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Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve

Simon Jubert, Chenxi Yin

TL;DR

This work establishes a relative uniform Yau–Tian–Donaldson correspondence for projective bundles $Y=\mathbb{P}(E)$ over a curve by introducing compatible test configurations induced from the fiber $X=\mathbb{P}(\oplus_j \mathbb{C}^{d_j})$ and gluing them via horospherical symmetry to the base bundle. It proves that relative uniform K-stability for these compatible configurations is equivalent to the existence of an extremal Kähler metric in the given class, with the metric being compatible, and to a weighted K-stability condition on the moment polytope $\Delta$ with explicit weights $(p v,\hat w)$. The strategy combines a fiberwise analysis on $X$ using a generalized Calabi construction and a global extension to $Y$ by a bundle construction, reducing stability questions to convex polytope functionals and Mabuchi energy coercivity on lifted polytopes. The results unify extremal existence, polytope stability, and horospherical symmetry, offering concrete criteria for projective bundles and enriching the YTD program with explicit, computable weights tied to the bundle's topological data.

Abstract

This paper is concerned with a relative uniform Yau--Tian--Donaldson correspondence, in terms of test configurations, for the projectivization \( \mathbb{P}(E) \) of a holomorphic vector bundle \( E \) over a smooth curve. For any Kähler class \( [ω] \) on \( \mathbb{P}(E) \), we construct Kähler test configurations, which we call \emph{compatible test configurations}. They are obtained by gluing horospherical test configurations from the fibers, arising from convex functions on a suitable moment polytope \( Δ\) following the construction of Delcroix, to the principal bundle associated with \( \mathbb{P}(E) \). Using the generalized Calabi ansatz of Apostolov--Calderbank--Gauduchon--Tønnesen-Friedman on these test configurations, we show that the relative uniform stability of \( (\mathbb{P}(E),[ω]) \) for compatible test configurations implies the existence of an extremal metric in this class, thereby establishing the equivalence. Along the way, we prove that these two conditions are equivalent to the weighted uniform stability of \( Δ\) for suitable explicit weight functions defined from the topological data of \( \mathbb{P}(E) \).

Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve

TL;DR

This work establishes a relative uniform Yau–Tian–Donaldson correspondence for projective bundles over a curve by introducing compatible test configurations induced from the fiber and gluing them via horospherical symmetry to the base bundle. It proves that relative uniform K-stability for these compatible configurations is equivalent to the existence of an extremal Kähler metric in the given class, with the metric being compatible, and to a weighted K-stability condition on the moment polytope with explicit weights . The strategy combines a fiberwise analysis on using a generalized Calabi construction and a global extension to by a bundle construction, reducing stability questions to convex polytope functionals and Mabuchi energy coercivity on lifted polytopes. The results unify extremal existence, polytope stability, and horospherical symmetry, offering concrete criteria for projective bundles and enriching the YTD program with explicit, computable weights tied to the bundle's topological data.

Abstract

This paper is concerned with a relative uniform Yau--Tian--Donaldson correspondence, in terms of test configurations, for the projectivization \( \mathbb{P}(E) \) of a holomorphic vector bundle over a smooth curve. For any Kähler class on \( \mathbb{P}(E) \), we construct Kähler test configurations, which we call \emph{compatible test configurations}. They are obtained by gluing horospherical test configurations from the fibers, arising from convex functions on a suitable moment polytope following the construction of Delcroix, to the principal bundle associated with \( \mathbb{P}(E) \). Using the generalized Calabi ansatz of Apostolov--Calderbank--Gauduchon--Tønnesen-Friedman on these test configurations, we show that the relative uniform stability of \( (\mathbb{P}(E),[ω]) \) for compatible test configurations implies the existence of an extremal metric in this class, thereby establishing the equivalence. Along the way, we prove that these two conditions are equivalent to the weighted uniform stability of for suitable explicit weight functions defined from the topological data of \( \mathbb{P}(E) \).
Paper Structure (44 sections, 41 theorems, 155 equations)

This paper contains 44 sections, 41 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results and strategy of proof
  4. Step 1: the fiber $X$
  5. Step 2: the fibration $Y$
  6. Organization of the paper
  7. Review on weighted csck metrics and weighted K-stability
  8. Weighted cscK metrics
  9. Weighted K-stability
  10. Weighted toric K-stability
  11. Compatible test configurations of projective spaces
  12. Blown-down structure and compatible Kähler metrics
  13. The Lie algebra and moment coordinates
  14. The spaces of integral functions
  15. Compatible symplectic potentials
  16. ...and 29 more sections

Key Result

Theorem A

If $(Y,[\omega_Y])$ is relatively uniformly K-stable on compatible test configurations, then there exists an extremal Kähler metric in $[\omega_Y]$. Moreover, any extremal Kähler metric in $[\omega_Y]$ is compatible.

Theorems & Definitions (89)

  • Theorem A
  • Corollary A
  • Theorem B
  • Corollary B
  • Theorem C
  • Example 2.1: Extremal Kähler metric
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5: Has16Li_2022
  • ...and 79 more