--- title: K-Stability of Spherical Varieties url: https://www.emergentmind.com/topics/k-stability-of-spherical-varieties type: topic --- # K-Stability of Spherical Varieties K-stability of spherical varieties is the algebro-geometric stability theory for varieties on which a Borel subgroup has an open orbit, and it is expressed in this setting through explicit combinatorial and valuative data attached to the open spherical homogeneous space. Across polarized, Fano, weighted, and affine-cone settings, the basic objects are the weight lattice, valuation cone, colors, moment polytope, and Duistermaat–Heckman density; these data turn Donaldson–Futaki, Mabuchi, Ding, and normalized-volume functionals into concrete integrals or linear forms on convex functions and valuations. In this sense, spherical varieties form the principal nonabelian extension of the toric theory, with toric, horospherical, symmetric, and group-compactification cases appearing as special subclasses [1608.01852], [2405.05833], [2509.08760]. ## 1. Spherical data, momentum polytopes, and combinatorial structure A normal \(G\)-variety \(X\) is spherical if a Borel subgroup \(B \subset G\) has an open dense orbit. Equivalently, in the affine case the coordinate ring is multiplicity free, and for polarized projective varieties \((X,L)\), the \(G\)-representations \(H^0(X,L^n)\) are multiplicity free for all \(n \ge 0\). The open orbit is \(G/H\) for a spherical subgroup \(H\), and the basic lattices are the weight lattice \(M\) of \(B\)-semi-invariant rational functions and its dual \(N=\operatorname{Hom}(M,\mathbb Z)\) [1809.08171]. The valuation cone \(\mathcal V \subset N \otimes \mathbb R\) is the cone of values of \(G\)-invariant valuations restricted to \(B\)-eigenfunctions. Colors are the \(B\)-stable prime divisors in the open orbit, and Luna–Vust theory organizes spherical embeddings by colored fans built from \(\mathcal V\), color valuations, and the corresponding orbit combinatorics. This structure governs both projective embeddings and degenerations. In particular, the moment polytope of a polarized spherical variety is the convex hull of dominant highest weights occurring in sections, and it admits descriptions both as a convex hull and as an intersection of half-spaces defined by facet normals and admissibility conditions [1809.08171]. For \(\mathbb Q\)-Fano spherical varieties, anticanonical geometry is encoded by the same data. The classification by reflexive or \(\mathbb Q\)-reflexive momentum triples gives a combinatorial description of Fano spherical varieties, while the anticanonical divisor is represented by a piecewise linear support function on the colored fan. This support function is the spherical analogue of the toric anticanonical support function and organizes admissible degenerations and K-stability computations [1809.08171]. A recurrent point in the literature is that spherical varieties generalize toric varieties. In the toric case the valuation cone and momentum data reduce to the standard fan and polytope description; in the spherical case the same role is played by moment polytopes together with valuation cones, colors, and root-theoretic weights [1608.01852]. ## 2. Equivariant test configurations and convex-analytic functionals For polarized spherical varieties, \(G\)-equivariant test configurations admit an explicit convex-geometric encoding. In one formulation, a \(G\)-equivariant test configuration for \((X,L)\) is encoded by a convex, piecewise rational affine function \[ f=\sup(c_1-v_1,\dots,c_k-v_k) \] on the translated moment polytope, with slopes \(v_j\) in the valuation cone \(\mathcal V\). The number of linearity domains of \(f\) equals the number of irreducible components of the central