--- title: Yau–Tian–Donaldson Conjecture url: https://www.emergentmind.com/topics/yau-tian-donaldson-conjecture type: topic --- # Yau–Tian–Donaldson Conjecture The **Yau–Tian–Donaldson (YTD) conjecture** is a correspondence between canonical Kähler metrics on polarized projective varieties and algebro-geometric stability under degenerations. In its constant-scalar-curvature form, it predicts that a smooth polarized projective manifold \((X,L)\) admits a constant scalar curvature Kähler (cscK) metric in \(c_1(L)\) if and only if \((X,L)\) is \(K\)-polystable. The analytic side is governed by the Mabuchi \(K\)-energy, while the algebraic side is expressed through test configurations, the Donaldson–Futaki invariant, and their non-Archimedean extensions. Precise equivalences have been established in important settings, including Fano varieties, singular \(\mathbb Q\)-Fano varieties, toric and spherical varieties, weighted soliton equations, and, in recent work, arbitrary smooth polarized projective varieties through finite-energy non-Archimedean methods. ## 1. Geometric formulation and historical scope Let \(X\) be a smooth projective variety of complex dimension \(n\), and let \(L\) be an ample line bundle. A Kähler form in the polarization class \(c_1(L)\) can be written as \[ \omega_\varphi=\omega+dd^c\varphi, \] where \(\omega\in c_1(L)\) is a reference Kähler form and \(\varphi\) is a Kähler potential. Its scalar curvature is denoted by \(S(\omega_\varphi)\). The metric is cscK when \[ S(\omega_\varphi)\equiv \overline S, \] where the average scalar curvature is determined cohomologically by \[ \overline S = n\frac{c_1(X)\cdot c_1(L)^{n-1}}{c_1(L)^n}. \] The classical cscK form of the conjecture is \[ \boxed{ c_1(L)\text{ contains a cscK metric} \quad\Longleftrightarrow\quad (X,L)\text{ is \(K\)-polystable}. } \] The qualification “polystable” is essential in the presence of automorphisms. Product test configurations induced by one-parameter subgroups of \(\operatorname{Aut}(X,L)\) may have zero Donaldson–Futaki invariant even when the associated degeneration is nontrivial as a family. Consequently, equality is permitted along product configurations rather than only along the completely trivial configuration. The Kähler–Einstein problem is a special case. If \(K_X=\lambda L\), then a cscK metric is automatically Kähler–Einstein and satisfies \[ \operatorname{Ric}(\omega_\varphi)=-\lambda\omega_\varphi. \] For Fano manifolds, \(L=-K_X\), and the cscK equation becomes the positive Kähler–Einstein equation. The Fano YTD correspondence was established in substantial generality through the work of Chen–Donaldson–Sun and Tian. The general polarized cscK problem is broader because \(c_1(L)\) need not be proportional to \(c_1(X)\). The modern formulation involves several related stability notions: - \(K\)-semistability requires nonnegativity of the Donaldson–Futaki invariant. - \(K\)-polystability additionally requires equality only for product configurations. - Uniform \(K\)-stability requires a positive lower bound proportional to a norm of the degeneration. - Relative or equivariant stability quotients out directions generated by automorphisms. - Filtration and non-Archimedean formulations test larger classes than finitely generated test configurations. The survey "Variational and non-Archimedean aspects of the Yau--Tian--Donaldson conjecture" emphasizes that the conjecture is not merely a formal equivalence between a fourth-order PDE and an algebraic inequality: it is a variational compactness, coercivity, and regularity problem [1805.03289]. ## 2. Test configurations and algebraic stability A test configuration for \((X,L)\) consists of a normal projective variety \(\mathcal X\), a \(\mathbb C^*\)-equivariant morphism \[ \pi:\mathcal X\longrightarrow\mathbb A^1, \] and a relatively ample or semiample \(\mathbb Q\)-line bundle \(\mathcal L\), together with an equivariant identification over \(\mathbb A^1\setminus\{0\}\): \[ (\mathcal X,\mathcal L)|_{\mathbb A^1\setminus\{0\}} \simeq (X,L^r)\times(\mathbb A^1\setminus\{0\}) \] for some exponent \(r>0\). The fiber \(\mathcal X_0\) is the central fiber and may be reducible or non-reduced. A product test configuration is induced by a one-parameter subgroup of \(\operatorname{Aut}(X,L)\). For the Hilbert polynomial and total weight polynomial, \[ h(k)=a_0k^n+a_1k^{n-1}+O(k^{n-2}), \] \[ w(k)=b_0k^{n+1}+b_1k^n+O(k^{n-1}), \] a standard normalization of the Donaldson–Futaki invariant is \[ \operatorname{DF}(\mathcal X,\mathcal L) = \frac{a_1b_0-a_0b_1}{a_0}, \] up to sign and normalization conventions. A polarized variety is \(K\)-semistable if \[ \operatorname{DF}(\mathcal X,\mathcal L)\geq 0 \] for every relevant test configuration. It is \(K\)-polystable if equality occurs only for product configurations. Uniform \(K\)-stability is expressed by an inequality such as \[ \operatorname{DF}(\mathcal X,\mathcal L) \geq \varepsilon\,J^{\mathrm{NA}}(\mathcal X,\mathcal L), \qquad \varepsilon>0, \] where \(J^{\mathrm{NA}}\) is a non-Archimedean norm measuring the nontriviality of the degeneration. The distinction between ordinary and relative stability reflects automorphism directions. If \(G\subset\operatorname{Aut}^0(X,L)\) is reductive and \(T\) is the identity component of its center, a \(G\)-equivariant test configuration can be twisted by a cocharacter \(\mu\in N_{\mathbb Q}(T)\). Relative uniform stability takes the form \[ \operatorname{DF}(\mathcal X,\mathcal L) \geq \sigma \inf_{\mu\in N_{\mathbb Q}(T)} J^{\mathrm{NA}}(\mathcal X_\mu,\mathcal L_\mu). \] The infimum removes precisely the product directions generated by the central torus. This is the appropriate formulation when cscK metrics are unique only modulo automorphisms [2605.30063]. For Fano and log-Fano pairs, special test configurations are particularly important. A special configuration has a normal irreducible central fiber and satisfies an anticanonical relation such as \[ \mathcal L\sim_{\mathbb Q} -\left(K_{\mathcal X/\mathbb A^1}+\mathcal D\right). \] In this setting, special-test-configuration reduction, relative minimal model programs, and valuative criteria can replace arbitrary test configurations. For admissible singular \(\mathbb Q\)-Fano varieties, \(K\)-polystability implies existence of a Kähler–Einstein metric, while the converse was already known [1711.09530]. ## 3. The Mabuchi functional, coercivity, and geodesics The Mabuchi \(K\)-energy \(M\) is characterized along a smooth path \(\varphi_t\) by \[ \frac{d}{dt}M(\varphi_t) = -\int_X \dot\varphi_t \bigl(S(\omega_{\varphi_t})-\overline S\bigr) \frac{\omega_{\varphi_t}^n}{n!}. \] Its critical points are precisely cscK metrics. The Monge–Ampère energy \(E\) satisfies \[ \frac{d}{dt}E(\varphi_t) = \int_X\dot\varphi_t\,\operatorname{MA}(\varphi_t), \qquad \operatorname{MA}(\varphi) = V^{-1}\omega_\varphi^n. \] The \(J\)-functional is a translation-invariant exhaustion functional, comparable to Darvas’s \(d_1\)-metric. In a standard normalization, \[ J(\varphi) = V^{-1}\int_X\varphi\,\omega^n-E(\varphi). \] The analytic form of properness is coercivity: \[ M(\varphi)\geq \varepsilon J(\varphi)-C. \] When automorphisms are present, \(J\) is replaced by a reduced functional obtained by minimizing over the automorphism group. The smooth space of Kähler potentials is completed in the \(d_1\)-metric by the finite-energy space \[ E^1\subset\operatorname{PSH}(X,\omega). \] The Mabuchi functional extends as a lower semicontinuous functional on \(E^1\), convex along weak geodesics. Weak geodesics are obtained pluripotential-theoretically from solutions of the homogeneous complex Monge–Ampère equation \[ (\pi_X^*\omega+dd^cU)^{n+1}=0. \] The variational framework yields the chain \[ \text{cscK metric} \Longrightarrow \text{coercivity of }M \Longrightarrow \text{uniform stability}. \] Conversely, if coercivity fails, one can often construct a nontrivial finite-energy geodesic ray along which the Mabuchi functional has nonpositive asymptotic slope. Such a ray is interpreted as an analytic destabilizing