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<p>We prove that the existence of a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-positive and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-critical Hermitian metric on a rank&#xa0;2 holomorphic vector bundle on a compact Kähler surface implies that the bundle is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-stable. As special cases, we obtain stability results for the deformed Hermitian Yang–Mills equation and the almost Hermite–Einstein equation for rank&#xa0;2 bundles over surfaces. We also provide examples of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-stable and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-unstable bundles, as well as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>-critical metrics, away from the large volume limit.</p>

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Z-critical equations for holomorphic vector bundles on Kähler surfaces

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Julien Keller,
  • Carlo Scarpa

摘要

We prove that the existence of a \(Z\) Z -positive and \(Z\) Z -critical Hermitian metric on a rank 2 holomorphic vector bundle on a compact Kähler surface implies that the bundle is \(Z\) Z -stable. As special 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 also provide examples of \(Z\) Z -stable and \(Z\) Z -unstable bundles, as well as \(Z\) Z -critical metrics, away from the large volume limit.