04.10.13 · algebraic-geometry / moduli

K-stability and the Yau-Tian-Donaldson conjecture

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Anchor (Master): Tian 1997 *Inventiones Math.* 130 (*Kähler-Einstein metrics with positive scalar curvature*); Donaldson 2002 *J. Differential Geom.* 62 (*Scalar curvature and stability of toric varieties*); Chen-Donaldson-Sun 2014 *J. Amer. Math. Soc.* 28 (three papers: *Kähler-Einstein metrics on Fano manifolds I, II, III*); Li-Xu 2014 *Ann. of Math.* 180 (*Special test configurations and K-stability*); Odaka 2013 *Ann. of Math.* 177 (*A generalization of the Ross-Thomas slope theory*); Boucksom-Hisamoto-Jonsson 2017 *Ann. Inst. Fourier* 67 (non-Archimedean K-stability); Mumford *Geometric Invariant Theory* 3rd ed. (Springer Ergebnisse 34, 1994) as the conceptual source of the stability picture

Intuition Beginner

A Fano variety is a smooth projective variety $X$ on which the anticanonical line bundle $- K_{X}$ is ample. The simplest examples are projective spaces, smooth quadrics, smooth cubic surfaces, and the smooth del Pezzo surfaces. These are the algebraic varieties of positive curvature class — the geometric counterpart of compact manifolds with positive Ricci curvature in Riemannian geometry.

A central question, raised by Calabi in 1954 and elevated to a conjecture by Yau, asks: when does a Fano variety carry a Kähler-Einstein metric of positive Ricci curvature? Kähler-Einstein metrics are the canonical Riemannian metrics on a complex manifold; their existence is the deepest possible compatibility between complex structure, symplectic structure, and curvature.

The answer, achieved by Chen, Donaldson, and Sun in 2014, is K-stability — an algebraic condition formulated by Tian in 1997 and recast by Donaldson in 2002. K-stability tests $X$ against every one-parameter algebraic degeneration of $X$ into a (possibly singular) central fibre; the degeneration produces a numerical invariant called the Donaldson-Futaki invariant, and $X$ is K-stable when this invariant is strictly positive across every non-product degeneration. The Yau-Tian-Donaldson theorem is the equivalence: K-stable iff Kähler-Einstein. Algebra detects geometry, and the bridge is built out of the same Hilbert-Mumford stability machinery that Mumford applied to moduli of curves.

Visual Beginner

A picture of a Fano variety $X$ on the left, a Kähler-Einstein metric — drawn as a smooth ellipsoidal envelope — on the right, and a test configuration shown as an arrow between them labelled with the Donaldson-Futaki invariant. K-stability is the assertion that the arrow always points to a positive number, and the Yau-Tian-Donaldson theorem says this is exactly the condition for the right-hand picture to exist.

The picture captures the central narrative: a Fano variety either admits a Kähler-Einstein metric or it admits a destabilising test configuration with $DF \leq 0$ , and the two possibilities are mutually exclusive. The arrow direction reverses in the obstructed case — a strictly negative Donaldson-Futaki invariant flags a destabilising degeneration, and the analytic continuity method along cone-angle metrics breaks down at the corresponding family.

Worked example Beginner

Consider a smooth cubic surface $X \subset P^{3}$ — the zero locus of a generic degree- $3$ homogeneous polynomial in four variables.

Step 1. Identify the anticanonical class. The adjunction formula on $P^{3}$ gives $K_{X} = (K_{P^{3}} + X) ∣_{X} = (- 4 + 3) \cdot H ∣_{X} = - H ∣_{X}$ where $H$ is the hyperplane class. So $- K_{X} = H ∣_{X} = O_{X} (1)$ , the hyperplane class restricted to $X$ .

Step 2. Verify Fano. Since $O_{X} (1)$ is the restriction of the very ample $O_{P^{3}} (1)$ to $X$ , it is itself very ample, hence ample. So $- K_{X}$ is ample and $X$ is Fano.

Step 3. Check automorphisms. A generic smooth cubic surface has a finite automorphism group. This rules out the Matsushima obstruction (which requires reductive infinite automorphism groups) right away. So there is no automorphic obstacle to Kähler-Einstein existence.

Step 4. Apply Tian's alpha-invariant test. Tian's 1987 alpha-invariant of $(X, - K_{X})$ satisfies $α (X) \geq 2/3$ for a generic smooth cubic surface, and the Tian criterion $α (X) > n / (n + 1) = 2/3$ for $n = dim X = 2$ delivers strict K-stability whenever the inequality is strict.

Step 5. Conclude. By Tian 1990, $X$ admits a Kähler-Einstein metric. By the Yau-Tian-Donaldson theorem, K-stability and Kähler-Einstein existence are equivalent, so $X$ is K-stable. Every test configuration of $X$ either has central fibre isomorphic to $X$ (a product configuration) or strictly positive Donaldson-Futaki invariant.

What this tells us: the smooth cubic surface — a concrete two-dimensional Fano — sits squarely inside the K-stable locus and carries the canonical Riemannian metric of positive curvature. The alpha-invariant test is the simplest practical method to certify K-stability without computing the Donaldson-Futaki invariant on every test configuration; the certification by an analytic argument is then validated by the Chen-Donaldson-Sun equivalence.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Donaldson-Futaki invariant of a test configuration is:

A. The dimension of the central fibre. B. A leading coefficient in the expansion of the weight of the central-fibre $C^{\times}$ -action on sections of high tensor powers of $L$ . C. The genus of the generic fibre. D. The Hilbert polynomial of the family at $t = 0$ .

Hint

Donaldson's 2002 reformulation extracts a numerical invariant from the asymptotic behaviour of a $C^{\times}$ -character on a tower of line-bundle cohomology spaces.

Answer

B. The Donaldson-Futaki invariant is computed from the asymptotic expansion of the weight $w (k)$ of the $C^{\times}$ -action on $det H^{0} (X_{0}, L_{0}^{k})$ and the dimension expansion $dim H^{0} (X_{0}, L_{0}^{k})$ . Both expand as polynomials in $k$ , and a specific normalised combination of leading and subleading coefficients gives the Donaldson-Futaki invariant.

Formal definition Intermediate+

Throughout this section let $X$ be a smooth projective variety over $C$ of complex dimension $n$ with $- K_{X}$ ample — a Fano manifold. Write $L = - K_{X}$ for the anticanonical $Q$ -line bundle (a multiple of it is genuinely a line bundle), and $ω \in c_{1} (L)$ for a Kähler class representative on $X$ .

Definition (Kähler-Einstein metric). A Kähler metric $ω$ on $X$ is Kähler-Einstein of positive Ricci curvature if it satisfies $$ \mathrm{Ric}(\omega) = \omega. $$ Existence of such an $ω$ in the class $c_{1} (L) = c_{1} (- K_{X})$ is the analytic question that the Yau-Tian-Donaldson conjecture answers in algebraic-geometric terms.

Definition (test configuration). A test configuration for $(X, L)$ of exponent $r \in Z_{> 0}$ is the data of:

(TC-1) a flat projective $C^{\times}$ -equivariant morphism $π : X \to A^{1}$ , with $C^{\times}$ acting on $A^{1}$ by multiplication;

(TC-2) a $C^{\times}$ -equivariant relatively ample line bundle $L$ on $X$ ;

(TC-3) an isomorphism of $C^{\times}$ -polarised varieties $$ (\mathcal{X}, \mathcal{L})|_{\mathbb{A}^1 \setminus {0}} \cong (X, L^r) \times (\mathbb{A}^1 \setminus {0}), $$ where the right-hand side carries the standard $C^{\times}$ -action on the second factor.

The central fibre $(X_{0}, L_{0}) := (π^{- 1} (0), L ∣_{π^{- 1} (0)})$ carries an induced $C^{\times}$ -action; this is the algebraic degeneration of $X$ . The test configuration is product if $(X, L) ≅ (X \times A^{1}, p_{1}^{*} L^{r})$ with the $C^{\times}$ -action coming from a one-parameter subgroup of $Aut (X, L)$ ; otherwise the configuration is a genuine degeneration in the geometric sense (the central fibre is not isomorphic to $X$ ).

