship with Geometric Invariant Theory (GIT)

The K-stability framework is deeply connected to GIT via the Hilbert–Mumford criterion, central charge constructions, and wall-crossing. For small boundary coefficients $c \ll 1$ and a fixed K-polystable $\mathbb Q$-Fano variety $X$, the locus

$$(X, cD) \text{ K-(semi/poly)stable} \iff D \text{ is GIT-(semi/poly)stable in } |L|$$

for $0 < c < \varepsilon_0$ (Liu et al., 2024). The non-proportional wall-crossing theory allows non-conventional boundaries, producing new moduli spaces and comparing GIT and K-stable loci via the structure of the CM line bundle.

The recent formalism of central charges/invariant theory axiomatizes K-stability as a special case of stability conditions on stacks, establishing analytic and algebraic correspondences through Kempf–Ness theory and moment maps (Dervan, 2022).

5. Non-Archimedean and Filtration-Theoretic Approaches

K-stability, originally formulated in terms of test configurations, is equivalently described using non-Archimedean pluripotential theory and filtrations. The space of admissible filtrations is approximated by Cauchy sequences of test configurations in the direct limit of Tits buildings; the normalized Donaldson–Futaki invariant becomes a continuous function on this metric space (Codogni, 2018).

Any graded norm or filtration of the section ring determines a non-Archimedean metric; uniform K-stability is equivalent to a uniform lower bound for the Mabuchi functional on the space of all filtrations (Yao, 20 Nov 2025, Boucksom et al., 2018, Boucksom et al., 2022). Quantization results establish convergence between destabilizers for Chow-stability and the maximal direction for K-instability via non-Archimedean functionals (Yao, 20 Nov 2025).

6. Relative Stability, Properness, and Families

In families of Fano or log Fano varieties, the relative stability threshold is computed by divisorial valuations. When special fibers are K-unstable but generic fibers are K-semistable, birational modifications by blowing up the minimizing divisorial places and running minimal model programs iteratively raise the threshold, proving properness of the K-moduli space by birational geometry alone (Blum et al., 7 Oct 2025). For relative settings over a base, Schur functor expansions, filtrations, and Chern class techniques compute stability in flag families (Isopoussu, 2013).

Openness results guarantee that the locus of divisorially stable (thus uniformly K-stable) pairs is open in the polarization parameter space, underlining the local constancy of moduli in wall-crossing (Boucksom et al., 2022, Liu et al., 2024).

7. Connections to Complex Differential Geometry and Physics

K-stable Fano varieties and log Fano pairs equivalently admit Kähler–Einstein or cscK metrics, under the Yau–Tian–Donaldson correspondence (Dervan et al., 2016, Liu et al., 2024, Rollin, 2013). In Sasaki–Einstein and AdS/CFT settings, K-stability is shown equivalent to stability for chiral rings and the existence of Ricci-flat cone metrics; deformations and stability in the algebraic side mirror physical constraints on operator rings (Collins et al., 2016).

Uniform K-stability is the algebro-geometric counterpart of coercivity for the Mabuchi functional; the quantification of gaps is explicit through the $\delta$-invariant and the lower bound for the $\beta$-invariant.

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Key Concepts Table

Concept	Definition / Role	Reference
Test configuration	Family over $\mathbb C$ with $C^*$-action	(Liu et al., 2024, Kaloghiros et al., 2020)
Donaldson–Futaki invariant	Intersects Hilbert/weight polynomials, measures instability	(Liu et al., 2024, Fujita, 2015)
K-(semi/poly)stability	Positivity/strict positivity of DF invariants	(Liu et al., 2024, Fujita, 2015)
$\beta$-/$\delta$-invariants	Valuative thresholds for stability	(Zhou et al., 2019, Boucksom et al., 2022)
Semi-algebraic chambers	Decomposition in boundary coefficients for constancy	(Liu et al., 2024)
Wall-crossing	Change in K-stability, birational moduli transformations	(Liu et al., 2024)
GIT-stability	Classical invariant theory; equivalence for small boundaries	(Liu et al., 2024, Dervan, 2022)
Filtrations / Tits buildings	Metric and topological encoding of test configurations	(Codogni, 2018, Yao, 20 Nov 2025)
Properness of K-moduli	Birational argument via divisorial minimizers	(Blum et al., 7 Oct 2025)

• Non-proportional wall crossing for K-stability (Liu et al., 2024)
• K-stability of Fano varieties via admissible flags (Abban et al., 2020)
• On Berman-Gibbs stability and K-stability of $\mathbb{Q}$-Fano varieties (Fujita, 2015)
• K-stability of relative flag varieties (Isopoussu, 2013)
• A non-Archimedean approach to K-stability, II: divisorial stability and openness (Boucksom et al., 2022)
• The maximal destabilizers for Chow and K-stability (Yao, 20 Nov 2025)
• Stability and coercivity for toric polarizations (Hisamoto, 2016)
• K stability and stability of chiral ring (Collins et al., 2016)
• On toric geometry and K-stability of Fano varieties (Kaloghiros et al., 2020)
• Relative stability theory and properness of K-moduli spaces (Blum et al., 7 Oct 2025)
• Stability conditions in geometric invariant theory (Dervan, 2022)
• K-stability for Kähler Manifolds (Dervan et al., 2016)
• K-stability and parabolic stability (Rollin, 2013)
• Some criteria for uniform K-stability (Zhou et al., 2019)
• Tits buildings and K-stability (Codogni, 2018)
• On uniform K-stability of pairs (Tian, 2018)
• A non-Archimedean approach to K-stability (Boucksom et al., 2018)