Continuity of HYM connections with respect to metric variations

Source abstract

Abstract We investigate the set of (real Dolbeault classes of) balanced metrics on a balanced manifold with respect to which a torsion‐free coherent sheaf on is slope stable. We prove that the set of all such is an open convex cone locally defined by a finite number of linear inequalities. When 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 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.