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In this paper, we first prove a Donaldson-Uhlenbeck-Yau theorem overprojective normal varieties smooth in codimension two. As a consequence wededuce the polystability of (dual) tensor products of stable reflexive sheaves,and we give a new proof of the Bogomolov-Gieseker inequality, along with aprecise characterization of the case of equality. This also improves severalpreviously known algebro-geometric results on normalized tautological classes.We study the limiting behavior of semistable bundles over a degenerating familyof projective normal varieties. In the case of a family of stable vectorbundles, we study the degeneration of the corresponding HYM connections andthese can be characterized from the algebro-geometric perspective. Inparticular, this proves another version of the singular Donaldson-Uhlenbeck-Yautheorem for the normal projective varieties in the central fiber. As anapplication, we apply the results to the degeneration of stable bundles throughthe deformation to projective cones, and we explain how our results are relatedto the Mehta-Ramanathan restriction theorem.

A Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles on degenerating families