fiber, and product configurations correspond, up to base change, to affine functions coming from the linear part \(\mathcal V \cap (-\mathcal V)\) [2509.08760]. In another formulation, used for polarized spherical varieties with \(W\)-invariant polytope \(P\), equivariant test configurations correspond to rational convex piecewise-linear functions \(u\) on \(P_+\) whose gradients lie in the dominant chamber \(V_+=-\mathcal V\). Product test configurations correspond to central affine functions. The Donaldson–Futaki invariant becomes a linear functional on such convex functions, and in the relative setting one obtains the reduced Futaki functional \[ \mathcal{L}_X(u) = \sum_{A=1}^{d_+} d_A \int_{F_A\cap P_+} u\, \langle y,\nu_A\rangle\, \rho\, d\sigma_0 + \int_{P_+} u\, (S+\theta_X)\, \rho\, dy - \int_{P_+} u\, \langle 2p,\nabla\rho\rangle\, dy. \] Here \(\rho\) is the Duistermaat–Heckman density, \(F_A\) are the outer facets, and \(\theta_X\) is the central affine potential of the extremal vector field [2111.04269]. The non-Archimedean Mabuchi functional is likewise explicit in the spherical setting. For the convex-geometric encoding above, one has \[ M^{NA}(\mathcal X)=\mathcal L(f), \] where \(\mathcal L\) is an integral functional involving boundary and interior terms weighted by root-theoretic polynomials \(P\) and \(Q\). In the smooth polarized spherical setting, existence of a cscK metric in \(c_1(L)\) is equivalent to the condition that \(\mathcal L(f)\ge 0\) for all admissible \(f\), with equality only for affine functions from \(\mathcal V\cap(-\mathcal V)\) [2509.08760]. This reduction is the basic reason spherical K-stability is computationally tractable: the stability problem is transferred from arbitrary test configurations to convex functions whose slopes are constrained by the valuation cone. ## 3. Explicit criteria across polarized, Fano, weighted, and cone settings Several distinct but compatible criteria are now available. They differ by polarization, by whether one works with the Fano or general cscK problem, and by whether the variety is projective or an affine cone. | Setting | Data | Criterion | |---|---|---| | Polarized spherical \((X,L)\) | Convex PL \(f\) with slopes in \(\mathcal V\) | K-polystable iff \(\mathcal L(f)\ge 0\) for all \(f\), with equality only for affine \(f\) from \(\mathcal V\cap(-\mathcal V)\) | | \(\mathbb Q\)-Fano spherical | Moment polytope \(\Delta^+\), valuation cone \(\mathcal V\), DH barycenter | K-polystable iff \(\mathrm{bar}_{DH}(\Delta^+) \in 2\rho_Q+\operatorname{Relint}(\Xi)\) | | Weighted \(\mathbb Q\)-Fano spherical | Weight \(g\), weighted barycenter \(b_g\) | Equivalent to uniformly \(g\)-Ding stable, uniformly \(g\)-K-stable, and equivariantly \(g\)-K-polystable | | Log spherical cone \((Y,D,\xi)\) | Slice \(\Delta_\xi\), angle \(\varpi_\gamma\), valuation cone \(V\) | K-stable iff \(\mathrm{bar}_{DH}(\Delta_\xi)-\varpi_\gamma \in \mathrm{RelInt}(({-V})^\vee)\) | For \(\mathbb Q\)-Fano spherical varieties, Delcroix’s barycenter criterion expresses \(G\)-equivariant K-stability in terms of the anticanonical moment polytope, the valuation cone, and a Duistermaat–Heckman density \[ P(p)\,dp=\prod_{\alpha\in \Phi_{Q^u}}\kappa(\alpha,p)\,dp. \] The Donaldson–Futaki invariant of a special \(G\)-equivariant test configuration is, up to a positive normalization constant, \[ \mathrm{DF}_{\zeta}(\mathcal X,\mathcal L) = C\, \Big\langle 2\rho_Q-\mathrm{bar}_{DH,\tilde\zeta}(\Delta^+),\, \tilde\xi \Big\rangle, \] and K-polystability is equivalent to the weighted barycenter lying in \(2\rho_Q+\mathrm{Relint}(\Xi)\) [1608.01852]. For weighted \(\mathbb Q\)-Fano spherical varieties, the Li–Li–Wang criterion replaces the ordinary barycenter by the \(g\)-weighted barycenter \[ b_g = \frac{1}{V_g}\int_{\Delta^+}\lambda\, g(t_{X_0}(\lambda))\,T(\lambda)\,d\lambda, \qquad V_g = n!