$Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces
Differential Geometry
2025-05-02 v3 Algebraic Geometry
Abstract
We prove that the existence of a -positive and -critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact K\"ahler 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.
Keywords
Cite
@article{arxiv.2405.03312,
title = {$Z$-critical equations for holomorphic vector bundles on K\"ahler surfaces},
author = {Julien Keller and Carlo Scarpa},
journal= {arXiv preprint arXiv:2405.03312},
year = {2025}
}
Comments
v3: 45 pages. We improved the main results of the paper, explaining more precisely the relation between the existence of a Z-positive metric and positivity properties of the bundle. We streamlined the discussion of some examples, introduced new ones, and updated references