  This is the first of a series of papers where we relate tangent cones of
Hermitian-Yang-Mills connections at an isolated singularity to the complex
algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally
modelled on the pull-back of a holomorphic vector bundle from the projective
space. In this paper we shall impose an extra assumption that the graded sheaf
determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle
is reflexive. In general we conjecture that the tangent cone is uniquely
determined by the double dual of the associated graded object of a
Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a
certain torsion-free sheaf on the projective space. In this paper we also prove
this conjecture when there is an algebraic tangent cone which is locally free
and stable.