Definition (Donaldson-Futaki invariant). For a test configuration $(X, L)$ of exponent $r$ , define two asymptotic expansions in $k \in Z_{> 0}$ for the central fibre: $$ d(k) := \dim H^0(\mathcal{X}_0, \mathcal{L}_0^{\otimes k}) = b_0 k^n + b_1 k^{n-1} + O(k^{n-2}), $$ $$ w(k) := \text{weight of } \mathbb{C}^\times \text{ on } \det H^0(\mathcal{X}_0, \mathcal{L}_0^{\otimes k}) = a_0 k^{n+1} + a_1 k^n + O(k^{n-1}). $$ The expansions exist by the equivariant Riemann-Roch theorem applied to the polarised central fibre, with $b_{0} = L^{n} / n!$ (independent of the test configuration), $b_{1} = - K_{X} \cdot L^{n - 1} / (2 (n - 1)!) = L^{n} / (2 (n - 1)!)$ when $L = - K_{X}$ . The Donaldson-Futaki invariant is $$ \mathrm{DF}(\mathcal{X}, \mathcal{L}) := \frac{a_1 b_0 - a_0 b_1}{b_0^2}. $$ Different normalisations appear in the literature; all preserve the sign and are linearly equivalent up to a positive multiplicative constant.

Definition (K-stability). The pair $(X, L)$ is:

K-semistable if $DF (X, L) \geq 0$ for every test configuration $(X, L)$ ;
K-stable if $DF (X, L) > 0$ for every non-product test configuration with $X_{0}$ reduced (i.e., having no multiple components);
K-polystable if K-semistable and the locus $DF = 0$ consists only of product configurations;
uniformly K-stable (Boucksom-Hisamoto-Jonsson 2017) if there exists $δ > 0$ such that $DF (X, L) \geq δ \cdot ∥ X ∥_{NA}$ for every non-product configuration, where $∥ \cdot ∥_{NA}$ is a non-Archimedean norm on test configurations.

For Fano $X$ with $L = - K_{X}$ , K-stability of $X$ refers to K-stability of the pair $(X, - K_{X})$ .

Counterexamples to common slips

A product test configuration is not the same as a plain product test configuration. A product configuration carries a non-identity $C^{\times}$ -action coming from a one-parameter subgroup of $Aut (X, L)$ ; only when this subgroup is the identity is the configuration genuinely a product of objects with no internal $C^{\times}$ -action. The Futaki invariant of a product configuration with non-identity automorphism action is the classical Futaki invariant from 1983, and need not vanish.
The exponent $r$ in the definition of a test configuration is not a stability label; it records how the test configuration polarises. Replacing $r$ by $r s$ for any positive integer $s$ rescales $DF$ by a factor of $s^{n} / s^{n + 1} = 1/ s$ , but does not change the sign. K-stability is exponent-independent.
A test configuration with $X_{0}$ not reduced (i.e., with multiple components in the central fibre) can have $DF = 0$ for spurious reasons; Li-Xu 2014 show that K-stability tested only on special test configurations (those with normal $Q$ -Fano central fibre) characterises K-stability for Fano varieties.
K-stability is a property of the polarised pair $(X, L)$ , not of $X$ alone. A variety can be K-stable in one polarisation and K-unstable in another. For Fano varieties, the canonical polarisation $L = - K_{X}$ is the one of analytic interest.

Key theorem with proof Intermediate+

Theorem (Yau-Tian-Donaldson; Chen-Donaldson-Sun 2014). Let $X$ be a smooth Fano manifold over $C$ . Then $X$ admits a Kähler-Einstein metric in the class $c_{1} (- K_{X})$ if and only if $(X, - K_{X})$ is K-polystable.

The full proof spans the three Chen-Donaldson-Sun papers (J. Amer. Math. Soc. 28, 2015) and rests on the continuity method along Kähler-Einstein metrics with cone singularities. The argument proves the harder direction (K-polystable implies Kähler-Einstein); the converse direction (Kähler-Einstein implies K-polystable) had earlier been established by Tian 1997 and Stoppa 2009. The architecture below records the converse direction in full and outlines the converse-to-conjecture direction in the Master tier.

Proof of the easier direction (Kähler-Einstein implies K-polystable). Suppose $X$ admits a Kähler-Einstein metric $ω$ with $Ric (ω) = ω$ . Pick a test configuration $(X, L)$ of exponent $r$ with central fibre $(X_{0}, L_{0})$ carrying a $C^{\times}$ -action $ρ : C^{\times} \to Aut (X_{0}, L_{0})$ .

Step 1 — Mabuchi functional. The Mabuchi $K$ -energy $M : H_{ω} \to R$ on the space of Kähler potentials $H_{ω} = {φ \in C^{\infty} (X) : ω + i \partial \overset{ˉ}{\partial} φ > 0}$ is defined by its variational derivative $$ \delta \mathcal{M}(\varphi) \cdot \dot\varphi = -\int_X \dot\varphi \cdot (S(\omega_\varphi) - \bar S) \cdot \omega_\varphi^n, $$ where $ω_{φ} = ω + i \partial \overset{ˉ}{\partial} φ$ , $S (ω_{φ})$ is its scalar curvature, and $\overset{ˉ}{S}$ is the average scalar curvature. Critical points of $M$ are constant-scalar-curvature Kähler (cscK) metrics; on a Fano in the anticanonical class, cscK in $c_{1} (- K_{X})$ is Kähler-Einstein.

Step 2 — Mabuchi functional along a test configuration. The test configuration $(X, L)$ induces a path of Kähler potentials $φ_{t} = φ (t)$ for $t \in R$ via the equivariant resolution and Bergman embedding. Tian and Phong-Sturm showed $$ \lim_{t \to -\infty} \frac{d \mathcal{M}(\varphi_t)}{dt} = -\mathrm{DF}(\mathcal{X}, \mathcal{L}) \cdot C $$ for a positive constant $C$ depending only on $dim X$ and the polarisation. This is the slope formula identifying the Donaldson-Futaki invariant with the asymptotic slope of the Mabuchi $K$ -energy along the path of the test configuration.

Step 3 — convexity of Mabuchi $K$ -energy. Berman, Berndtsson, and Boucksom (2010–2017) prove the Mabuchi $K$ -energy is convex along weak geodesics in $H_{ω}$ . The path induced by a test configuration is a weak geodesic in this sense, and the limit as $t \to - \infty$ is the asymptotic slope.

Step 4 — Kähler-Einstein hypothesis. A Kähler-Einstein metric in $H_{ω}$ is a minimum of $M$ ; if $X$ has finite automorphism group, this minimum is unique. Convexity gives the slope $lim_{t \to - \infty} d M / d t \geq 0$ along every path emerging from a minimum, equivalently $DF (X, L) \geq 0$ for every test configuration — this is K-semistability.

Step 5 — equality case and polystability. The equality $DF (X, L) = 0$ holds precisely when the path stays on a geodesic ray along which $M$ is constant. By the uniqueness theory of geodesic rays from Mabuchi minima (Berman-Berndtsson 2017, Donaldson-Sun 2014), such a ray exists iff $(X, L)$ is a product configuration coming from a one-parameter subgroup of $Aut (X, L)$ .

Therefore: every non-product test configuration of $X$ has $DF > 0$ , equivalently $X$ is K-polystable. $□$

Bridge. The Mabuchi-slope identification is the foundational reason K-stability detects Kähler-Einstein existence, and the central insight is that the Donaldson-Futaki invariant is exactly the Hilbert-Mumford weight in infinite-dimensional GIT: this is exactly the Kempf-Ness picture of 04.10.04 generalised to the infinite-dimensional space of Kähler metrics. The bridge builds toward 04.10.14 K-moduli of Fano varieties, where the chamber-and-wall structure on the moduli of K-semistable Fanos appears again in the wall-crossing of K-stability across one-parameter families of polarisations, and the Mumford-style construction of a projective good moduli space rests on Birkar's BAB boundedness. The forward-promise here is geometric — K-stability generalises Mumford's slope stability for vector bundles (a finite-dimensional Hilbert-Mumford picture), is dual to the analytic Kähler-Einstein existence problem, and identifies algebraic stability with the analytic minimisation of the Mabuchi functional. Putting these together, the Yau-Tian-Donaldson theorem is the Kempf-Ness theorem for the infinite-dimensional GIT of Fano polarisations, and the bridge is the slope formula of Step 2.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the Donaldson-Futaki invariant of the plain product test configuration $(X \times A^{1}, p_{1}^{*} L)$ where the $C^{\times}$ -action on $X \times A^{1}$ is the identity on $X$ and acts by multiplication on $A^{1}$ . Show that $DF = 0$ .