direction. In the Fano setting, the Ding functional \[ D=L-E \] is often more convenient than the Mabuchi functional. For twisted Kähler–Einstein currents, the twisted Ding functional is \[ D_\theta(u)=L_\theta(u)-E(u), \] and a critical point is a finite-energy solution of the twisted Monge–Ampère equation. Coercivity, \[ D_\theta(u)\geq\varepsilon J(u)-C, \] implies existence of a twisted Kähler–Einstein current. Under suitable uniqueness hypotheses, existence also implies coercivity. The variational approach therefore replaces a continuity method by convexity, geodesic-ray extraction, and minimization in \(E^1\) [1509.04561]. A weighted version treats equations of the form \[ g(m_\varphi)\frac{(dd^c\varphi)^n}{n!} = e^{-(\varphi-\varphi_0)}d\mu_0, \] where \(m_\varphi\) is a moment map and \(g\) is a positive function on the moment polytope. The associated weighted Ding functional is \[ D_{g,\Theta}(\varphi) = -E_g(\varphi)+L_\Theta(\varphi). \] For suitable torus actions and reductive groups, existence of generalized Kähler–Ricci \(g\)-solitons is equivalent to equivariant uniform twisted \(g\)-Ding stability [2006.00903]. ## 4. Non-Archimedean geometry and slope formulas The non-Archimedean side is constructed over the trivially valued field \((\mathbb C,|\cdot|_0)\). The Berkovich analytification \(X_{\mathrm{NA}}\) consists of semivaluations, with divisorial valuations forming a dense algebro-geometric subset. A normal ample test configuration determines a non-Archimedean metric or potential \(\phi\). Its non-Archimedean energy is intersection-theoretic: \[ E^{\mathrm{NA}}(\phi) = \frac{(\overline{\mathcal L}^{\,n+1})}{(n+1)V}, \] and its non-Archimedean \(J\)-functional measures distance from the trivial metric. The non-Archimedean Mabuchi functional has the structure \[ M^{\mathrm{NA}} = \overline S\,E^{\mathrm{NA}} + R^{\mathrm{NA}} + H^{\mathrm{NA}}, \] where \(H^{\mathrm{NA}}\) is defined using the log-discrepancy function \(A_X\). For test configurations with reduced central fiber, \[ M^{\mathrm{NA}}(\phi_{\mathcal X,\mathcal L}) = \operatorname{DF}(\mathcal X,\mathcal L). \] In general, the Donaldson–Futaki invariant dominates the non-Archimedean Mabuchi functional, with equality when the central fiber is reduced. The fundamental bridge is the association \[ E^1_{\mathrm{NA}} \hookrightarrow E^1_{\mathrm{rad}}, \] where \(E^1_{\mathrm{rad}}\) is the asymptotic cone of the finite-energy space of Kähler potentials. A non-Archimedean metric determines a maximal weak geodesic ray, and for the relevant functionals, \[ \lim_{t\to\infty}\frac{M(\varphi_t)}{t} = M^{\mathrm{NA}}(\phi). \] Similarly, \[ \lim_{t\to\infty}\frac{E(\varphi_t)}{t} = E^{\mathrm{NA}}(\phi), \qquad \lim_{t\to\infty}\frac{J(\varphi_t)}{t} = J^{\mathrm{NA}}(\phi). \] Entropy is the technically difficult term. The entropy slope theorem identifies the Archimedean entropy slope with non-Archimedean entropy for finite-energy non-Archimedean directions and assigns infinite slope to rays not represented in that space. This makes the Donaldson–Futaki invariant the asymptotic Mabuchi slope of a degeneration [2602.10800]. The non-Archimedean formulation is stronger than testing only ordinary test configurations. Classical \(K\)-stability tests positivity on the space of ample test configurations, whereas finite-energy or \(\widehat K\)-stability tests positivity on a completion containing general filtrations and non-Archimedean potentials. A central regularization question is whether finite-energy non-Archimedean metrics can be approximated by ample test configurations while preserving entropy. Special Fujita approximations establish this entropy approximation for smooth projective varieties [2605.30063]. The resulting uniform correspondence for smooth polarized varieties is: \[ \boxed{ c_1(L)\text{ contains a cscK metric} \Longleftrightarrow (X,L)\text{ is }\operatorname{Aut}^{\circ}(X,L)\text{-uniformly \(K\)-stable}. } \] The automorphism group is understood modulo the irrelevant fiberwise scalar action, and