  We consider a sufficiently smooth semi-stable holomorphic vector bundle over
a compact K\"ahler manifold. Assuming the automorphism group of its graded
object to be abelian, we provide a semialgebraic decomposition of a
neighbourhood of the polarisation in the K\"ahler cone into chambers
characterising (in)stability. For a path in a stable chamber converging to the
initial polarisation, we show that the associated HYM connections converge to
an HYM connection on the graded object.