Hint

The central fibre is $(X, L)$ with identity $C^{\times}$ -action. What is the weight of an identity action on $det H^{0} (X, L^{k})$ ?

Answer

The central fibre is $(X_{0}, L_{0}) = (X, L)$ . The $C^{\times}$ -action induced from the action on $A^{1}$ acts identically on the fibre (since the second factor is collapsed), so the action on $H^{0} (X, L^{k})$ is the identity for every $k$ , hence the weight of $C^{\times}$ on $det H^{0} (X, L^{k})$ vanishes for every $k$ . Therefore $w (k) = 0$ for all $k$ , and the expansion $w (k) = a_{0} k^{n + 1} + a_{1} k^{n} + O (k^{n - 1})$ has $a_{0} = a_{1} = 0$ . The Donaldson-Futaki invariant $$ \mathrm{DF} = \frac{a_1 b_0 - a_0 b_1}{b_0^2} = \frac{0 \cdot b_0 - 0 \cdot b_1}{b_0^2} = 0. $$ This is consistent with the K-stability definition: product test configurations with identity action are the "stationary" configurations that contribute zero to the stability test. Genuine product configurations with non-identity automorphism actions can have non-vanishing classical Futaki invariant (Futaki 1983), but the plain product configuration with identity action always has $DF = 0$ . Rubric: full credit for the vanishing-of-action argument and the formal computation.

Exercise 4 (medium, symbolic).

State the slope formula identifying the asymptotic slope of the Mabuchi $K$ -energy along a test configuration with the Donaldson-Futaki invariant. Sketch the equivariant Riemann-Roch input that makes the formula computable.

Hint

The equivariant Riemann-Roch theorem expresses the weight $w (k)$ and dimension $d (k)$ as integrals over the central fibre involving the Chern character, equivariant Todd class, and equivariant first Chern class of the polarising line bundle.

Answer

Slope formula (Tian, Donaldson, Phong-Sturm, Paul-Tian). For a test configuration $(X, L)$ of $(X, L)$ , the path $φ_{t} \in H_{ω}$ induced by the test configuration via equivariant Bergman embeddings satisfies $$ \lim_{t \to -\infty} \frac{d \mathcal{M}(\varphi_t)}{dt} = C_n \cdot \mathrm{DF}(\mathcal{X}, \mathcal{L}), $$ where $C_{n}$ is a normalisation constant depending only on $n = dim X$ . The constant is positive, so the sign of the slope agrees with the sign of $DF$ .

Equivariant Riemann-Roch input. The expansions $$ d(k) = b_0 k^n + b_1 k^{n-1} + O(k^{n-2}), \quad w(k) = a_0 k^{n+1} + a_1 k^n + O(k^{n-1}) $$ are special cases of the equivariant Riemann-Roch theorem applied to the polarised central fibre $(X_{0}, L_{0})$ with $C^{\times}$ -action. Specifically, $$ \chi^{\mathbb{C}^\times}(\mathcal{X}_0, \mathcal{L}0^{\otimes k}) = \int{\mathcal{X}_0} \mathrm{ch}^{\mathbb{C}^\times}(\mathcal{L}_0^{\otimes k}) \cdot \mathrm{td}^{\mathbb{C}^\times}(T \mathcal{X}_0), $$ where $ch^{C^{\times}}$ is the equivariant Chern character and $td^{C^{\times}}$ is the equivariant Todd class. Expanding in $k$ and reading off the weight-zero and weight-one coefficients of the $C^{\times}$ -action gives the four numbers $a_{0}, a_{1}, b_{0}, b_{1}$ ; the Donaldson-Futaki invariant is then the algebraic combination $(a_{1} b_{0} - a_{0} b_{1}) / b_{0}^{2}$ .

Geometric content. $b_{0}$ is the volume $L^{n} / n!$ (Riemann-Roch leading term); $b_{1}$ is half the average scalar curvature times the volume (Riemann-Roch subleading); $a_{0}$ is the Futaki-like volume of the $C^{\times}$ -action on the central fibre; $a_{1}$ records the equivariant subleading term, equivalent to a weighted-volume integral. The combination $(a_{1} b_{0} - a_{0} b_{1}) / b_{0}^{2}$ is the normalised average over the test family of the difference between the scalar-curvature integral and its product-configuration value — exactly the slope of the Mabuchi $K$ -energy.

Rubric: full credit for the slope-formula statement, the equivariant Riemann-Roch expansion identifying $a_{i}, b_{i}$ , and the geometric interpretation as a scalar-curvature-related weighted average. Partial credit for the slope formula alone.

Exercise 5 (medium, short-answer).

State Tian's alpha-invariant criterion for K-stability and apply it to verify that a generic smooth cubic surface in $P^{3}$ is K-stable.

Hint

Tian's criterion (1987 Inventiones Math. 89) states that if $α (X) > n / (n + 1)$ for a Fano $n$ -fold $X$ , then $X$ admits a Kähler-Einstein metric. The alpha invariant is the supremum of $α \geq 0$ such that for every effective $Q$ -divisor $D \sim_{Q} - K_{X}$ , the pair $(X, α D)$ has Kawamata log terminal singularities.

Answer

Tian's alpha-invariant criterion. Let $X$ be a Fano $n$ -fold. The alpha invariant of $X$ is $$ \alpha(X) = \sup\left{ \alpha \in \mathbb{Q}{\geq 0} ;\middle|; (X, \alpha D) \text{ is klt for every effective } \mathbb{Q}\text{-divisor } D \sim{\mathbb{Q}} -K_X \right}, $$ where "klt" stands for Kawamata log terminal singularities. Tian's criterion states: if $α (X) > n / (n + 1)$ , then $X$ admits a Kähler-Einstein metric and is therefore K-polystable by the Yau-Tian-Donaldson theorem. In fact, a slightly weaker bound $α (X) \geq n / (n + 1)$ together with finiteness of $Aut (X)$ suffices for K-polystability.

Application to the smooth cubic surface. Let $X \subset P^{3}$ be a generic smooth cubic surface. The known alpha-invariant bound is $$ \alpha(X) \geq 2/3 = \frac{n}{n+1} ;; \text{at} ;; n = 2, $$ established for generic cubic surfaces by Cheltsov, Park, and Shramov (Russ. Math. Surv. 63, 2008) by case analysis on the configurations of $(- 1)$ -curves on $X$ . Equality is achieved only at specific symmetric cubic surfaces (e.g., the Cayley cubic) and at the Eckardt-point configurations; for a generic cubic surface the inequality is strict, $α (X) > 2/3$ . Tian's criterion delivers K-stability immediately: $$ \alpha(X) > 2/3 = \frac{n}{n+1} ;; \Longrightarrow ;; X \text{ is K-stable}. $$

Combined with the finiteness of $Aut (X)$ for a generic cubic surface, this gives K-polystability and hence Kähler-Einstein existence by Yau-Tian-Donaldson. The cubic surface result was first proved by Tian 1990 via a direct analytic continuity argument before K-stability was even formulated; the alpha-invariant route gives a uniform method that works for all Fano surfaces with $α > 2/3$ .

Rubric: full credit for the alpha-invariant definition with klt singularities, the strict bound $n / (n + 1)$ , and the cubic-surface application citing Cheltsov-Park-Shramov; partial credit for the criterion statement alone.

Exercise 6 (medium, short-answer).

Reduction to special test configurations: state the Li-Xu theorem and explain why it simplifies the K-stability test.

Hint

Li-Xu 2014 Ann. of Math. 180 reduces K-stability to test configurations whose central fibre is itself a normal $Q$ -Fano variety.

Answer

Theorem (Li-Xu 2014, Ann. of Math. 180). Let $X$ be a $Q$ -Fano variety, i.e., a normal projective variety with $- K_{X}$ ample $Q$ -Cartier and log canonical singularities. Then $X$ is K-stable (respectively K-semistable, K-polystable) if and only if $DF (X, L) > 0$ (respectively $\geq 0$ , $> 0$ or product) for every $C^{\times}$ -equivariant special test configuration $(X, L)$ , where special means $X_{0}$ is a normal $Q$ -Fano variety (with klt singularities) and $L = - K_{X / A^{1}}$ .

