Donaldson-Sun Theory in the Conic Case

Abstract: We extend the Donaldson-Sun theory of metric tangent cones to non-collapsing Gromov-Hausdorff limits of conical Kahler-Einstein pairs whose boundary coefficients lie in a fixed finite subset of Q, and prove uniqueness of the log metric tangent cone. Furthermore, under a mild lc compatibility condition, we relate this with the Li-Xu and Li-Liu-Xu stable degeneration machinery. We also construct polarized smooth Kaehler metrics on CP^2 with a uniform lower Ricci bound and volume non-collapsing, whose limit has nonunique tangent cones, showing that the Kahler-Einstein assumption is crucial for rigidity.

• The paper extends Donaldson–Sun theory to non-collapsing limits of conical Kähler–Einstein pairs, showing that each limit and metric tangent cone carries a canonical rational boundary divisor and weak conical Kähler–Einstein metric.
• The paper constructs a logarithmic two-step algebraic degeneration whose boundary cycle matches the metric blow-up of the original divisor, while finite generation of the metric filtration recovers the tangent-cone structure.
• The paper proves that log metric tangent cones are unique and demonstrates through a polarized CP^2 example that this rigidity depends essentially on the Kähler–Einstein condition rather than lower Ricci bounds alone.

Setting and main result

The paper extends the Donaldson–Sun theory of metric tangent cones to non-collapsing Gromov–Hausdorff limits of conical Kähler–Einstein pairs. The objects of study are conical Kähler–Einstein metrics on pairs $(X,\Delta)$ with $\Delta=\sum_a(1-\beta_a)D_a$, where $-(K_X+\Delta)$ is a Kähler $\mathbb Q$-class and the cone angles $2\pi\beta_a$ have denominators dividing a fixed integer $N_{\mathcal F}$ and are uniformly bounded away from $0$ and $2\pi$. The family also carries uniform diameter and volume bounds. A polarized approximation lemma shows that each such pair can be approximated by smooth polarized Kähler metrics with a uniform Ricci lower bound, so that Liu–Székelyhidi's algebraicity theorem applies: the limit $X_\infty$ is homeomorphic to a normal projective variety.

The main theorem establishes a logarithmic version of the Donaldson–Sun tangent-cone picture (2608.18432):

1. The limit metric determines an effective rational Weil divisor $\Delta_\infty=\sum_a(1-\beta_a)D_a$ on $\Delta=\sum_a(1-\beta_a)D_a$0; the pair $\Delta=\sum_a(1-\beta_a)D_a$1 is log $\Delta=\sum_a(1-\beta_a)D_a$2-Gorenstein and $\Delta=\sum_a(1-\beta_a)D_a$3 satisfies the weak conical Kähler–Einstein equation in the Bedford–Taylor sense.
2. Every metric tangent cone $\Delta=\sum_a(1-\beta_a)D_a$4 at $\Delta=\sum_a(1-\beta_a)D_a$5 is a normal affine variety carrying a canonical effective rational Weil divisor $\Delta=\sum_a(1-\beta_a)D_a$6, with $\Delta=\sum_a(1-\beta_a)D_a$7 log $\Delta=\sum_a(1-\beta_a)D_a$8-Gorenstein and $\Delta=\sum_a(1-\beta_a)D_a$9 a weak conical Ricci-flat Kähler cone metric.
3. The metric filtration of $-(K_X+\Delta)$0 is finitely generated; setting $-(K_X+\Delta)$1, there is a logarithmic two-step degeneration

$-(K_X+\Delta)$2

and the boundary cycle produced by the second degeneration agrees, with generic multiplicities, with the metric blow-up of $-(K_X+\Delta)$3.

1. The log affine cone $-(K_X+\Delta)$4 is independent of the subsequence defining the tangent cone; hence the metric tangent cone at $-(K_X+\Delta)$5 is unique.

The last point is the strongest claim of the paper. It is not a formal consequence of the general lower-Ricci framework: the appendix constructs smooth polarized Kähler metrics on $-(K_X+\Delta)$6 in one fixed integral class, with uniform Ricci lower bound and volume non-collapsing, whose limit has a nontrivial continuous family of pairwise nonisometric tangent cones (all with underlying variety $-(K_X+\Delta)$7). Thus the Kähler–Einstein condition is a genuine rigidity input, not merely a convenience.

Local structure away from codimension four

The technical core separating the divisorial conical locus from higher metric singularities rests on three inputs. First, the fixed finite set of cone angles yields a uniform gap in volume density, so the metric regular set coincides with a fixed quantitative $-(K_X+\Delta)$8-regular set. Second, at points of $-(K_X+\Delta)$9, where some tangent cone splits off a flat cone $\mathbb Q$0, a tubular trapping argument using the Liu–Székelyhidi good-coordinate theorem confines the approximating divisors near the axis $\mathbb Q$1. Third, a transverse holonomy computation—adapting the Chern–Levine–Nirenberg argument from Chen–Donaldson–Sun—bounds the transverse degree of the divisors: since each angle defect $\mathbb Q$2 lies in $\mathbb Q$3, the degrees are uniformly bounded and the limiting angle satisfies $\mathbb Q$4.

