On polynomial convergence to tangent cones for singular Kähler–Einstein metrics

1 Introduction

In this paper, we study the geometry of singular Kähler–Einstein metrics. Such metrics arise from two main sources: Gromov–Hausdorff limits of smooth Kähler–Einstein metrics and pluripotential theory. Similar to other problems in geometric analysis, a crucial approach to understanding the structure of a singular Kähler–Einstein metric near its singularity involves examining its tangent cones. This process typically involves three steps:

1. the existence of tangent cones;

2. the uniqueness of tangent cones;

3. relating the geometry of 𝑍 near 𝑝 with the geometry on the tangent cone at 𝑝.

Firstly, we consider singular Kähler–Einstein metrics coming from the Gromov–Hausdorff limits. Let ( Z , p , d Z ) be a pointed Gromov–Hausdorff limit of complete non-collapsing Kähler–Einstein manifolds with uniformly bounded Ricci curvature. According to [9, 1, 10, 18], there is a decomposition Z = Z reg ⊔ Z sing , where the regular part Z reg consists of points around which a neighborhood is a smooth Riemannian manifold. This Z reg is an open, connected, smooth manifold, and the restriction of d Z to Z reg is induced by a Kähler–Einstein metric ( g Z , J Z , ω Z ) , which is referred to as the singular Kähler–Einstein metric on 𝑍.

If one further assumes that the smooth Kähler–Einstein metrics are polarized, i.e., the Kähler forms are curvature forms of line bundles, and that the canonical bundle is homomorphically trivial for Ricci-flat metrics, then Donaldson and Sun [23, 24] showed that 𝑍 has the structure of a normal analytic space with log terminal singularities and the tangent cone 𝒞 at 𝑝 is unique. Furthermore, a two-step degeneration exists, proceeding from the local ring O Z , p through an intermediate K-semistable cone𝑊 to 𝒞. This is referred to as Donaldson–Sun’s two-step degeneration theory. It is an open problem whether Donaldson–Sun’s two-step degeneration theory holds without the polarization condition.

In this paper, we assume 𝑝 is an isolated singularity and that, near 𝑝, the curvature grows at most quadratically. That is, there exists a constant C > 0 such that the following holds near 𝑝:

however, the polarization condition is not required. The main result is as follows.

We refer to Section 2.3 for the definition of Fano cones and to Section 3 for further details on Donaldson–Sun’s two-step degeneration theory. We will simply use C = W to denote that 𝑊 is isomorphic to 𝒞 as Fano cones in the following. We remark that, assuming (1.1), Donaldson–Sun’s two-step degeneration theory in this context (without the polarized condition) can be directly achieved using results in [47] and following the original argument in [24]. The essence of the result lies in establishing the algebraic criterion for the polynomial convergence rate. We also remark that, as a consequence of the results in [16, 17], condition (1.1) is equivalent to stating that a tangent cone at 𝑝 has a smooth link, which is an a priori weaker condition.

In this paper, we use the terms “being conical” and “polynomially close to a Calabi–Yau cone” interchangeably, which means the following definition holds for some δ > 0 .

It is conjectured in [24] that both 𝑊 and 𝒞 depend only on the germ of the singularity ( Z , p ) , rather than the metric. For algebraic singularities, this conjecture has been confirmed and generalized by Li–Xu and Li–Wang–Xu; see [46, 45] and the references therein. By this result in algebraic geometry and considering cones over strictly K-semistable Fano manifolds, we obtain examples ( Z , p ) that are not conical.

This highlights the sharpness of the logarithmic rate result proved by Colding–Minicozzi [17] within the context of Kähler geometry, as initially predicted by Hein–Sun [39]. We remark that, in [17], the logarithmic convergence rate is also established for tangent cones at infinity; however, it is proved in [53] that every complete Calabi–Yau metric with Euclidean volume growth and quadratic curvature decay is polynomially close to its tangent cone at infinity. This reveals distinctions between the two settings.

