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The maximal destabilizers for Chow and K-stability

Yi Yao

TL;DR

The paper develops a program to relate maximal Chow-destabilizers and maximal K-destabilizers in polarized manifolds by non-Archimedean pluripotential theory. It builds a bridge from Archimedean quantization (balanced norms) to their non-Archimedean counterparts, expressing Chow-weights via NA energies and proving existence/uniqueness of maximal Chow-destabilizers, with symmetry properties mirroring embedding symmetries. Under idealized assumptions (A1–A4), it shows that suitably normalized maximal Chow-destabilizers quantize the maximal K-destabilizer, i.e., the NA metric associated to the Chow picture converges to the maximal K-destabilizer in the NA limit, tying the steepest descent of K-energy to the maximal destabilizing Chow data. The work outlines a concrete quantization pathway—via FS/SN maps, NA energies, and finite-dimensional approximations $(k o ty)$—to connect destabilization in Chow theory with K-stability, aiming to illuminate how unstable manifolds decompose along dominant NA directions. This framework advances the understanding of stability in Kähler geometry by unifying two central destabilization notions under a common variational/NA paradigm, with toric and symmetry considerations guiding explicit computations. The significance lies in a principled route to quantify K-destabilization through Chow data, potentially informing the structure of degenerations and the formation of canonical pieces in unstable settings.

Abstract

Donaldson showed that the constant scalar curvature Kähler metrics can be quantized by the balanced Hermitian norms on the spaces of global sections. We explore an analogous problem in the unstable situation. For a K-unstable manifold $(X,L)$, its projective embedding via $\left|kL\right|$ will be Chow-unstable when $k$ is sufficiently large and divisible. There is a unique filtration on $\mathrm{H}^{0}(X,kL)$, that corresponds to the maximal destabilizer for Chow-stability of the embedded variety. On the other hand, there is a maximal destabilizer for K-stability after the work of Xia and Li, which corresponds to the steepest descent direction of K-energy. Based on Boucksom-Jonsson's non-Archimedean pluripotential theory and some idealistic assumptions, we provide a route to show that maximal K-destabilizers are quantized by the maximal Chow-destabilizers.

The maximal destabilizers for Chow and K-stability

TL;DR

The paper develops a program to relate maximal Chow-destabilizers and maximal K-destabilizers in polarized manifolds by non-Archimedean pluripotential theory. It builds a bridge from Archimedean quantization (balanced norms) to their non-Archimedean counterparts, expressing Chow-weights via NA energies and proving existence/uniqueness of maximal Chow-destabilizers, with symmetry properties mirroring embedding symmetries. Under idealized assumptions (A1–A4), it shows that suitably normalized maximal Chow-destabilizers quantize the maximal K-destabilizer, i.e., the NA metric associated to the Chow picture converges to the maximal K-destabilizer in the NA limit, tying the steepest descent of K-energy to the maximal destabilizing Chow data. The work outlines a concrete quantization pathway—via FS/SN maps, NA energies, and finite-dimensional approximations —to connect destabilization in Chow theory with K-stability, aiming to illuminate how unstable manifolds decompose along dominant NA directions. This framework advances the understanding of stability in Kähler geometry by unifying two central destabilization notions under a common variational/NA paradigm, with toric and symmetry considerations guiding explicit computations. The significance lies in a principled route to quantify K-destabilization through Chow data, potentially informing the structure of degenerations and the formation of canonical pieces in unstable settings.

Abstract

Donaldson showed that the constant scalar curvature Kähler metrics can be quantized by the balanced Hermitian norms on the spaces of global sections. We explore an analogous problem in the unstable situation. For a K-unstable manifold , its projective embedding via will be Chow-unstable when is sufficiently large and divisible. There is a unique filtration on , that corresponds to the maximal destabilizer for Chow-stability of the embedded variety. On the other hand, there is a maximal destabilizer for K-stability after the work of Xia and Li, which corresponds to the steepest descent direction of K-energy. Based on Boucksom-Jonsson's non-Archimedean pluripotential theory and some idealistic assumptions, we provide a route to show that maximal K-destabilizers are quantized by the maximal Chow-destabilizers.

Paper Structure

This paper contains 40 sections, 29 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Space of Hermitian norms and non-Archimedean norms
  3. Symmetric space of Hermitian norms
  4. Relative volumes
  5. Asymptotic cone of $\mathcal{H}(V)$
  6. Negatively curved path of Hermitian norms
  7. Space of non-Archimedean norms
  8. The limit NA norm associated to a ray
  9. Geodesics in $\mathcal{N}(V)$ and $\hat{\mathcal{H}}(V)$
  10. Finite-energy metrics and geodesic rays
  11. Finite-energy metrics and energy functionals
  12. Finite-energy geodesic rays
  13. Chow-stability and its maximal destabilizers
  14. Chow-stability and the Kempf--Ness functional
  15. Maximal destabilizers for Chow-unstable varieties
  16. ...and 25 more sections

Key Result

Theorem 1

For any projective manifold $X\subset\mathbb{P}V^{*}$, consider the minimization problem for $L^{p}$-normalized Chow-weight where $\left\Vert \chi\right\Vert _{p\in[1,\infty]}$ is the $p$-norm (eq:p norm). (1) there exists a minimizer for any $p\in[1,\infty]$. (2) when $X$ is Chow-unstable (i.e. $\inf M_{X}<0$) and $p\in(1,\infty)$, the minimizer is unique up to scaling.

Theorems & Definitions (70)

  • Theorem 1: Theorem \ref{['thm:existence-uniqueness of Chow']}
  • Corollary 2: Theorem \ref{['thm:sym of Chow-dest']}
  • Theorem 3: Theorem \ref{['thm: NA express Chow']}, \ref{['thm: property of Chow-destab']}
  • Theorem 4
  • Theorem 5: Theorem \ref{['thm:quantize L2 norm']}
  • Proposition 6
  • proof
  • Remark 7
  • Remark 8
  • Definition 9: limit NA norm
  • ...and 60 more