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Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability

Kai Zheng

TL;DR

The paper proves that the existence of constant scalar curvature Kähler cone metrics is equivalent to the properness of the log K-energy and to geodesic stability, extending Chen–Cheng's smooth results to cone singularities. It develops a conical analogue of the Chen–Chen–Cheng framework through an approximation scheme that replaces cone problems with twisted smooth problems, coupled with robust a priori estimates to pass to the limit. A key advance is the construction of singular cscK metrics in big cohomology classes, together with an openness and approximation theory for the cscK cone path, mirroring the continuity method used for Kähler–Einstein problems. The results rely on cone-specific convexity, generalized cone geodesics, and an extended alpha-invariant theory in big classes, and they apply to normal complex spaces, enabling a YTD-type program in singular settings. Collectively, these contributions deepen the understanding of stability notions in complex geometry and offer a viable route to canonical cone metrics in broader cohomological contexts.

Abstract

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture of the cscK problem for smooth Kähler metrics by Chen-Cheng to the setting of cscK cone metrics. One applications of our main results is that we introduce and construct singular cscK metrics with possible degeneration in big cohomology class. As another application, we also prove both openness and approximation property for the path of cscK cone metrics, which are paralleling to Donaldson's continuity method through Kähler-Einstein cone metrics in the resolution of Yau-Tian-Donaldson conjecture for Fano Kähler-Einstein metrics.

Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability

TL;DR

The paper proves that the existence of constant scalar curvature Kähler cone metrics is equivalent to the properness of the log K-energy and to geodesic stability, extending Chen–Cheng's smooth results to cone singularities. It develops a conical analogue of the Chen–Chen–Cheng framework through an approximation scheme that replaces cone problems with twisted smooth problems, coupled with robust a priori estimates to pass to the limit. A key advance is the construction of singular cscK metrics in big cohomology classes, together with an openness and approximation theory for the cscK cone path, mirroring the continuity method used for Kähler–Einstein problems. The results rely on cone-specific convexity, generalized cone geodesics, and an extended alpha-invariant theory in big classes, and they apply to normal complex spaces, enabling a YTD-type program in singular settings. Collectively, these contributions deepen the understanding of stability notions in complex geometry and offer a viable route to canonical cone metrics in broader cohomological contexts.

Abstract

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log -energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture of the cscK problem for smooth Kähler metrics by Chen-Cheng to the setting of cscK cone metrics. One applications of our main results is that we introduce and construct singular cscK metrics with possible degeneration in big cohomology class. As another application, we also prove both openness and approximation property for the path of cscK cone metrics, which are paralleling to Donaldson's continuity method through Kähler-Einstein cone metrics in the resolution of Yau-Tian-Donaldson conjecture for Fano Kähler-Einstein metrics.

Paper Structure

This paper contains 73 sections, 88 theorems, 539 equations.

Table of Contents

  1. Introduction
  2. Constant scalar curvature Kähler cone metrics
  3. Hölder spaces and geometrically polyhomogeneous
  4. Geodesics in the space of Kähler cone metrics
  5. CscK cone metrics: definition and regularity
  6. Reference metric
  7. Definition of cscK cone metric
  8. Regularity of cscK cone metrics: geometrical polyhomogeneity
  9. Uniqueness of cscK cone metrics and automorphism group
  10. Log $K$-energy and convexity
  11. Entropy
  12. $J$-functionals
  13. Log $K$-energy
  14. $I$ and $J$-functionals
  15. Properness implies existence
  16. ...and 58 more sections

Key Result

Theorem 1.2

Assume that $\Omega$ is a Kähler class and the cone angle satisfies $0<\beta\leq 1$. Then $\Omega$ admits a constant scalar curvature Kähler cone metric with cone angle $\beta$ is equivalent to the properness of the log $K$-energy.

Theorems & Definitions (213)

  • Definition 1.1
  • Theorem 1.2: Log properness theorem
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: Log geodesic stability theorem
  • Remark 1.6
  • Theorem 1.7: Approximation theorem
  • Theorem 1.8: Openness theorem
  • Definition 2.1: MR4020314
  • Theorem 2.2: Cone geodesic Theorem 1.1 in MR4020314
  • ...and 203 more