Continuity of HYM connections with respect to metric variations

We investigate the set of (real Dolbeault classes of) balanced metrics Θ 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 [Θ] ∈ Hn - 1,n - 1(X,R) is a convex cone defined by a finite number of linear inequalities. For this, we prove that the set of all c₁(S) for S ⊂ E saturated and coherent is finite. 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.