Then let us compare Theorem 1.1 with some results in the literature. In [39], assuming the germ ( Z , p ) is biholomorphic to ( C , o ) and 𝒞 has a smooth link, Hein–Sun proved that the ( Z , p ) is polynomially close to ( C , o ) , and indeed, in (1.2), Φ can be chosen to be a biholomorphism. In [15], assuming ( Z , p ) is a polarized limit and the germ ( Z , p ) is biholomorphic to ( C , o ) , Chiu–Székelyhidi proved a polynomial rate closeness on the level of Kähler potentials without assuming any smoothness on 𝒞. Although we assume 𝒞 has a smooth link, we are able to address the case where the germ ( Z , p ) is not biholomorphic to ( C , o ) and provide a characterization for the polynomial closeness. The direction from the polynomial closeness to C = W is established directly by constructing holomorphic functions using the Hörmander L 2 -method. The direction from C = W to polynomial closeness uses an approach similar to that in [53] and relies on the results in [39, 15]. See Section 5 for more details.

Then we move to singular Kähler–Einstein metrics coming from pluripotential theory. Singular Kähler–Einstein metrics with non-positive Ricci curvature on log terminal Kähler varieties have been constructed in [28]; see [5, 6, 43] and references therein for further studies. Understanding the metric behavior of the singular Kähler–Einstein metrics near the singular locus is a major open problem. The main progress in this direction is due to Hein–Sun [39]. Following the approach in [39], we extend their results to a broader class of singular Kähler–Einstein metrics.

Let ( X , p ) be a germ of an isolated log terminal algebraic singularity. Motivated by the results in [24], there is a well-developed theory for local stability of (Kawamata) log terminal singularities; see [44, 46, 45, 58] and the references therein. Specifically, this local stability theory establishes the existence of a K-semistable Fano cone 𝑊 and a K-polystable Fano cone 𝒞, canonically associated with the singularity. Then we have the following.

We remark that, as in Theorem 1.1, the isomorphism between 𝒞 and 𝑊 is also a necessary condition for the existence of a conical singular Kähler–Einstein metric. If the germ ( X , p ) itself is biholomorphic to a Calabi–Yau cone, then the assumption in the theorem is satisfied, and we can show that Φ can be chosen to be a biholomorphism. This makes this result a generalization of [39]. The proof follows the same idea as in [39], with some new key insights from [14, 45]. Chen–Wang’s compactness and regularity results in [14], generalizing Cheeger–Colding’s results, ensure that we can take limits for Kähler–Einstein metrics with mild singularities. Li–Wang–Xu’s result in [45] ensures that we can determine the tangent cone for the limiting metric from the underlying algebraic structure. See Section 7 for more details. Unfortunately, we are unable to prove similar results for singular Kähler–Einstein metrics with positive Ricci curvature. We will briefly discuss the related issues in Section 7.3.

This paper is organized as follows. In Section 2, we collect some definitions and preliminary results necessary for this paper. In Section 3, we give a review for Donaldson–Sun’s two-step degeneration theory. In Section 4, the direction from the polynomial convergence to C = W is proved. In Section 5, the other direction from C = W to polynomial convergence is established. In Section 6, we construct examples claimed in Corollary 1.3. In Section 7, we give the proof of Theorem 1.4. In the appendices, we give proofs of two results needed in this paper, which are also well-studied in the literature.

2 Preliminary

3 Donaldson–Sun two-step degeneration theory

Let ( Z , p , d Z ) be a pointed Gromov–Hausdorff limit of complete non-collapsing Kähler–Einstein manifolds with uniformly bounded Ricci curvature. This means that there is a sequence of complete Kähler manifolds ( X i , h i ) and points q i ∈ X i satisfying Ric ⁡ ( h i ) = λ i ⁢ h i with | λ i | ≤ 1 and Vol ⁡ ( B ⁢ ( q i , 1 ) ) ≥ κ for some constant 𝜅 independent of 𝑖 such that ( X i , q i , h i ) converges to ( Z , p , d Z ) in the pointed Gromov–Hausdorff topology. Indeed, since we are interested in the local geometry of 𝑍 near 𝑝, completeness of ( X i , h i ) is not important as long as B ⁢ ( q i , 1 ) is compactly contained in X i .

