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Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles

Yoshinori Hashimoto, Julien Keller

TL;DR

The paper introduces the Quot-scheme limit of Fubini--Study metrics to connect slope stability with the asymptotics of Donaldson's functional, establishing coercivity of the functional on FS metrics under slope stability. It develops the renormalised Quot-scheme limit and the non-Archimedean Donaldson functional, linking analytic energy growth to algebro-geometric filtrations and their degrees. The results provide a PDE-free route to the slope stability implies Hermitian--Einstein implication and outline an approach (in a sequel HK2) toward a PDE-light proof of the Donaldson--Uhlenbeck--Yau theorem. Overall, the work generalises stability-analogue concepts from cscK geometry to vector bundles, offering a unifying framework via filtrations, Quot-schemes, and NA functionals that quantify stability through energy slopes.

Abstract

For a holomorphic vector bundle $E$ over a polarised Kähler manifold, we establish a direct link between the slope stability of $E$ and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldson's functional is coercive on the set of Fubini-Study metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein metrics implying slope stability.

Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles

TL;DR

The paper introduces the Quot-scheme limit of Fubini--Study metrics to connect slope stability with the asymptotics of Donaldson's functional, establishing coercivity of the functional on FS metrics under slope stability. It develops the renormalised Quot-scheme limit and the non-Archimedean Donaldson functional, linking analytic energy growth to algebro-geometric filtrations and their degrees. The results provide a PDE-free route to the slope stability implies Hermitian--Einstein implication and outline an approach (in a sequel HK2) toward a PDE-light proof of the Donaldson--Uhlenbeck--Yau theorem. Overall, the work generalises stability-analogue concepts from cscK geometry to vector bundles, offering a unifying framework via filtrations, Quot-schemes, and NA functionals that quantify stability through energy slopes.

Abstract

For a holomorphic vector bundle over a polarised Kähler manifold, we establish a direct link between the slope stability of and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldson's functional is coercive on the set of Fubini-Study metrics if is slope stable, and give a new proof of Hermitian-Einstein metrics implying slope stability.

Paper Structure

This paper contains 19 sections, 34 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Slope stability
  4. Filtrations and Quot-scheme limit
  5. Fubini--Study metrics on vector bundles
  6. $Q^*$ as a $C^{\infty}$-quotient map
  7. Donaldson's functional and the Hermitian--Einstein equation
  8. Quot-scheme limit of Fubini--Study metrics
  9. Filtrations and metric duals
  10. Renormalised Quot-scheme limit of Fubini--Study metrics
  11. Estimates for Donaldson's functional
  12. Preliminaries
  13. Strategy of the proof of Proposition \ref{['sumprop']}
  14. Proof of Proposition \ref{['lemdtmd']}
  15. Proof of Proposition \ref{['sumprophat']}
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that we take $k \in \mathbb{N}$ so that $\mathcal{E}(k) := \mathcal{E} \otimes \mathcal{O}_X(k)$ is globally generated. Let $\{ h_{\sigma_{t}} \}_{t \ge 0}$ be a family of Fubini--Study metrics on $\mathcal{E}$, emanating from a reference metric $h_{\mathrm{ref}}$, defined by a 1-parameter s where $\mu (\mathcal{E})$ (resp. $\mu (\mathcal{E}_{\le q})$) is the slope of $\mathcal{E}$ (resp.

Theorems & Definitions (88)

  • Theorem
  • Definition 2.1
  • Definition 2.2: Slope stability
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • ...and 78 more