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$Z$-critical equations for holomorphic vector bundles on Kähler surfaces

Julien Keller, Carlo Scarpa

TL;DR

This work develops a non-asymptotic bridge between differential-geometric $Z$-critical metrics on rank-$2$ holomorphic bundles over compact Kähler surfaces and algebraic stability notions. By recasting the $Z$-critical equation as a twisted vector bundle Monge–Ampère equation, it introduces $Z$-positivity and shows that, for $\alpha>0$, a $Z$-positive $Z$-critical metric implies $Z$-stability, while $Z$-stability aligns with twisted Monge–Ampère stability. The paper establishes a Bogomolov-type inequality under a volume-form hypothesis and proves openness under perturbations of the central charge, yielding non-asymptotic existence results; it also develops a $Z$-polystability theory for decomposable bundles and connects these notions to Gieseker stability through almost Hermitian–Einstein theory. Together, these results provide a concrete framework for predicting and constructing $Z$-positive, $Z$-critical metrics and illuminate the interplay between analytic and algebraic stability in the rank-$2$ setting. The findings have implications for broader stability programs, including deformed Hermitian Yang–Mills, almost Hermite–Einstein equations, and their non-asymptotic regimes on complex surfaces.

Abstract

We prove that the existence of a $Z$-positive and $Z$-critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact Kähler surface implies that the bundle is $Z$-stable. As particular cases, we obtain stability results for the deformed Hermitian Yang-Mills equation and the almost Hermite-Einstein equation for rank 2 bundles over surfaces. We show examples of $Z$-unstable bundles and $Z$-critical metrics away from the large volume limit.

$Z$-critical equations for holomorphic vector bundles on Kähler surfaces

TL;DR

This work develops a non-asymptotic bridge between differential-geometric -critical metrics on rank- holomorphic bundles over compact Kähler surfaces and algebraic stability notions. By recasting the -critical equation as a twisted vector bundle Monge–Ampère equation, it introduces -positivity and shows that, for , a -positive -critical metric implies -stability, while -stability aligns with twisted Monge–Ampère stability. The paper establishes a Bogomolov-type inequality under a volume-form hypothesis and proves openness under perturbations of the central charge, yielding non-asymptotic existence results; it also develops a -polystability theory for decomposable bundles and connects these notions to Gieseker stability through almost Hermitian–Einstein theory. Together, these results provide a concrete framework for predicting and constructing -positive, -critical metrics and illuminate the interplay between analytic and algebraic stability in the rank- setting. The findings have implications for broader stability programs, including deformed Hermitian Yang–Mills, almost Hermite–Einstein equations, and their non-asymptotic regimes on complex surfaces.

Abstract

We prove that the existence of a -positive and -critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact Kähler surface implies that the bundle is -stable. As particular cases, we obtain stability results for the deformed Hermitian Yang-Mills equation and the almost Hermite-Einstein equation for rank 2 bundles over surfaces. We show examples of -unstable bundles and -critical metrics away from the large volume limit.
Paper Structure (10 sections, 21 theorems, 202 equations)

This paper contains 10 sections, 21 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. The Z-critical and vector bundle Monge-Ampère equations
  3. Some positivity conditions for Hermitian metrics
  4. Z-stability and Monge-Ampère stability
  5. Consequences of Monge-Ampère positivity
  6. Other stability conditions
  7. Small variations of the charge and Mumford stability
  8. Some examples, Gieseker stability, and Z-polystability
  9. Z-polystability
  10. Z-stability and Gieseker stability

Key Result

Theorem 1.1

Let $X$ be a compact Kähler surface, and let $Z$ be a polynomial central charge. For any rank $2$ indecomposable vector bundle $E\to X$, if there exists $h\in\mathcal{H}^+(E)$ that is $Z$-positive and solves the $Z$-critical equation, then for any sub-bundle $S\subset E$ such that $0<\mathrm{rk}(S)< If the bundle moreover satisfies condition eq:Z_alphapositive below, this inequality holds for any

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7: Pingali_vbMA, Proposition $3.1$
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 57 more