Note that we do not assume the Kähler–Einstein metrics are polarized; therefore, Donaldson–Sun’s theory cannot be directly applied here. We assume the following curvature condition holds near 𝑝:

Liu’s result [47, Proposition 2.5] is the crucial step in establishing Donaldson–Sun’s two-step degeneration theory in our context. The curvature condition (3.1) is needed there to obtain pointwise bound from integral bound for functions. It is worth noting that, although Liu’s work primarily concerns tangent cones at infinity, the proof can be directly applied here without much modification.

Let ( Z j , p j , J j , g j , ω j , d j ) denote the rescaling of ( Z , p , J Z , g Z , ω Z , d Z ) by a factor 2 j , j ∈ Z ≥ 1 . This means that we take J j = J , p j = p , g j = 2 2 ⁢ j ⁢ g Z , ω j = 2 2 ⁢ j ⁢ ω Z and d j = 2 j ⁢ d Z . Let r j be the distance to p j with respect to d j . Recall that we denote by Ψ ⁢ ( ϵ ; … ) any non-negative functions depending on 𝜖 and some additional parameters such that, when these parameters are fixed, lim ϵ → 0 Ψ ⁢ ( ϵ ; … ) = 0 .

Proposition 3.1 implies that there are (arbitrary small) neighborhoods of 𝑝 in 𝑍, which have strictly pseudoconvex boundary. In the regular part of 𝑍, we have smooth convergence of Kähler–Einstein metrics ( X i , q i , h i ) ; therefore, there are neighborhoods of q i with strictly pseudoconvex boundary, converging in the Gromov–Hausdorff sense. Then we can use the Hörmander L 2 -estimate to construct holomorphic functions in a neighborhood of q i ; therefore, repeating the argument in [23, 24], one can show that 𝑍 has a normal complex analytic space structure.

Let ( C , o , J C , g C , ω C , d C ) denote a tangent cone of ( Z , p , g Z , J Z , ω Z , d Z ) at 𝑝. As a consequence of (3.1), 𝒞 is a Calabi–Yau cone with a smooth link. Since 𝑍 admits a normal analytic space structure with an isolated singularity 𝑝, we can also do the Hörmander L 2 -method in a neighborhood of p ∈ Z . In particular, we have the following [23].

Donaldson–Sun’s two-step degeneration theory can be established following their original argument [24]. In particular, the tangent cone 𝒞 is unique. This theory describes the Gromov–Hausdorff convergence of Z j to 𝒞 using complex analytic data and we summarize their results as follows.

Let B j denote the unit ball in Z j centered at p j , and 𝐵 the unit ball in 𝒞 centered at 𝑜. Fix a distance d j on B j ⊔ B that realizes the Gromov–Hausdorff convergence of B j to 𝐵. More precisely, this means that the Hausdorff distance between B j and 𝐵 under d j is Ψ ⁢ ( j − 1 ) and d j ⁢ ( p j , p ) = Ψ ⁢ ( j − 1 ) . Moreover, for any compact set 𝐾 contained in the regular part of 𝐵, we can find, for large enough 𝑗, open embeddings χ j of an open neighborhood of 𝐾 into B j such that d j ⁢ ( x , χ j ⁢ ( x ) ) = Ψ ⁢ ( j − 1 ) for all x ∈ K and ( χ j ∗ ⁢ g j , χ j ∗ ⁢ J j ) converges smoothly over 𝐾 to ( g C , J C ) .

