From Calabi's extremal metrics to scalar-flat Kähler cones

Abstract: We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal Kähler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in arXiv:1906.04827.

• The paper proves that for a polarized Kähler manifold with an extremal metric, the associated high-dimensional cone admits a scalar-flat Kähler metric when the join dimension is sufficiently large.
• It employs weighted Futaki invariants and Einstein–Hilbert functionals to reduce the existence problem to a weighted cscK equation on the product space.
• The work extends Sasaki geometry by demonstrating that the join of a quasi-regular extremal Sasaki manifold with high-dimensional spheres attains constant transversal scalar curvature.

From Calabi's Extremal Metrics to Scalar-Flat Kähler Cones: An Expert Analysis

Introduction and Context

The intersection of extremal Kähler metrics and scalar-flat Kähler cone geometry is a focal area at the confluence of complex differential geometry, Sasaki geometry, and analytic aspects of the constant scalar curvature Kähler (cscK) problem. The paper "From Calabi's extremal metrics to scalar-flat Kähler cones" (2603.29911) by Apostolov, Lahdili, and Pan constructs an explicit bridge from the existence of extremal Kähler metrics to the existence of scalar-flat Kähler cone metrics, yielding new results on the geometric structure of both Kähler and Sasaki manifolds.

Main Results and Statements

The core theorem asserts that given a smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which supports an extremal Kähler metric in $c_1(L_X)$, the complex cone over $X \times \mathbb{P}^k$ associated with $L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1)$ admits a scalar-flat Kähler cone metric for all sufficiently large $k$ (with effective bound depending on the polarized data). In the Sasaki framework, this translates into the existence, for the unweighted Sasaki join of a compact quasi-regular extremal Sasaki manifold $S$ with a standard Sasaki sphere $\mathbb{S}^{2k+1}$ of large enough dimension, of a Sasaki structure with constant positive transversal scalar curvature.

This gives a definitive answer to an asymptotic version of a question due to Boyer, Huang, Legendre, and Tønnesen-Friedman [BHLT0], circumventing various known obstructions (such as those arising from K-stability in the irregular setting).

Technical Framework

Kähler Cones, Sasaki Geometry, and Polarizations

Let $Y$ be a smooth complex affine cone, equipped with the action of a maximal compact torus $\hat{T}$, with $(X, L_X)$0 smooth away from the cone point. The equivariant existence theory for scalar-flat cone Kähler metrics is governed by the notion of the Sasaki–Reeb cone $(X, L_X)$1 and the associated linear Futaki-type invariants for polarizations. Unlike traditional cscK settings, vanishing of these invariants is generically satisfied for sufficiently large classes in the cone, especially when moving to higher join dimensions.

A core analytic mechanism is the reduction of the existence problem for a scalar-flat cone Kähler metric on $(X, L_X)$2 to a weighted cscK-type equation on the base $(X, L_X)$3, formulated in the weighted Futaki formalism (see [Lahdili_2019, ACL, AC]). The weights trace the geometric data of the construction and have explicit asymptotic behavior as $(X, L_X)$4.

Einstein–Hilbert Functionals and Asymptotic Analysis

A novel aspect is the use of a family of weighted Einstein–Hilbert-type functionals $(X, L_X)$5. For each $(X, L_X)$6, these functionals are defined on the finite-dimensional space of affine-linear functions over the momentum polytope associated with the torus action. Their critical points correspond to choices of weights and geometric parameters so that the corresponding weighted Futaki invariant vanishes, ensuring the solvability of the weighted cscK equation. As $(X, L_X)$7, these functionals converge (in a suitable $(X, L_X)$8 sense on compacta) to the entropy functional associated with weighted cscK geometry, as studied by Inoue [Inoue_2022].

As such, the asymptotic maximizers correspond to the extremal direction in the polytope, aligning with the unique affine-linear function associated with the extremal metric (from Calabi’s theory).

Deformation–Openness and Join Constructions

The analytic pillar allowing these methods to function is a LeBrun–Simanca-type openness theorem for the equation, ensuring the existence of nearby weighted cscK metrics as the weights deform (see [Apostolov_Lahdili_Legendre_2024]). The join construction is exploited to pass from $(X, L_X)$9 to $c_1(L_X)$0, controlling the Futaki invariants and ensuring that for sufficiently large $c_1(L_X)$1, the relevant cones admit scalar-flat Kähler cone metrics.

Implications and Noteworthy Features

Unobstructed Existence for Large Joins: The strong, explicit claim is that for any quasi-regular polarized Kähler manifold with an extremal metric, as soon as $c_1(L_X)$2 exceeds a computable threshold, a scalar-flat cone metric exists on the corresponding high-dimensional join. The proof is non-constructive regarding the quantitative value of $c_1(L_X)$3 but is robust and works in full generality.

Sasaki Geometry Consequences: These results yield a vast new family of compact Sasaki manifolds with constant transversal scalar curvature (positive and negative), accessible via explicit Sasaki join operations. This extends prior results (notably [BT1]) to all unweighted joins with sufficiently large spheres.

Obstruction Avoidance: While $c_1(L_X)$4-stability and related invariants provide highly nontrivial obstructions in fixed dimension, the framework here demonstrates these can be circumvented asymptotically. The weighted theory and the behavior of the entropy functionals reveal that in the large join regime, the analytic barriers vanish.

Relation to Extremal Metrics: The result is strictly an "extremal implies scalar-flat cone"; it does not imply that every scalar-flat cone metric arises in this way nor does it address the existence problem for other classes (such as non-extremal cscK).

Role in Geometric Analysis and Moduli Theory: The results fit into the broader program studying moduli and degenerations in Kähler geometry, and the behavior of special metrics on singular or non-compact spaces (see e.g., [Boucksom_Jonsson_Trusiani_2024], [Pan_To_Trusiani_2023]). The asymptotic existence claims suggest new directions for metric limit geometry and the structure of the moduli spaces of scalar-flat Kähler and Sasaki-manifolds.

Future Directions and Theoretical Developments

• Effective Bounds: The paper leaves open the question of explicit, effective lower bounds for $c_1(L_X)$5. The analytic dependence of the thresholds on quantities such as moment polytope geometry, Futaki weights, and underlying automorphism group data remains a rich topic.
• Regularity and Moduli: Understanding the regularity and uniqueness properties of the constructed scalar-flat cone metrics, especially near the join point, and their moduli, is a natural next step.
• Extension to Singular and Non-Compact Bases: The theory is developed for smooth or orbifold $c_1(L_X)$6. Extending analogous results to singular (e.g., klt) bases or to the analytic setting of non-algebraic complex spaces is of independent interest.
• Interplay with K-Stability: While asymptotic methods avoid obstruction, understanding quantitatively the threshold at which $c_1(L_X)$7-instability disappears and its interplay with the geometry of the join is significant.
• Bridging Sasaki and Kähler Flows: Since Ricci solitons and extremal metrics relate to flows (Kähler–Ricci and Calabi, respectively), a flow-theoretic interpretation of the join construction and weight asymptotics could offer new analytic approaches.

Conclusion

This work provides a rigorous and general framework connecting the existence of Calabi extremal metrics to the existence of scalar-flat Kähler cone metrics in high-dimensional geometric join constructions. The analytic machinery, via asymptotic analysis of weighted Einstein–Hilbert functionals and variations of Futaki invariants, underpins the affirmative existence theorems. The approach yields expansive new families of Sasaki and Kähler manifolds with prescribed scalar curvature properties and illuminates the landscape of special metrics in complex geometry from the perspective of large-dimension limits and weighted geometric analysis (2603.29911).