$Z$-critical equations for holomorphic vector bundles on Kähler surfaces

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 Kähler 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.

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