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$Z$-critical connections and Bridgeland stability conditions

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arxiv 2012.10426 v5 pith:T5D5MBPB submitted 2020-12-18 math.DG math.AG

Ruadhaí Dervan , John Benjamin McCarthy , Lars Martin Sektnan This is my paper · ORCID
classification math.DGmath.AG
keywords connectionscriticalequationsbridgelandconditionsdeformedhermitianholomorphic
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abstract

We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 26 citations worldwide. 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 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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