Why this is a simplification. The general K-stability definition asks for $DF \geq 0$ over the entire category of test configurations — a parameter space encompassing all flat degenerations of $X$ with arbitrarily singular central fibres (e.g., non-reduced, non-normal, with arbitrarily bad singularities). The Li-Xu reduction restricts to the subcategory of special test configurations, where the central fibre is a normal $Q$ -Fano with klt singularities and the polarisation is the canonical one $L = - K_{X / A^{1}}$ . This subcategory is much more tractable:

(LX-1) The central fibres are themselves K-stability test objects (normal $Q$ -Fano varieties).

(LX-2) The polarisation $L = - K_{X / A^{1}}$ is canonical, removing the choice of $L$ .

(LX-3) The Donaldson-Futaki invariant reduces to a log canonical threshold computation involving the central fibre $X_{0}$ alone via Odaka's interpretation (Odaka 2013).

Proof strategy of Li-Xu. Given any test configuration $(X, L)$ , run the minimal model programme with scaling on $X / A^{1}$ to convert $X$ into a special test configuration $X^{sp}$ with normal $Q$ -Fano central fibre and polarisation $- K_{X^{sp} / A^{1}}$ . The Donaldson-Futaki invariant satisfies $DF (X^{sp}) \leq DF (X)$ , so if $DF (X^{sp}) > 0$ for every special $X^{sp}$ then $DF (X) > 0$ for every test configuration. The MMP step uses Birkar-Cascini-Hacon-McKernan 2010 (existence of MMP for klt pairs) and the cone theorem.

Significance. The Li-Xu reduction is the algebraic foundation of K-moduli theory: it identifies K-stability with a finite-type stability condition on a tractable parameter space (the space of special degenerations), and makes the moduli space of K-semistable Fano varieties amenable to standard algebro-geometric tools. Xu's later construction of the K-moduli space (Xu 2020 Ann. of Math. 191) builds directly on this reduction.

Rubric: full credit for the theorem statement, the MMP-based proof sketch, and the reduction to log-canonical-threshold computations. Partial credit for the theorem alone.

Exercise 7 (hard, short-answer).

Show that the Donaldson-Futaki invariant of a non-product test configuration of $P^{1}$ is strictly positive, hence $P^{1}$ is K-stable in the canonical polarisation $- K_{P^{1}} = O (2)$ .

Hint

$P^{1}$ has automorphism group $PGL_{2}$ , so non-product test configurations are genuine degenerations of $P^{1}$ . Use that the central fibre is a curve degeneration of $P^{1}$ , hence a singular curve, and compute the asymptotic weight directly.

Answer

Setup. $X = P^{1}$ , $L = - K_{X} = O_{P^{1}} (2)$ , $n = 1$ . A test configuration $(X, L)$ is a flat family $π : X \to A^{1}$ with $X_{t} ≅ P^{1}$ for $t \neq = 0$ .

Geometry of the central fibre. The Picard scheme of curves of arithmetic genus $0$ is rigid in the smooth case; a flat family of arithmetic-genus- $0$ curves with smooth generic fibre has central fibre either $P^{1}$ (in which case the family is locally a product and the test configuration is product) or a chain of $P^{1}$ 's meeting at nodes (a singular reduced curve of arithmetic genus $0$ ). Non-product test configurations have central fibres of the latter form.

Computation of the Donaldson-Futaki invariant. For a curve of arithmetic genus $0$ polarised by an $O (r)$ line bundle, the Riemann-Roch leading coefficient is $b_{0} = r$ and the subleading is $b_{1} = r$ (Euler characteristic of $O_{X} (r k)$ for $X$ of arithmetic genus $0$ is $r k + 1$ , giving $b_{0} = r$ , $b_{1} = 1$ after normalisation). The $C^{\times}$ -action on the central fibre breaks the chain of $P^{1}$ 's into weight components; standard computation (Donaldson 2002 §4 for $P^{1}$ specifically; Wang 2012 for general genus-0 curves) gives $$ a_0 = 0, \quad a_1 = \frac{1}{2} \sum_i (w_i^+ - w_i^-)^2 / r > 0, $$ where the sum is over the nodes of the central fibre and $w_{i}^{\pm}$ are the $C^{\times}$ -weights on the two branches at each node. Each squared difference $(w_{i}^{+} - w_{i}^{-})^{2}$ is strictly positive whenever the two branches carry distinct weights, which is the case for any non-product configuration (in a product configuration there are no nodes and the central fibre is smooth $P^{1}$ ).

Donaldson-Futaki invariant. $$ \mathrm{DF}(\mathcal{X}, \mathcal{L}) = \frac{a_1 b_0 - a_0 b_1}{b_0^2} = \frac{a_1 \cdot r - 0 \cdot 1}{r^2} = \frac{a_1}{r} > 0. $$ Strict positivity holds for every non-product test configuration of $P^{1}$ , confirming K-stability.

Verification via the Tian alpha invariant. Independently, the alpha invariant of $P^{1}$ in the polarisation $- K = O (2)$ is $α (P^{1}) = 1/2$ , and Tian's criterion $α > n / (n + 1) = 1/2$ for $n = 1$ is not strict; the alpha-invariant route is borderline. The direct DF computation above is the cleaner certification. Alternatively, $P^{1}$ admits the Fubini-Study Kähler-Einstein metric, so K-polystability follows from the Yau-Tian-Donaldson theorem applied in reverse.

Note on automorphisms. $Aut (P^{1}) = PGL_{2}$ is a $3$ -dimensional reductive group; product test configurations coming from one-parameter subgroups of $PGL_{2}$ have $DF = 0$ but are product configurations, hence the polystable definition is satisfied (vanishing of DF on product configurations is allowed in K-polystability).

Rubric: full credit for the central-fibre analysis (chain of $P^{1}$ 's), the explicit formula $a_{1} = (1/2) \sum (w_{i}^{+} - w_{i}^{-})^{2} / r > 0$ , and the DF computation; partial credit for the qualitative argument.

Exercise 8 (hard, short-answer).

State Odaka's algebraic K-stability criterion via log discrepancies and explain how it relates the Donaldson-Futaki invariant to the minimal model programme.

Hint

Odaka 2013 Ann. of Math. 177 reformulates the Donaldson-Futaki invariant as an integral over the central fibre involving log discrepancies (a singularity invariant from the minimal model programme).

Answer

Theorem (Odaka 2013, Ann. of Math. 177). Let $X$ be a $Q$ -Fano variety and $(X, L)$ a special test configuration (Li-Xu sense) with central fibre $X_{0}$ . Then $$ \mathrm{DF}(\mathcal{X}, \mathcal{L}) = \frac{1}{(-K_X)^n} \cdot \left( -K_{\mathcal{X}_0/\mathbb{A}^1} \right)^{n+1} $$ up to normalisation. Equivalently, $DF > 0$ iff the pair $(X_{0}, - K_{X_{0} / A^{1}})$ has all log discrepancies strictly bounded above the threshold defined by the central fibre's klt singularity type.

Log discrepancy. For a pair $(X, Δ)$ with $Δ$ a $Q$ -divisor and a prime divisor $E$ on a log resolution $f : Y \to X$ , the log discrepancy of $(X, Δ)$ along $E$ is $$ a(E; X, \Delta) := 1 + \mathrm{coeff}_E(K_Y - f^*(K_X + \Delta)). $$ The pair $(X, Δ)$ is klt (Kawamata log terminal) if $a (E; X, Δ) > 0$ for every $E$ ; it is log canonical if $a (E) \geq 0$ .

Odaka's reformulation. For a special test configuration, the Donaldson-Futaki invariant is a numerical integration of the log discrepancy of the central fibre. Strict K-stability becomes the condition: every special test configuration's central fibre has strictly positive total log discrepancy excess in the precise sense above.

Connection to MMP. Odaka's reformulation places K-stability inside the framework of the minimal model programme:

(O-1) K-stability is detected by the singularity type of test-configuration central fibres, computed via log discrepancies and intersection numbers — purely algebraic data.

(O-2) The Li-Xu reduction (Exercise 6) converts arbitrary test configurations to special ones via MMP with scaling, so K-stability is fully algebraic: testable without analysis, without metrics, without Mabuchi functionals.

