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

Julien Keller; Carlo Scarpa

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arXiv · 2405.03312

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

Julien Keller·Carlo Scarpa

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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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Julien Keller, Carlo Scarpa. 2024-05-06. $Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces. https://arxiv.org/abs/2405.03312

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