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A valuative criterion of K-polystability

Linsheng Wang

TL;DR

This work develops a valuative, torus-equivariant criterion for K-polystability of log Fano pairs by constructing a refined data pair $(X_r,\Delta_r;W^{X_r}_\bullet)$ from a maximal torus action and showing that, under the Futaki obstruction vanishing on the cocharacter lattice, stability reduces to the delta-invariant $\delta(X_r,\Delta_{X_r};W^{X_r}_\bullet)$ exceeding 1. The paper proves a sharp Abban–Zhuang-type estimate, builds the toric-refined sequence of models, and proves the main equivalence between K-polystability (or semistability) of $(X,\Delta)$ and the corresponding delta condition on $(X_r,\Delta_r)$; it also provides an almost complete linear-series variant linking to a fixed divisor on $X_r$. Applications include the existence of $g$-solitons for arbitrary weight functions and weight-insensitive stability phenomena in several Fano $\mathbb{T}$-varieties, with concrete moduli-interpretations via GIT. An alternative construction via qdlt Fano-type models is developed to realize the same framework, extending the toolkit for torus-equivariant K-stability analyses.

Abstract

For any log Fano pair with a torus action, we associate a computable invariant to it, such that the pair is (weighted) K-polystable if and only if this invariant is greater than one. As an application, we present examples of Fano varieties admitting $g$-solitons for any weight function $g$.

A valuative criterion of K-polystability

TL;DR

This work develops a valuative, torus-equivariant criterion for K-polystability of log Fano pairs by constructing a refined data pair from a maximal torus action and showing that, under the Futaki obstruction vanishing on the cocharacter lattice, stability reduces to the delta-invariant exceeding 1. The paper proves a sharp Abban–Zhuang-type estimate, builds the toric-refined sequence of models, and proves the main equivalence between K-polystability (or semistability) of and the corresponding delta condition on ; it also provides an almost complete linear-series variant linking to a fixed divisor on . Applications include the existence of -solitons for arbitrary weight functions and weight-insensitive stability phenomena in several Fano -varieties, with concrete moduli-interpretations via GIT. An alternative construction via qdlt Fano-type models is developed to realize the same framework, extending the toolkit for torus-equivariant K-stability analyses.

Abstract

For any log Fano pair with a torus action, we associate a computable invariant to it, such that the pair is (weighted) K-polystable if and only if this invariant is greater than one. As an application, we present examples of Fano varieties admitting -solitons for any weight function .
Paper Structure (16 sections, 25 theorems, 118 equations)

This paper contains 16 sections, 25 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Conventions
  4. Special test configurations and special divisorial valuations
  5. Valuative criterion of K-stability
  6. Toric divisors over Fano ${\mathbb T}$-varieties
  7. The $\xi$-twist of valuations
  8. Multi-graded linear series and refinements
  9. The sharpness of Abban-Zhuang estimate
  10. A Valuative Criterion of (Weighted) K-polystability
  11. Construction of $(X_r,\Delta_r,W^{X_r}_\bullet)$
  12. Proof of the main theorem
  13. Applications: existence of $g$-solitons
  14. Almost complete condition
  15. Existence of $g$-solitons
  16. ...and 1 more sections

Key Result

Theorem 1.1

If ${\mathbb T}\subseteq {\rm Aut}(X,\Delta)$ is a maximal subtorus and ${\rm Fut}|_N=0$, then the log Fano pair $(X,\Delta)$ is K-polystable (K-semistable) if and only if

Theorems & Definitions (71)

  • Theorem 1.1: Theorem \ref{['Theorem: main thm']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Theorem \ref{['Theorem: Sharpness of AZ']}
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['Theorem: main thm of almost complete']}
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 61 more