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$Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces

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arxiv 2405.03312 v3 pith:3OKWXBO5 submitted 2024-05-06 math.DG math.AG

Julien Keller , Carlo Scarpa This is my paper · ORCID
classification math.DGmath.AG
keywords bundlescriticalahlerbundleequationhermitianholomorphicrank
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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 K\"ahler 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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Perturbations of Vector Bundle whose Curvature Form Solves a Polynomial Equation

    math.DG 2025-07 conditional novelty 8.0 of 10

    Local existence of solutions to polynomial curvature equations is equivalent to local polystability, with local filtrations and a Kempf-Ness homeomorphism.

  2. Deformed Hermitian-Yang-Mills equation on the manifold of full flags

    math.DG 2026-07 accept novelty 7.0 of 10

    First irreducible rank-2 dHYM connections are constructed on the full flag manifold F₂, and rank-1 solutions outside the supercritical regime disprove conjectured stability conditions.

  3. Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups

    math.AG 2025-06 accept novelty 6.0 of 10

    P-critical connections generalize Z-critical connections; on toric varieties P-positivity is checked finitely, and uniform P-positivity survives point blow-ups.

  4. Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves

    math.AG 2025-06 conditional novelty 6.0 of 10

    Bayer's polynomial Bridgeland stability on coherent sheaves reduces to lexicographic slope-vector stability exactly when the stability vector is adapted, and a dHYM-type example destabilizes the trivial bundle while a...

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