the stability inequality is reduced along automorphism-generated product directions [2605.30063]. ## 5. Valuative, singular, and relative extensions For Fano and log-Fano varieties, stability can be expressed through divisorial valuations. If \(E\) is a prime divisor over \(X\), its log discrepancy is \(A(E)\), and its expected vanishing order is \[ S_L(E) = \frac{1}{\operatorname{vol}(L)} \int_0^\infty \operatorname{vol}(\pi^*L-xE)\,dx. \] The Fujita–Odaka or Blum–Jonsson stability threshold is \[ \delta(L;\theta) = \inf_E\frac{A_\theta(E)}{S_L(E)}. \] For the Fano polarization \(L=-K_X\), \[ \delta(-K_X)>1 \quad\Longleftrightarrow\quad (X,-K_X)\text{ is uniformly \(K\)-stable}. \] In the twisted setting, \(\delta(L;\theta)>1\) gives coercivity of a twisted Ding functional and existence of a twisted Kähler–Einstein current [2102.02438]. The greatest twisted Ricci lower bound can be expressed as \[ \beta_\theta(X,L) = \min\{\delta_\theta(X,L),s_\theta(X,L)\}, \] where \(s_\theta\) is a nef threshold. The two terms represent distinct obstructions: \(s_\theta\) is cohomological, while \(\delta_\theta\) is valuative and algebro-geometric [1509.04561]. For singular \(\mathbb Q\)-Fano varieties, the log-Fano condition requires \(X\) to be normal, \(-K_X\) to be ample and \(\mathbb Q\)-Cartier, and \(X\) to have klt singularities. For admissible \(\mathbb Q\)-Fano varieties possessing a log resolution with discrepancies \[ -11 \] is equivalent to uniform \(\mathbf D\)-log Ding stability and to existence and uniqueness of the prescribed-singularity Kähler–Einstein metric. The associated stability locus is strongly open in the space of generalized \(b\)-divisors [2302.07213]. ## 6. Symmetric, weighted, toric, and effective forms Symmetry can reduce the YTD problem to finite-dimensional or convex-geometric analysis. For a compact toric manifold, a torus-invariant Kähler metric is encoded by a strictly convex symplectic potential \(u\) on the moment polytope \(P\). The scalar curvature is given by Abreu’s formula \[ S(u) = -\sum_{i,j} \frac{\partial^2u^{ij}}{\partial x_i\partial x_j}, \] where \((u^{ij})\) is the inverse Hessian of \(u\). The relevant Donaldson functional is \[ \mathcal L_V(f) = \int_{\partial P}f\,d\sigma - \int_P(\overline S+V)f\,dx, \] where \(V\) is the extremal affine function. Rational piecewise affine convex functions correspond to toric test configurations, and affine functions represent product configurations generated by torus automorphisms. For polarized toric manifolds, \[ \boxed{ \text{\(S\)-invariant extremal metric} \Longleftrightarrow \text{uniform relative \(K\)-polystability} \Longleftrightarrow \text{\(T\)-coercivity of the relative \(K\)-energy}. } \] The stronger filtration version tests all convex functions on \(P\), not only rational piecewise affine functions. In the toric and homogeneous toric-bundle settings, filtration stability is equivalent to uniform stability, and the corresponding prescribed-scalar-curvature or generalized Abreu equation has a smooth solution [2110.08491] [2110.10386]. Rank-one spherical and cohomogeneity-one manifolds admit a similar reduction to one-dimensional convex functions. Their stability functional is an explicit boundary-minus-interior integral involving the Duistermaat–Heckman density and root-theoretic correction terms. For rank-one spherical varieties, stability can be detected by special equivariant test configurations and a single combinatorial condition. These results also demonstrate that not every Kähler class on a cohomogeneity-one manifold admits an extremal metric: explicit examples have stable classes admitting cscK metrics and unstable classes admitting no extremal metric [2011.07135]. Weighted soliton equations generalize both Kähler–Einstein and cscK equations. Given a torus action with moment polytope \(P\) and a positive weight \(g\), the weighted Monge–Ampère equation \[ g(m_\varphi)\frac{(dd^c\varphi)^n}{n!