(O-3) The Birkar-Cascini-Hacon-McKernan 2010 framework provides the technical inputs (existence of canonical models, termination of flips in the relevant codimension).

Practical computation. For a Fano variety $X$ and a candidate test configuration $X$ , Odaka's criterion reduces K-stability to: compute the log discrepancies of $X_{0}$ relative to $- K_{X_{0} / A^{1}}$ ; if all are bounded below by a threshold determined by the dimension and volume of $X$ , then $DF > 0$ and $X$ is K-stable along this test configuration. The threshold is the delta invariant of Fujita-Odaka 2018, and uniform K-stability is the condition $δ (X) > 1$ .

Significance. Odaka's criterion connects three previously separate threads: the Tian-Donaldson algebraic stability picture, the Mori-style MMP framework of birational geometry, and the analytic Kähler-Einstein theory. The K-moduli space of K-semistable Fano varieties — constructed by Xu, Li-Xu, Liu-Xu, Blum-Halpern-Leistner-Liu-Xu in 2017-2022 — uses Odaka's algebraic reformulation as the foundation.

Rubric: full credit for the Odaka formula, the log discrepancy definition, the MMP connection, and the delta-invariant link to uniform K-stability; partial credit for the criterion alone.

Advanced results Master

The non-Archimedean reformulation

Boucksom-Hisamoto-Jonsson 2017 (Ann. Inst. Fourier 67) gave a unifying non-Archimedean reformulation of K-stability that subsumes the test-configuration definition and the Odaka criterion. The framework treats the rational function field $Frac (O_{X, A^{1}})$ as a discretely valued field and identifies test configurations with non-Archimedean Fubini-Study metrics on $- K_{X}^{NA}$ — the pull-back of $- K_{X}$ to the Berkovich analytification of $X$ over the non-Archimedean field.

Theorem (Boucksom-Hisamoto-Jonsson 2017). Let $X$ be a $Q$ -Fano variety. The Donaldson-Futaki invariant of a test configuration $(X, L)$ equals the non-Archimedean Mabuchi functional $M^{NA} (ϕ_{X, L})$ evaluated at the corresponding non-Archimedean Fubini-Study metric $ϕ_{X, L}$ . Uniform K-stability of $X$ is equivalent to coercivity of $M^{NA}$ over the non-Archimedean energy space.

The non-Archimedean Mabuchi functional decomposes as $M^{NA} = H^{NA} - n \cdot E^{NA}$ , with $H^{NA}$ a non-Archimedean entropy and $E^{NA}$ a non-Archimedean Monge-Ampère energy. Both functionals are defined entirely algebraically: $H^{NA}$ via log discrepancies (the Odaka link), $E^{NA}$ via intersection numbers on the test configuration's compactification. The framework subsumes Odaka's criterion and clarifies why uniform K-stability — strictly stronger than K-stability — is the analytically natural condition that the Chen-Donaldson-Sun proof actually establishes.

Connection to the Calabi conjecture and the Tian-Yau theorem

The Calabi conjecture (1954) asked whether every Kähler class on a compact Kähler manifold contains a metric of prescribed Ricci curvature satisfying the topological constraint. Yau 1978 (Comm. Pure Appl. Math. 31) proved the case of zero or negative Ricci: every Calabi-Yau manifold (with $K_{X} = 0$ ) carries a Kähler-Einstein metric with $Ric = 0$ , and every variety of general type ( $K_{X}$ ample) carries a Kähler-Einstein metric with $Ric = - ω$ (the Aubin-Yau theorem, 1976-78).

The positive case — Fano varieties — was the open question. The Futaki invariant (Futaki 1983) gives a Lie-algebraic obstruction; the Matsushima theorem (1957) gives a reductive-automorphism obstruction. Tian-Yau (1987 Mathematical Aspects of String Theory) proved Kähler-Einstein existence on Fano varieties of complex dimension $2$ with vanishing Futaki invariant; Tian 1990 (Inventiones Math. 101) extended to del Pezzo surfaces, settling the surface case completely. The general case waited until Chen-Donaldson-Sun 2014 with the K-stability framework.

The historical thread runs Calabi 1954 → Yau 1978 → Aubin-Yau 1976-78 → Matsushima 1957 → Futaki 1983 → Tian 1987 → Tian 1990 → Tian 1997 → Donaldson 2002 → Chen-Donaldson-Sun 2014. The Yau-Tian-Donaldson theorem closes the positive case of the Calabi conjecture in the Fano setting and identifies the obstruction precisely as failure of K-stability.

K-moduli compactifications

Building on Li-Xu's reduction to special test configurations and Odaka's algebraic reformulation, the moduli space of K-semistable Fano varieties of fixed dimension and volume has been constructed as a proper Deligne-Mumford stack with projective good moduli space:

Theorem (Xu 2020 Ann. of Math. 191; Liu-Xu, Xu-Liu, Blum-Halpern-Leistner-Liu-Xu, ABHLX 2022). For each pair $(n, V)$ with $n \in Z_{> 0}$ and $V \in Q_{> 0}$ , the moduli stack $M_{n, V}^{Kss}$ of K-semistable Fano varieties of dimension $n$ and volume $V$ is a finite-type Artin stack with projective good moduli space $M_{n, V}^{Kps}$ parametrising K-polystable Fano varieties.

The construction uses Birkar's BAB conjecture (Birkar 2016, awarded the 2018 Fields Medal): the family of K-semistable Fano varieties of fixed dimension and volume is bounded — that is, parametrised by a finite-type scheme. Birkar's later work on BAB and on log Calabi-Yau pairs (Birkar 2021 — 2022 Fields Medal lecture) extends to the moduli of K-semistable log Fano pairs.

K-moduli compactifications generalise the GIT moduli of Mumford 1965, replacing the finite-dimensional GIT setup with the infinite-dimensional K-stability setup. The chambers of the K-moduli space correspond to the wall-crossings of K-stability as the polarisation varies, providing a continuous bridge from finite-dimensional GIT (the Mumford-Fogarty-Kirwan theory) to infinite-dimensional GIT (the Chen-Donaldson-Sun setting).

Extension to log Fano pairs

A log Fano pair is a pair $(X, Δ)$ with $X$ a normal projective variety, $Δ$ an effective $Q$ -divisor with coefficients in $[0, 1]$ , the pair klt, and $- (K_{X} + Δ)$ ample $Q$ -Cartier. The Donaldson-Futaki invariant extends to test configurations of $(X, Δ)$ with appropriately defined boundary degenerations $(X, D)$ .

Theorem (Berman 2016, Li-Tian-Wang 2017). A log Fano pair $(X, Δ)$ admits a Kähler-Einstein cone metric along $Δ$ (with cone angle $2 π (1 - coeff_{Δ})$ at each component of $Δ$ ) if and only if $(X, Δ)$ is K-polystable in the log sense.

The log K-stability framework is necessary for the moduli theory: K-moduli compactifications naturally include boundary strata corresponding to log Fano pairs, and the log Yau-Tian-Donaldson equivalence ensures the moduli space carries a continuous Kähler-Einstein structure across the boundary. Chi Li, Xu, Liu, and others (2015-2020) have developed the log theory in detail; the connection to mirror symmetry (Calabi-Yau pairs as degenerations of log Fano pairs at cone angle zero) is an active area of current research.

Test configurations as one-parameter degenerations

The conceptual placement of K-stability inside Mumford's GIT framework — visible already in the Tian 1997 paper and made explicit in Donaldson 2002 — is the following: a test configuration is exactly a one-parameter subgroup in the infinite-dimensional space of Fano polarisations. The Donaldson-Futaki invariant is exactly the Hilbert-Mumford weight for this infinite-dimensional GIT. The Yau-Tian-Donaldson equivalence is exactly the Kempf-Ness theorem (see 04.10.04) for the symplectic reduction by the gauge group of the Mabuchi $K$ -energy. This identifies K-stability with the natural infinite-dimensional extension of Mumford's GIT, and the bridge is built out of the slope formula linking the Donaldson-Futaki invariant to the asymptotic slope of the Mabuchi functional.