\int_{\Delta^+} g(t_{X_0}(\lambda))\,T(\lambda)\,d\lambda. \] The condition \[ b_g \in \kappa_P+\operatorname{RelInt}(-\mathcal V(G/H)) \] is equivalent to \(G\times \operatorname{Aut}_G(X)\)-uniform \(g\)-Ding stability, uniform \(g\)-K-stability, and equivariant \(g\)-K-polystability; by the cited analytic results, it is also equivalent to existence of a Kähler–Ricci \(g\)-soliton [2208.02708]. For affine log spherical cones, the criterion is formulated in terms of the slice \[ \Delta_\xi=\{p\in C_Y^\vee:\langle p,\xi\rangle=n\}, \] its DH barycenter, and the angle functional \(\varpi_\gamma\). The Donaldson–Futaki invariant is \[ \mathrm{Fut}_\xi(Y,D,\nu) = \Big\langle -\frac{\langle \varpi_\gamma,\xi\rangle}{n}\,\mathrm{bar}_{DH}(\Delta_\xi)+\varpi_\gamma,\, \nu \Big\rangle, \] and for a Ricci-flat cone metric, where \(\langle\varpi_\gamma,\xi\rangle=n\), the K-stability condition becomes positivity of \(\langle \mathrm{bar}_{DH}(\Delta_\xi)-\varpi_\gamma,\nu\rangle\) on \(({-V})\setminus\operatorname{lin}(V)\) [2405.05833]. These criteria recover classical special cases. In the toric case, \(\Phi_{Q^u}=\varnothing\), so \(P\equiv 1\) and \(2\rho_Q=0\), giving the ordinary Euclidean barycenter condition. In the horospherical case, the valuation cone is the full space, so the criterion simplifies to a barycenter equality [1608.01852]. ## 4. Semistability, homogeneous Monge–Ampère, and optimal degenerations The spherical framework does not stop at numerical criteria; it also yields a structural description of strict semistability. In the variational approach, if \(\mathcal L_X(u)\ge 0\) for all admissible convex \(u\) and \(\mathcal L_X(u_0)=0\) for some non-central affine \(u_0\), then \(u_0\) is a generalized solution of the homogeneous Monge–Ampère equation \[ \det(D^2u)=0 \] in the Alexandrov sense. The minimizer is therefore developable rather than strictly convex, and its contact set with a supporting affine function has extreme points on the boundary of the polytope [2111.04269]. In rank two, this statement becomes much more explicit. If strict semistability occurs, then there exists a \(W\)-dominate simple piecewise-linear function \(\tilde u\), not central affine, with \(\mathcal L_X(\tilde u)=0\). For \(\mathbb Q\)-Fano spherical varieties, strict K-semistability is equivalent to the existence of a fundamental weight \(w\) such that \[ \mathcal L(l_w)=0, \qquad l_w(y)=\langle w,y\rangle, \] which is the same as saying that the \(\rho\)-barycenter lies on the corresponding wall [2111.04269]. The rank-two theory also identifies polystable degenerations. If \(M\) is strictly K-semistable, \(\operatorname{rank}(M)=2\), \(\dim V_z=1\), and \(P_+=P\cap V_+\), then there is a unique polystable degeneration to a \(\mathbb Q\)-Fano horospherical variety. In this case, the zero set of the Futaki functional is precisely \[ \{u\in\mathcal C_{1,W}\mid \mathcal L(u)=0\} = \{\langle A,y\rangle+c\mid A\in V_+,\ c\in\mathbb R\}, \] and the limit satisfies the barycenter identity \(b_\rho(P_+)=a\) [2111.04269]. An affine-cone analogue is established for spherical log cones: any K-semistable spherical log cone admits a \(G\)-equivariant special degeneration to a K-stable spherical log cone, unique up to \(G\)-equivariant isomorphism preserving the Reeb field. The existence argument degenerates along the vanishing locus of the Futaki invariant, while uniqueness uses the uniqueness of K-stable central fibers in the sense of LWX theory [2405.05833]. A common misconception is that semistability always leaves a large family of optimal limits. In the spherical literature, the