} = e^{-(\varphi-\varphi_0)}d\mu_0 \] includes twisted Kähler–Einstein metrics, Kähler–Ricci solitons, and Mabuchi solitons. The corresponding generalized Ding invariant is \[ D_{g,\Theta}^{\mathrm{NA}} = -E_g^{\mathrm{NA}} + L_\Theta^{\mathrm{NA}}. \] Under suitable reductivity, torus-equivariance, and positivity hypotheses, finite-energy solutions are equivalent to equivariant uniform twisted \(g\)-Ding stability [2006.00903]. The effective YTD problem asks whether, for fixed dimension and Picard rank, cscK existence can be decided by testing only test configurations of bounded complexity, such as configurations with a bounded number of irreducible central-fiber components. For spherical varieties, equivariant test configurations correspond to convex piecewise affine functions on a moment polytope, and the number of linearity domains equals the number of irreducible components of the central fiber. In rank-one cases, the criterion can reduce to a single functional or a barycenter condition. No general bound of the form \(m=m(n,\rho)\) is known [2509.08760]. A manuscript claiming a counterexample to the cscK YTD conjecture has been circulated, but its supplied material contains malformed formulas, apparent logical errors, and unverified external inputs. It therefore establishes only a claim rather than a verified disproof; the broader correspondence should not be regarded as refuted on that basis [2608.19301]. ## 7. Scope, limitations, and current formulation The YTD conjecture has several nonidentical formulations, and conclusions must be matched to their hypotheses. For smooth polarized varieties, the strongest general result in the supplied literature identifies cscK existence with \(\operatorname{Aut}^{\circ}(X,L)\)-uniform \(K\)-stability. Its proof combines: - coercivity of the Mabuchi functional; - finite-energy pluripotential theory; - weak geodesic rays; - non-Archimedean metrics and entropy; - slope formulas; - special Fujita approximations; - regularization of big models by ample test configurations. In the Fano case, Ding stability, \(K\)-stability, and valuative thresholds provide especially effective criteria. For singular \(\mathbb Q\)-Fano varieties, uniform and group-relative versions have been established under klt hypotheses, with the automorphism group incorporated through reductive equivariant stability. Several distinctions remain essential: - **Polystability versus uniform stability**: polystability allows zero invariants along product directions, whereas uniform stability imposes a positive quantitative gap away from those directions. - **Test configurations versus filtrations**: finitely generated filtrations correspond to algebraic test configurations, while arbitrary filtrations and finite-energy non-Archimedean metrics form larger spaces. - **Ordinary versus relative stability**: continuous automorphism groups require quotienting the stability norm by automorphism-generated directions. - **Fano versus general polarization**: Ding functionals and valuative thresholds are naturally adapted to Fano geometry, while the general cscK problem is governed by the Mabuchi functional. - **Existence versus regularity**: finite-energy minimizers must be promoted to smooth metrics in the smooth polarized case, whereas singular settings generally yield bounded or weak currents that are smooth only on the appropriate regular locus. - **Algebraic criteria versus computability**: even when YTD is established, testing all relevant degenerations or valuations can remain difficult. The conceptual structure is summarized by \[ \boxed{ \text{canonical metric} \Longleftrightarrow \text{coercivity of an Archimedean energy} \Longleftrightarrow \text{positive asymptotic slopes} \Longleftrightarrow \text{non-Archimedean stability}. } \] Test configurations provide algebraic directions at infinity in the space of Kähler metrics, while the Donaldson–Futaki invariant is the corresponding asymptotic Mabuchi slope. This identification is the central organizing principle of the Yau–Tian–Donaldson theory. Source: https://www.emergentmind.com/topics/yau-tian-donaldson-conjecture