Continuity of HYM connections with respect to metric variations

We investigate the set of (real Dolbeault classes of) balanced metrics $\Theta$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $E$ on $X$ is slope stable. We prove that the set of all such $\left[\Theta \right]\in H^{n-1,n-1}(X,R)$ is an open convex cone locally defined by a finite number of linear inequalities. When $E$ is a Hermitian vector bundle, the Kobayashi–Hitchin correspondence provides associated Hermitian Yang–Mills connections, which we show depend continuously on the metric, even around classes with respect to which $E$ is only semi-stable. In this case, the holomorphic structure induced by the connection is the holomorphic structure of the associated graded object. The method relies on semi-stable perturbation techniques for geometric PDEs with a moment map interpretation and is quite versatile, and we hope that it can be used in other similar problems.