A key lemma produces a genuine local logarithmic gauge for the approximating volume forms,

$\mathbb Q$5

with $\mathbb Q$6 nowhere-vanishing holomorphic and uniformly bounded above and below after shrinking. The lower bound uses Hurwitz's theorem together with uniform integrability of the singular weights, which holds because the total angle defect along a transverse disk is strictly less than one. Passing to the limit via Bedford–Taylor continuity gives the local structure theorem: near every point of $\mathbb Q$7 the codimension-one singular set is a smooth divisor $\mathbb Q$8, the density is exactly $\mathbb Q$9 with $2\pi\beta_a$0 rational of bounded denominator, and $2\pi\beta_a$1 is uniformly equivalent to the standard cone metric $2\pi\beta_a$2.

From this, the paper identifies the irreducible codimension-one components $2\pi\beta_a$3 of the analytic singular set, assigns to each a locally constant coefficient $2\pi\beta_a$4 defined on the dense good locus $2\pi\beta_a$5, and proves that $2\pi\beta_a$6 is $2\pi\beta_a$7-Cartier by identifying the limiting polarization with $2\pi\beta_a$8 as reflexive sheaves. An analogous statement holds on any tangent cone $2\pi\beta_a$9, proved there by gluing local log-canonical frames across a codimension-two exceptional set and extending via reflexivity; the same argument shows $N_{\mathcal F}$0 is klt, which supplies the integrability needed later.

Boundary convergence under blow-up is handled by Kuratowski convergence of supports rather than Hausdorff-measure semicontinuity. Both directions use the density gap: inclusion of limits of divisor points into $N_{\mathcal F}$1 follows from comparing the tubular-volume lower bound $N_{\mathcal F}$2 against the Cheeger–Naber estimate $N_{\mathcal F}$3 for the singular-set neighborhood; the reverse direction uses a holonomy/Thom-transversality argument showing that a transverse disk avoiding the boundary would have trivial limiting monodromy, contradicting $N_{\mathcal F}$4.

Analytic inputs on the space of cones

For a fixed $N_{\mathcal F}$5, the paper studies the compact connected space $N_{\mathcal F}$6 of tangent cones. Each $N_{\mathcal F}$7 admits a $N_{\mathcal F}$8-equivariant affine embedding by homogeneous holomorphic functions, and the log-canonical trivializing section $N_{\mathcal F}$9 is a $0$0-eigensection because the only regular units on a normal affine cone are constants. This defines a rational normalization hyperplane $0$1 in the Reeb cone. The variational formulae for the normalized volume functional carry over verbatim from Donaldson–Sun, since the calculation does not involve the boundary; combined with vanishing of the log Futaki invariant at the conical Ricci-flat metric, this makes $0$2 an isolated critical point of the rationally defined functional $0$3. Clearing denominators and applying the fact that isolated zeros of polynomial systems over $0$4 have algebraic coordinates yields algebraicity of the Reeb vector, the discreteness input for the spectrum.

Two further tools adapt to the logarithmic setting. A grafting lemma, modeled on the Donaldson–Sun $0$5/grafting argument but using cutoffs that treat $0$6 via the conic charts and reserve codimension-four smallness for the residual stratum, allows holomorphic functions to be transported between nearby cones. This gives rigidity of the holomorphic spectrum and Hilbert function across $0$7. The metric order

$0$8

then exists in the common spectrum plus $0$9 conventions, satisfies the semivaluation identities, and its kernel ideal descends to a genuine valuation on the quotient ring. Note that at this stage $2\pi$0 is only a semivaluation on $2\pi$1 itself; it becomes a valuation only after quotienting by the ideal of infinite-order functions.

Filtration, boundary tracking, and uniqueness

Following Donaldson–Sun, adapted sequences of bases produce decompositions $2\pi$2 with $2\pi$3, finite generation of $2\pi$4, and a $2\pi$5-equivariant degeneration $2\pi$6 inside the multigraded Hilbert scheme $2\pi$7, which is projective by the positivity of the Reeb grading (Haiman–Sturmfels).

The genuinely new ingredient is tracking the boundary through these degenerations. Since $2\pi$8 is only a Weil divisor whose components may merge or split under degeneration, ambient Hilbert convergence does not determine the limiting boundary. The paper instead works with height-one prime ideals $2\pi$9—the correct local datum when components need not be Cartier at $X_\infty$0—defines initial ideals $X_\infty$1, and forms the associated graded cycle $X_\infty$2 as a Weil cycle. The pair is then placed in a projective Hilbert–Chow incidence space $X_\infty$3 parametrizing schemes together with supporting codimension-one cycles of fixed degree.