The non-Archimedean Boucksom-Hisamoto-Jonsson reformulation can be read in this light as the non-Archimedean GIT of Berkovich spaces: the Berkovich analytification replaces the finite-dimensional projective variety, and the K-stability condition becomes coercivity of a Mabuchi-style functional over the corresponding non-Archimedean space. This perspective extends to the Yau-Tian-Donaldson conjecture for general cscK metrics (constant scalar curvature Kähler metrics in arbitrary polarisations, not only Fano), which remains open as of 2026 and is an active subject of current research (Chen-Cheng 2021, Berman-Berndtsson-Sjöström-Dyrefelt).

Synthesis. K-stability is the foundational reason a Fano variety admits a canonical Riemannian metric of positive Ricci curvature: this is exactly the algebraic condition that detects the analytic Yau-Tian-Donaldson equivalence, and the central insight is that test configurations play the role of one-parameter subgroups in infinite-dimensional GIT, with the Donaldson-Futaki invariant identifying as the corresponding Hilbert-Mumford weight via the equivariant Riemann-Roch theorem on central fibres. Three apparently distinct constructions — the analytic Mabuchi functional and its asymptotic slope along test configurations, the algebraic Donaldson-Futaki invariant via leading-coefficient expansions of $C^{\times}$ -weights, and the algebraic-geometric Odaka log-discrepancy invariant via the minimal model programme — fit into a single equivalence: putting these together, the slope formula identifies the analytic Mabuchi slope with the algebraic DF invariant, Li-Xu reduction simplifies the algebraic test to special configurations, and Odaka recasts the reduced test as an MMP question about central-fibre discrepancies. This bridge appears again in 04.10.14 K-moduli of Fano varieties, where the Xu-Liu-Birkar construction of the projective good moduli space of K-polystable Fanos rests on the BAB-boundedness of the K-semistable locus, and generalises the Mumford GIT construction of 04.10.01 moduli of curves to the infinite-dimensional setting; it also generalises the slope-stability picture of 04.10.06 moduli of vector bundles via the Kobayashi-Hitchin correspondence to the algebraic-geometric setting of Fano varieties. The Yau-Tian-Donaldson theorem stands as the modern apex of the Calabi-Yau-Aubin-Tian programme initiated by the Calabi conjecture, and the bridge to non-Archimedean GIT (Boucksom-Hisamoto-Jonsson) shows the framework extends to Berkovich-analytic moduli theory.

The synthesis is structural: every step in the proof of the Yau-Tian-Donaldson conjecture corresponds to a precise dictionary entry between algebra and analysis. The Mabuchi $K$ -energy is dual to the Donaldson-Futaki invariant via the slope formula. The Kähler-Einstein equation is dual to the algebraic stability test via the Chen-Donaldson-Sun continuity method. K-polystability is dual to the existence of an analytic minimum of the Mabuchi functional via geodesic-ray uniqueness. K-moduli compactifications are dual to the analytic Gromov-Hausdorff limit space of Kähler-Einstein Fano varieties via Cheeger-Colding-Tian compactness. Every algebraic notion has an analytic counterpart, and the equivalence is the theorem. This duality is the deepest known instance of the algebraic-analytic dictionary in higher-dimensional algebraic geometry, and the Chen-Donaldson-Sun proof is one of the great mathematical achievements of the 2010s.

Full proof set Master

Theorem (Yau-Tian-Donaldson, Kähler-Einstein implies K-polystable; Tian 1997, Stoppa 2009 completion). Proof given in the Intermediate-tier section: the Mabuchi $K$ -energy $M$ on the space of Kähler potentials has the Kähler-Einstein metric as its (unique up to automorphisms) minimum; convexity of $M$ along weak geodesics (Berman-Berndtsson-Boucksom 2017) gives non-negative asymptotic slope along every geodesic ray; the slope formula of Tian and Phong-Sturm identifies this slope with the Donaldson-Futaki invariant up to a positive constant; non-product test configurations correspond to non-product geodesic rays from a Kähler-Einstein minimum, and the geodesic-ray uniqueness theorem of Berman-Berndtsson 2017 and Donaldson-Sun 2014 forces strict positivity of the slope on such rays. Hence Kähler-Einstein implies K-polystable. $□$

**Theorem (Yau-Tian-Donaldson, K-polystable implies Kähler-Einstein; Chen-Donaldson-Sun 2014), full proof in J. Amer. Math. Soc. 28 ^{[source pending]}). Architecture of the proof.

(YTD-1) Continuity method along cone-angle Kähler-Einstein metrics. Fix a smooth divisor $D \in ∣ - m K_{X} ∣$ for some large $m$ , and consider the family of metrics $ω_{β}$ solving $Ric (ω_{β}) = β ω_{β} + (1 - β) [D]$ for cone angles $β \in (0, 1]$ . The endpoint $β = 1$ is the desired Kähler-Einstein metric on $X$ ; the endpoint $β \in (0, 1)$ corresponds to a Kähler-Einstein cone metric along $D$ .

(YTD-2) Existence at small $β$ . For $β$ sufficiently small, the cone-angle Kähler-Einstein metric exists by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi 2010 and Berman 2016, since the cone angle dominates the analytic obstruction.

(YTD-3) Openness. The set ${β : ω_{β} exists}$ is open by the implicit function theorem on the relevant Banach space of cone-Kähler metrics — a substantive analytic estimate due to Brendle 2013 and Donaldson 2012.

(YTD-4) Closedness. The set is closed by Cheeger-Colding-Tian compactness theory applied to the family $(X, ω_{β})$ as $β$ approaches a limit value $β_{\infty}$ . The compactness theorem produces a Gromov-Hausdorff limit $(X_{\infty}, ω_{\infty})$ — a metric measure space which Donaldson-Sun show is algebraic: $X_{\infty}$ is a normal $Q$ -Fano variety with klt singularities, and $ω_{\infty}$ is a weak Kähler-Einstein metric on $X_{\infty}$ .

(YTD-5) Algebraic identification of the limit. If $X_{\infty} \neq = X$ , the family $(X, ω_{β}) \to (X_{\infty}, ω_{\infty})$ defines a test configuration $X \to A^{1}$ with $X_{0} = X_{\infty}$ . Chen-Donaldson-Sun compute the Donaldson-Futaki invariant of this configuration: it equals the energy gap between the cone-angle Kähler-Einstein metrics on $X$ and the limit metric on $X_{\infty}$ , and an analytic argument shows this energy gap is non-positive — i.e., $DF (X, L) \leq 0$ .

(YTD-6) K-polystability gives the contradiction. By K-polystability, $DF (X, L) \leq 0$ forces $(X, L)$ to be a product configuration, which means $X_{\infty} = X$ — but the failure assumed $X_{\infty} \neq = X$ . Hence $β_{\infty} = 1$ and the Kähler-Einstein metric exists on $X$ .

The Cheeger-Colding-Tian compactness step (YTD-4) is the analytic heart and uses Perelman's $W$ -functional, the volume non-collapsing estimate, and the algebraic regularity of Gromov-Hausdorff limits in complex dimension $n \leq 7$ (extended later). The algebraic identification (YTD-5) uses the Bergman kernel asymptotics in the equivariant setting and the Donaldson-Sun 2014 algebraicity theorem (Acta Math. 213): Gromov-Hausdorff limits of polarised Kähler-Einstein Fano manifolds are algebraic and polarised. $□$

Proposition (Donaldson-Futaki invariant is well-defined; Donaldson 2002). The Donaldson-Futaki invariant $DF (X, L) = (a_{1} b_{0} - a_{0} b_{1}) / b_{0}^{2}$ is independent of the choice of polynomial-expansion normalisation, depends only on the test configuration up to base change $t \mapsto t^{k}$ (rescaling $DF$ by $1/ k$ ), and equals zero on every plain product test configuration with identity $C^{\times}$ -action.

Proof. The expansions of $w (k)$ and $d (k)$ are extracted from equivariant Riemann-Roch on the polarised central fibre $(X_{0}, L_{0})$ : $$ \chi^{\mathbb{C}^\times}(\mathcal{X}_0, \mathcal{L}_0^{\otimes k}) = d(k) + \zeta \cdot w(k) + O(\zeta^2), $$ where $ζ$ is the equivariant parameter of the $C^{\times}$ -action. Equivariant Riemann-Roch gives both $d (k)$ and $w (k)$ as polynomial integrals on $X_{0}$ of equivariant Chern characters and Todd classes. The leading and subleading coefficients are intrinsic to $(X_{0}, L_{0})$ and the $C^{\times}$ -action, hence intrinsic to $(X, L)$ .

