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

Abstract

In this paper, we first prove a Donaldson–Uhlenbeck–Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov–Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable vector bundles, we study the degeneration of the corresponding HYM connections and these can be characterized from the algebro-geometric perspective. In particular, this proves another version of the singular Donaldson–Uhlenbeck–Yau theorem for the normal projective varieties in the central fiber. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta–Ramanathan restriction theorem.