The central comparison theorem states that the algebraic Chow limit $X_\infty$4 equals the metric blow-up cycle $X_\infty$5. The proof exploits the fact that the adapted-basis maps are change-of-adapted-basis maps, so the equations cutting out $X_\infty$6 are precisely the leading terms of the rescaled analytic equations for $X_\infty$7; both limits are cut out by the same homogeneous ideal on $X_\infty$8. Coefficients are compared at generic transverse disks, where the Monge–Ampère equation forces the exponent of $X_\infty$9 to equal the total angle defect of collapsing branches—the same computation as in the tubular trapping lemma. The paper explicitly warns that this identification is special to cycles arising from filtered ideals in $\Delta_\infty=\sum_a(1-\beta_a)D_a$0; arbitrary cycles in the incidence space need not be metric blow-ups of anything.

Uniqueness then follows the Donaldson–Sun scheme applied to the pair rather than the underlying cone: all log tangent-cone points lie in the orbit closure $\Delta_\infty=\sum_a(1-\beta_a)D_a$1 within $\Delta_\infty=\sum_a(1-\beta_a)D_a$2; choosing a point of minimal stabilizer dimension, log polystability closes the orbit, log Matsushima makes the stabilizer reductive, and Luna's slice theorem with the equivariant projective-slice rigidity argument forces all nearby points onto a single orbit. Iterating over the compact connected set $\Delta_\infty=\sum_a(1-\beta_a)D_a$3 gives a single orbit, and uniqueness of conical Ricci-flat metrics with fixed Reeb field—a Bando–Mabuchi-type statement proved via Berndtsson convexity of the log Ding functional—upgrades orbit-isomorphism to isometry of metric cones.

Relation to stable degeneration

Under the closed-orbit hypothesis, the Hilbert–Mumford–Birkes theorem upgrades the orbit-closure relation to a genuine one-parameter degeneration $\Delta_\infty=\sum_a(1-\beta_a)D_a$4, yielding a flat family with relative boundary cycle. Kollár's $\Delta_\infty=\sum_a(1-\beta_a)D_a$5 space refines this so the boundary is retained as a Mumford divisor. However, the paper is explicit that this does not by itself give a log $\Delta_\infty=\sum_a(1-\beta_a)D_a$6-Gorenstein degeneration: the missing information concerns whether the reflexive log-pluricanonical sheaves commute with taking associated graded objects. Kollár's hull theory provides locally closed strata on which this compatibility holds, but the paper does not prove that the specific Donaldson–Sun degenerations factor through such a stratum. This is isolated as an explicit assumption. Under it, the family is a $\Delta_\infty=\sum_a(1-\beta_a)D_a$7-equivariant special test configuration of the log Fano cone $\Delta_\infty=\sum_a(1-\beta_a)D_a$8, and the Li–Wang–Xu machinery applies: $\Delta_\infty=\sum_a(1-\beta_a)D_a$9 is K-polystable by the Ricci-flat cone/Ding-polystability correspondence, $\Delta=\sum_a(1-\beta_a)D_a$00 is K-semistable, and the central fiber is the unique stable degeneration.

Limitations and open questions

Three limitations are conceded in the paper. First, the stable-degeneration comparison depends entirely on the associated-graded log-canonical compatibility assumption; whether this is automatic for the metric filtration, or follows from singularity-theoretic hypotheses on the pairs, is left open. Second, the uniform density gap from the fixed finite coefficient set degenerates as some $\Delta=\sum_a(1-\beta_a)D_a$01; the Chen–Donaldson–Sun second approach for the single-smooth-divisor regime (stronger smooth approximation, Ricci flow, pseudolocality) is not extended here to general log pairs, particularly when several coefficients tend to zero simultaneously. Third, the role of polarization is unresolved: the paper asks whether a non-collapsed Gromov–Hausdorff limit of smooth projective Kähler manifolds with only a uniform Ricci lower bound, diameter bound, and volume lower bound—but no controlled polarization—can fail to be homeomorphic to any complex projective variety, and whether such behavior can occur for conical metrics with a fixed cone angle.

One should also note structural caveats internal to the results: the semivaluation $\Delta=\sum_a(1-\beta_a)D_a$02 is a valuation only modulo the ideal of infinite-order germs, the algebraic central boundary cycle is defined only subsequentially until the comparison theorem removes the ambiguity, and the appendix example shows the uniqueness conclusion fails without the Einstein equation even with polarization, noncollapse, and a lower Ricci bound.

Conclusion

The paper supplies the logarithmic analogue of the Donaldson–Sun tangent-cone theory: for non-collapsing limits of conical Kähler–Einstein pairs with coefficients in a fixed finite subset of $\Delta=\sum_a(1-\beta_a)D_a$03, the boundary divisor is recovered from the metric, survives both steps of the algebraic degeneration with matching generic multiplicities, and the resulting log metric tangent cone is unique. The proof combines tubular trapping and transverse holonomy estimates adapted from Chen–Donaldson–Sun with the Liu–Székelyhidi lower-Ricci framework, and the appendix demonstrates by construction on $\Delta=\sum_a(1-\beta_a)D_a$04 that the Kähler–Einstein hypothesis is indispensable for the rigidity conclusion.