It is proved in [24] that there exist holomorphic embeddings

such that

1. F ∞ = ( h 1 , … , h N ) , where each h i is a d i -homogeneous holomorphic function with d i > 0 . Under the embedding F ∞ , ξ = J C ⁢ ( r ⁢ ∂ r ) extends to a linear vector field on C N of the form Re ⁡ ( − 1 ⁢ ∑ i d i ⁢ z i ⁢ ∂ z i ) , which we also denote by 𝜉. In particular, the dilation action Λ : r → 2 ⁢ r on 𝒞 extends to the diagonal linear transformation of C N given by

   Λ ⁢ ( z 1 , … , z N ) = ( 2 d 1 ⁢ z 1 , … , 2 d N ⁢ z N ) .

2. F j = Λ j ∘ F j − 1 | B j = Λ j ∘ ⋯ ∘ Λ 2 ∘ F 1 | B j , where Λ j ∈ G ξ . Here G ξ denotes the subgroup of GL ⁢ ( N ; C ) consisting of elements that commute with the actions generated by 𝜉. Moreover, Λ j → Λ as j → ∞ .

3. d j ⁢ ( x j , x ) → 0 if and only if F j ⁢ ( x j ) → F ∞ ⁢ ( x ∞ ) ∈ C N .

Moreover, there is an intrinsic intermediate K-semistable cone𝑊 associated to ( Z , p ) that can be characterized as follows. Let O p be the ring of germs of holomorphic functions on 𝑍 at 𝑝. For any non-zero function f ∈ O p , one defines its order of vanishing

Such a limit exists and is in 𝒮, the holomorphic spectrum of 𝒞. We list them in order as 0 = d 0 < d 1 < ⋯ . Then, for any k ≥ 0 , one can define an ideal

We obtain a filtration O p = I 0 ⊃ I 1 ⊃ ⋯ and an associated graded ring ⨁ k ≥ 0 I k / I k + 1 . Then the intermediate K-semistable cone is defined to be

which is a normal affine variety and also admits a natural 𝕋-action, which defines a polarized affine cone structure on 𝑊.

The relation between 𝑊 and 𝒞 can be described as follows. Let W j denote the weighted tangent cone of F j ⁢ ( B j ) with respect to the weight ( d 1 , … , d N ) . It is proved in [24] that each W j is isomorphic to 𝑊 as polarized affine cones. As Λ j ∈ G ξ , commuting with 𝜉, we have W j = Λ j ⁢ ( W j − 1 ) . Moreover, W j converge to F ∞ ⁢ ( C ) in a certain multi-graded Hilbert scheme Hilb (see [24]). It is shown in [24] that transverse automorphisms of the cone 𝒞, i.e., automorphisms of 𝒞 that preserve the Reeb vector field ξ = J C ⁢ ( r ⁢ ∂ r ) , form a reductive complex Lie group. Then a variant of Luna’s slice theorem, proved by Donaldson [27], implies the existence of a one-parameter subgroup λ ⁢ ( t ) of G ξ such that C = lim t → 0 λ ⁢ ( t ) . W . That is, 𝑊 admits an equivariant degeneration to 𝒞. Since 𝒞 has only an isolated singularity and 𝑊 is a cone, it follows that 𝑊 also has only an isolated singularity. Since 𝒞 has an isolated log terminal singularity and has a quotient singularity in dimension 2, we know that 𝑊 also has log terminal singularity by [40, Proposition 9.1.9 and Theorem 9.1.19]. Therefore, 𝑊 admits a Fano cone structure, justifying the terminology intermediate K-semistable cone.

4 Polynomial convergence implies C = W

Suppose ( g Z , J Z , ω Z ) is conical of order δ > 0 ; we show in this section that 𝒞 is isomorphic to 𝑊 as Fano cones. This is achieved using the Hörmander L 2 -method to construct J Z -holomorphic functions, whose rescalings converge directly to holomorphic functions on 𝒞.