opposite is frequently true: in rank-two projective cases and in the log-cone setting, strict semistability often leads to a unique equivariant polystable degeneration. ## 5. Valuations, normalized volume, and compatible divisors A complementary approach replaces convex functions by valuations. For a \(\mathbb Q\)-Fano spherical variety \(X\), the equivariant stability threshold is \[ \delta_G(X) = \inf_{v\in \operatorname{DivVal}_X^G\setminus\{0\}} \frac{A_X(v)}{S(-K_X;v)}. \] The compatible-divisor theory shows that there is a unique effective \(B\)-invariant anticanonical \(\mathbb Q\)-divisor \(D_X^B\) such that \[ S(-K_X;v)=v(D_X^B) \] for all \(G\)-invariant divisorial valuations, hence \[ \delta_G(X) = \inf_{v\in \operatorname{DivVal}_X^G\setminus\{0\}} \frac{A_X(v)}{v(D_X^B)}. \] This divisor is defined as the barycenter of the \(B\)-invariant real linear series, and it is independent of the choice of linearization [2601.00054]. The key representation-theoretic input is that for spherical \(X\), the decomposition \[ H^0(X,mL)=\bigoplus_\lambda H^0(X,mL)_\lambda \] is compatible with every \(G\)-invariant divisorial valuation: each such valuation is constant on every isotypic component and equals the value on the unique \(B\)-semi-invariant line inside that component. This makes the expected vanishing orders \(S_m(L;v)\) and \(S(L;v)\) computable from a single averaged \(B\)-invariant divisor. By piecewise \(\mathbb R_{\ge 0}\)-linearity of the discrepancy on the valuation cone, the infimum reduces to a finite minimum over a prescribed finite set of \(G\)-invariant divisorial valuations [2601.00054]. The affine-cone theory connects these ideas to normalized volume. For a log spherical cone \((Y,D,\xi)\), \[ \widehat{\operatorname{vol}}_{(Y,D)}(\xi) = A_{(Y,D)}(w_\xi)^n\,\operatorname{vol}_Y(\xi) = \langle \varpi_\gamma,\xi\rangle^n\,\operatorname{vol}_Y(\xi), \] and the Donaldson–Futaki invariant is the directional derivative of \(\log \widehat{\operatorname{vol}}\) at \(\xi\), up to normalization. The K-stable Reeb field is therefore the minimizer of normalized volume on the Reeb cone [2405.05833]. This suggests a unified picture: projective Fano criteria, weighted barycenter criteria, and valuative \(\delta\)-invariants are not separate theories but different projections of the same spherical combinatorics. ## 6. Examples, classifications, and failures of naive toric analogies Concrete classifications and case studies are a major part of the subject. For \(\mathbb C^3\) with spherical symmetry under \(SO_3(\mathbb R)\times S^1\), the only complete Calabi–Yau metrics with maximal volume growth are the standard flat metric and the Li–Conlon–Rochon–Székelyhidi AC metrics with horospherical asymptotic cone \(A_1\times \mathbb C\). In the same work, an affine smoothing is exhibited that admits no \(K\)-invariant Calabi–Yau metric asymptotic to the cone, and the asymptotic cone of a complete \(K\)-invariant Calabi–Yau metric on an affine \(G\)-spherical manifold is shown to be unique up to \(G\)-equivariant isomorphism preserving the Reeb field [2405.05833]. Low-rank projective examples show how explicit the criteria can become. For rank-one spherical varieties, K-stability can reduce to a single scalar inequality. In the note on rank-one spherical Fano fourfolds, the blowup of \(\mathbf P^2\times \mathbf P^2\) along the diagonal is studied in arbitrary Kähler classes. After normalization, the problem becomes positivity of an explicit scalar \(C(a,b)\) derived from one-dimensional integrals against the Duistermaat–Heckman density \(P(t)=-2t^3-3t^2-t\). Boundary factorizations give positivity near the walls of the ample cone, and numerical evaluation indicates positivity on the full triangle \(0