Replacing $L$ by $L^{\otimes s}$ (an exponent change) rescales $L_{0}$ by $s$ , so $b_{0} \to s^{n} b_{0}$ , $b_{1} \to s^{n - 1} b_{1}$ , $a_{0} \to s^{n + 1} a_{0}$ , $a_{1} \to s^{n} a_{1}$ . The combination $a_{1} b_{0} - a_{0} b_{1}$ scales by $s^{2 n + 1}$ and $b_{0}^{2}$ scales by $s^{2 n}$ , so $DF$ scales by $s$ . Up to this positive multiplicative factor (which preserves sign), $DF$ is independent of the choice of exponent.

For the plain product configuration $(X \times A^{1}, p_{1}^{*} L)$ with identity $C^{\times}$ -action on $X$ , the weight on $H^{0} (X, L^{k})$ is identically zero, so $w (k) \equiv 0$ and $a_{0} = a_{1} = 0$ , hence $DF = 0$ . $□$

Proposition (alpha-invariant criterion implies K-stability; Tian 1987, Demailly-Kollár 2001). Let $X$ be a Fano manifold of dimension $n$ . If $α (X) > n / (n + 1)$ , then $X$ is K-stable.

Proof. The strict alpha-invariant bound implies coercivity of the Mabuchi $K$ -energy on the space of Kähler potentials in $c_{1} (- K_{X})$ (Tian 1997, Inventiones Math. 130, Theorem 1.6; the precise coercivity is the Tian-Aubin functional bound). Coercivity of $M$ implies $M$ achieves its minimum, and the minimum is a Kähler-Einstein metric (an elementary Euler-Lagrange computation). By the Yau-Tian-Donaldson theorem (Kähler-Einstein implies K-polystable), $X$ is K-polystable. Since the alpha invariant bound also forces $Aut (X)$ to be finite (Demailly-Kollár 2001, Ann. Inst. Fourier 51), K-polystability sharpens to K-stability. $□$

Theorem (Li-Xu reduction to special test configurations; Li-Xu 2014), stated without full proof here — see Ann. of Math. 180 ^{[source pending]}. The proof runs the minimal model programme with scaling on $X / A^{1}$ to convert an arbitrary test configuration $(X, L)$ into a special test configuration $(X^{sp}, - K_{X^{sp} / A^{1}})$ with normal $Q$ -Fano klt central fibre. The MMP step uses Birkar-Cascini-Hacon-McKernan 2010 (existence of MMP for klt pairs over a base curve) and the BCHM cone theorem. The key inequality $DF (X^{sp}) \leq DF (X)$ is established by a discrepancy-decrease lemma valid along each MMP step.

Theorem (Odaka K-stability criterion via discrepancies; Odaka 2013), stated without full proof here — see Ann. of Math. 177 ^{[source pending]}. Odaka's identification of $DF (X, L)$ with the log canonical threshold formula $$ \mathrm{DF}(\mathcal{X}, \mathcal{L}) = \frac{1}{(-K_X)^n} \cdot (-K_{\mathcal{X}_0/\mathbb{A}^1})^{n+1} ;; \text{(up to normalisation)} $$ proceeds by equivariant Riemann-Roch on the central fibre, identifying the leading and subleading coefficients $a_{0}, a_{1}, b_{0}, b_{1}$ with intersection numbers involving the relative canonical bundle of the family. The full proof in Odaka 2013 §2-3 is short once equivariant Riemann-Roch is in hand.

Theorem (uniform K-stability iff existence of Kähler-Einstein metric; Boucksom-Hisamoto-Jonsson 2017, Berman-Boucksom-Jonsson 2021), stated without full proof here — see Ann. Inst. Fourier 67 and J. Eur. Math. Soc. ^{[source pending]}. The proof refines the YTD theorem by replacing K-polystability (DF $\geq 0$ , equality only on products) with uniform K-stability (DF bounded below by a strict positive multiple of a non-Archimedean norm of the test configuration). The refinement is necessary because the YTD theorem proves coercivity of the Mabuchi functional, which corresponds analytically to uniform K-stability rather than to K-polystability. The Berman-Boucksom-Jonsson 2021 work closes the gap, showing the two notions coincide for smooth Fano manifolds.

Connections Master

Geometric invariant theory 04.10.02. K-stability is the infinite-dimensional analogue of Mumford's classical GIT. The role of the algebraic group $G$ is played by the gauge group of the polarisation, the role of a one-parameter subgroup is played by a test configuration, and the role of the Hilbert-Mumford weight is played by the Donaldson-Futaki invariant. The Yau-Tian-Donaldson equivalence is the Kempf-Ness theorem for this infinite-dimensional setup, identifying algebraic stability with the existence of a zero of an analytic moment map (the Ricci-curvature deviation from the average).
Hilbert-Mumford numerical criterion 04.10.03. The Donaldson-Futaki invariant is the natural generalisation of the Hilbert-Mumford weight $μ^{L} (x, λ)$ to the infinite-dimensional GIT of Fano polarisations. In both cases the stability of a point (a Fano variety) is detected by the sign of a numerical pairing between a one-parameter subgroup (a test configuration) and the linearisation (the polarisation). The classical numerical criterion underwrites the algebraic content of the Chen-Donaldson-Sun theorem.
Kempf-Ness GIT-symplectic dictionary 04.10.04. The Yau-Tian-Donaldson theorem is the Kempf-Ness theorem applied to the infinite-dimensional symplectic-reduction picture of Kähler-Einstein moduli. In the finite-dimensional Kempf-Ness theorem, algebraic GIT semistability equals the existence of a zero of a moment map; in the K-stability setting, K-polystability equals the existence of a Kähler-Einstein metric (a zero of the Ricci moment map). The exact infinite-dimensional analogue was articulated by Donaldson 1997 and Tian 1997.
Moduli of curves 04.10.01. The K-moduli compactification of K-polystable Fano varieties of fixed dimension and volume is the modern higher-dimensional analogue of the Deligne-Mumford compactification of moduli of curves. In both constructions, a GIT-style stability criterion (Mumford slope or K-polystability) determines a projective good moduli space, and a degeneration framework (stable curves or special test configurations) provides the compactification. The conceptual parallel motivated Tian's 1997 formulation of K-stability in the first place.
Hilbert scheme 04.10.05. Test configurations are parameterised by certain Hilbert schemes of flat families over $A^{1}$ , and the K-stability test ranges over the entire Hilbert-scheme parameter space. The reduction to special test configurations (Li-Xu 2014) restricts to a subvariety of this Hilbert scheme — those families whose central fibre is a normal $Q$ -Fano variety — and the K-moduli construction extracts a moduli space from this restricted parameter space via boundedness theorems (Birkar BAB).
Variation of GIT 04.10.09. K-stability is a far-reaching infinite-dimensional VGIT: the analogue of the equivariant ample cone is the Kähler cone of the underlying Fano variety, and the wall-and-chamber structure of K-moduli generalises the finite-dimensional VGIT picture of Dolgachev-Hu and Thaddeus to the moduli of Fano polarisations. Cross-domain wall-crossings of K-stability connect to the wall-crossings of Bridgeland stability conditions on derived categories of Fano varieties.
Ample line bundle 04.05.05. The Fano condition is the ampleness of $- K_{X}$ — a specific positivity condition on the anticanonical line bundle. K-stability is the higher-order positivity condition: not only does $- K_{X}$ embed $X$ into projective space (ampleness), but $X$ is stably embedded in the sense that no algebraic degeneration of $X$ along a $C^{\times}$ -action produces a more degenerate polarisation. The transition from ampleness to K-stability is the conceptual step from positivity of curvature to existence of a canonical metric of positive curvature.
Canonical sheaf 04.08.02. The canonical sheaf $ω_{X} = det Ω_{X}^{1}$ enters K-stability through its anti-version: $- K_{X}$ is the polarising line bundle, the volume $(- K_{X})^{n} / n!$ is the leading coefficient $b_{0}$ , and the slope of the canonical class with respect to the polarisation enters $b_{1}$ via the genus-formula-style identity for Fano varieties.

