Limiting properties of Hermitian-Einstein metrics under variation of the Kähler class

We study the behaviour of Hermitian-Einstein metrics on a holomorphic vector bundle over a compact Kähler manifold, as the underlying Kähler metric varies and its Kähler class changes. In particular, if the bundle stops being stable at some point, the Hermitian-Einstein metrics blow up, and their rescaling can be used to construct a destabilizing subsheaf, following the argument of Uhlenbeck and Yau. Furthermore, we study the behaviour of Chern connections of these metrics, and show, that after passing to a subsequence, and applying suitable gauge transforms these connections converge, and identify the blow-up locus of their Uhlenbeck limit.