Let Φ denote the diffeomorphism given in Definition 1.2. Using the diffeomorphism Φ, we will identify the neighborhood U p of 𝑝 with U o , which, without loss of generality, we may assume to be 𝐵, the unit ball in 𝒞 centered at the vertex. We recall that d j denotes the rescaling of d Z by a factor 2 j , and B j denotes the unit ball in Z j , and Λ denotes the dilation r → 2 ⁢ r on 𝒞. Then, using (1.2), one can easily show the following result.

As mentioned in [24], the notion of convergence of holomorphic functions depends on the choice of the metric on B j ⊔ B , which realizes the pointed Gromov–Hausdorff convergence. Moreover, the argument in [24] works for any such choice. The above lemma implies that

realizes the pointed Gromov–Hausdorff convergence as j → ∞ . Therefore, in the following, when we talk about the convergence of holomorphic functions, we are considering convergence under this Gromov–Hausdorff approximation.

Let 𝜓 denote the plurisubharmonic function log ⁡ r 2 − ( − log ⁡ r 2 ) 1 2 on 𝒞. Note that

Then it follows that

Since ( g Z , J Z , ω Z ) is Kähler–Einstein and conical of order δ > 0 , as a consequence of (4.1), we know that, in a neighborhood of 𝑜, for any ϵ > 0 , there exists a constant c ϵ > 0 such that

There are homogeneous holomorphic functions ( h 1 , … , h N ) on 𝒞, generating R ⁢ ( C ) and giving an embedding F = ( h 1 , … , h N ) : C → C N such that the Reeb vector field 𝜉 extends to a linear diagonal vector field Re ⁡ ( − 1 ⁢ ∑ i d i ⁢ z i ⁢ ∂ z i ) and the dilation Λ on 𝒞 extends to a diagonal dilation on C N , Λ ⁢ ( z 1 , … , z N ) = ( 2 d 1 ⁢ z 1 , … , 2 d N ⁢ z N ) , where d i = deg ⁡ ( h i ) > 0 .

Then, as ( g Z , J Z , ω Z ) is conical of order δ > 0 , we obtain that, for all l ≥ 0 ,

We choose a neighborhood 𝑈 of 𝑜 with smooth strictly pseudoconvex boundary with respect to the complex structure J Z such that (4.2) holds on 𝑈, and choose the weight

and the background Kähler metric ω Z . Using Theorem 2.12, we can solve the ∂ ̄ J Z -equation

and get a solution u i of (4.4) with the estimate

Let f i = h i − u i . Then we have the following.

This lemma says that, with respect to the Gromov–Hausdorff approximation Λ − j , the holomorphic function 2 j ⁢ d i ⁢ f i | B j converges to h i | B on 𝒞 as j → ∞ . Therefore, it implies that C = W in Donaldson–Sun’s two-step degeneration theory.

5 C = W implies polynomial convergence

The overall idea of the proof is similar to [53]. Firstly, using C = W , we get an almost Kähler–Einstein metric ( ω ̃ , J C , g ̃ ) in a neighborhood of o ∈ C , which is (by construction) polynomially close to the singular Kähler–Einstein metric ( ω Z , J Z , g Z ) . Then, solving a complex Monge–Ampère equation, we get a Calabi–Yau metric ( ω ̄ , J C , g ̄ ) in a neighborhood of o ∈ C , which is again polynomially close to the singular Kähler–Einstein metric ( ω Z , J Z , g Z ) . Finally, we can apply Chiu–Székelyhidi’s result [15] to get that ( ω ̄ , J C , g ̄ ) is polynomially close to the Calabi–Yau cone metric ( ω C , J C , g C ) .

We follow the notation in Section 3. Moreover, in the following, we identify 𝒞 with its image F ∞ ⁢ ( C ) ⊂ C N , and as we only care about the local geometry near 𝑝, we also identify 𝑍 with its image F 1 ⁢ ( B 1 ) ⊂ C N .