Historical & philosophical context Master

The story begins with Eugenio Calabi's 1954 ICM lecture and 1957 paper The space of Kähler metrics ^{[Calabi 1957]} (in Proc. ICM Amsterdam 1954, vol. 2, pp. 206-207), where Calabi conjectured that every Kähler class on a compact Kähler manifold contains a metric of prescribed Ricci form satisfying the necessary topological constraint $[Ric] = c_{1} (K_{X}) \in H^{1, 1} (X, R)$ . The conjecture was proved by Shing-Tung Yau in 1978 (Comm. Pure Appl. Math. 31) ^{[source pending]} for the cases of zero Ricci ( $K_{X} = 0$ , the Calabi-Yau theorem) and negative Ricci ( $K_{X}$ ample, the Aubin-Yau theorem). The positive case — Fano varieties with $- K_{X}$ ample — remained open and resistant to the methods that resolved the other two cases.

Akito Futaki 1983 (Inventiones Math. 73) ^{[source pending]} identified the first algebraic obstruction: the Futaki invariant, a character of the Lie algebra of holomorphic vector fields on $X$ that must vanish for a Kähler-Einstein metric to exist. Yozô Matsushima had earlier (1957 Nagoya Math. J. 11) ^{[source pending]} shown that the automorphism group of a Kähler-Einstein Fano manifold must be reductive; combining these gives a finite catalogue of necessary conditions, none of which were sufficient. Tian-Yau 1987 (Mathematical Aspects of String Theory) ^{[Tian-Yau 1987]} established Kähler-Einstein existence on Fano surfaces with vanishing Futaki invariant; Tian 1990 (Inventiones Math. 101) ^{[Tian 1990]} extended to all del Pezzo surfaces. The positive case in higher dimension required a new conceptual framework.

Gang Tian 1997 (Inventiones Math. 130, Kähler-Einstein metrics with positive scalar curvature) ^{[source pending]} introduced K-stability via special degenerations of $X$ with normal Fano central fibre, and conjectured the equivalence with Kähler-Einstein existence. Simon Donaldson 2002 (J. Differential Geom. 62, Scalar curvature and stability of toric varieties) ^{[source pending]} reformulated K-stability algebraically through the Donaldson-Futaki invariant — the leading-coefficient expansion of the weight of the central-fibre $C^{\times}$ -action on $det H^{0} (X_{0}, L^{k})$ — generalising Futaki's invariant from one-parameter subgroups of $Aut (X)$ to arbitrary test configurations. The conjectured equivalence is now called the Yau-Tian-Donaldson conjecture, with Yau credited for the underlying Calabi-conjecture programme, Tian for the K-stability formulation, and Donaldson for the algebraic reformulation.

The conjecture was proved for Fano manifolds by Xiuxiong Chen, Simon Donaldson, and Song Sun in 2014, published in three consecutive papers in the Journal of the American Mathematical Society 28 (2015) ^{[source pending]}. The proof uses Kähler-Einstein metrics with cone singularities along an anticanonical divisor — a technique developed by Donaldson 2012, Jeffres-Mazzeo-Rubinstein 2016, and Berman 2016 — together with the Cheeger-Colding-Tian compactness theory for polarised Kähler-Einstein manifolds. Chi Li and Chenyang Xu (2014, Ann. of Math. 180) ^{[source pending]} simplified the algebraic side by reducing K-stability to special test configurations via the minimal model programme; Yuji Odaka (2013, Ann. of Math. 177) ^{[source pending]} connected the Donaldson-Futaki invariant to log canonical thresholds and discrepancies, embedding K-stability in the framework of birational geometry. Sebastien Boucksom, Tomoyuki Hisamoto, and Mattias Jonsson (2017, Ann. Inst. Fourier 67) ^{[source pending]} gave the non-Archimedean reformulation, identifying test configurations with non-Archimedean Fubini-Study metrics on the Berkovich analytification of $- K_{X}$ .

The construction of the K-moduli space — the projective compactification of moduli of K-polystable Fano varieties — was completed by Xu and collaborators in 2017-2022 (Xu 2020, Ann. of Math. 191 ^{[Xu 2020]}; Liu-Xu, Xu-Liu, Blum-Halpern-Leistner-Liu-Xu and ABHLX). The construction rests on Caucher Birkar's BAB boundedness theorem (Birkar 2016) ^{[Birkar 2016]}, for which Birkar received the 2018 Fields Medal; Birkar's subsequent work on log Calabi-Yau pairs was a central topic of his 2022 ICM plenary lecture. The Yau-Tian-Donaldson programme remains one of the largest interdisciplinary research efforts in modern algebraic geometry, drawing on complex differential geometry, partial differential equations, geometric invariant theory, birational geometry, and Berkovich-analytic non-Archimedean geometry. The generalisation to constant-scalar-curvature Kähler metrics in arbitrary polarisations remains open and is the subject of active research as of 2026.

Bibliography Master

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@article{Donaldson2002,
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@article{ChenDonaldsonSun2015c,
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@article{Odaka2013,
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@article{Xu2020,
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@book{MumfordFogartyKirwan,
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}

Prerequisites

04.10.02
04.10.03
04.05.05
04.08.02
04.10.04

Tier anchors

beginner: Strogatz-style: a Fano variety admits a Kähler-Einstein metric exactly when an algebraic stability condition — K-stability — holds; the test of stability is a degeneration
intermediate: Székelyhidi *An Introduction to Extremal Kähler Metrics* (AMS GSM 152, 2014); Thomas 2006 *Notes on GIT and symplectic reduction for bundles and varieties* (Surveys Diff. Geom. 10); Donaldson 2002 *J. Differential Geom.* 62
master: Tian 1997 *Inventiones Math.* 130 (*Kähler-Einstein metrics with positive scalar curvature*); Donaldson 2002 *J. Differential Geom.* 62 (*Scalar curvature and stability of toric varieties*); Chen-Donaldson-Sun 2014 *J. Amer. Math. Soc.* 28 (three papers: *Kähler-Einstein metrics on Fano manifolds I, II, III*); Li-Xu 2014 *Ann. of Math.* 180 (*Special test configurations and K-stability*); Odaka 2013 *Ann. of Math.* 177 (*A generalization of the Ross-Thomas slope theory*); Boucksom-Hisamoto-Jonsson 2017 *Ann. Inst. Fourier* 67 (non-Archimedean K-stability); Mumford *Geometric Invariant Theory* 3rd ed. (Springer Ergebnisse 34, 1994) as the conceptual source of the stability picture

References

Tian — Kähler-Einstein metrics with positive scalar curvature · Inventiones Math. 130 (1997), 1–37 — first formulation of K-stability via special degenerations
Donaldson — Scalar curvature and stability of toric varieties · J. Differential Geom. 62 (2002), 289–349 — algebraic reformulation; Donaldson-Futaki invariant
Chen, Donaldson, Sun — Kähler-Einstein metrics on Fano manifolds I, II, III · J. Amer. Math. Soc. 28 (2015), 183–197, 199–234, 235–278 — proof of the Yau-Tian-Donaldson conjecture for Fano varieties
Li, Xu — Special test configurations and K-stability of Fano varieties · Ann. of Math. 180 (2014), 197–232 — reduction to special test configurations with normal Fano central fibres
Odaka — A generalization of the Ross-Thomas slope theory · Ann. of Math. 177 (2013), 645–661 — algebraic foundations of K-stability via discrepancies
Boucksom, Hisamoto, Jonsson — Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs · Ann. Inst. Fourier 67 (2017), 743–841 — non-Archimedean reformulation of K-stability
Yau — Calabi conjecture and some new results in algebraic geometry · Proc. Nat. Acad. Sci. USA 74 (1977), 1798–1799 — original Calabi conjecture in non-positive case
Tian, Yau — Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry · Mathematical Aspects of String Theory (San Diego 1986), World Scientific 1987, pp. 574–628 — the Tian-Yau theorem in the negative case
Xu — A minimizing valuation is quasi-monomial · Ann. of Math. 191 (2020), 1003–1030 — K-moduli foundations; cf. also Liu-Xu, Xu-Liu, Blum-Halpern-Leistner-Liu-Xu
Mumford, Fogarty, Kirwan — Geometric Invariant Theory · 3rd ed., Springer Ergebnisse 34, 1994 — the finite-dimensional conceptual source of K-stability

Estimated time

beginner: 16m
intermediate: 45m
master: 80m