The talk consists of two parts. First, I will discuss a variant of anti-self-dual connections in higher dimensions that is motivated by calibrated geometry and multipolarizations in Kaehler geometry. Such connections are shown to satisfy a version of Uhlenbeck weak compactness. In the case of Hermitian manifolds, we prove analogs of the Donaldson–Uhlenbeck–Yau and nonabelian Hodge theorems. Second, I describe an extension of the Donaldson–Uhlenbeck–Yau theorem to normal projective varieties. Taken together, the two parts give an analytic proof of Miyaoka’s version of the Bogomolov–Gieseker inequality, with a sharp result in the case of equality. This is joint work with Xuemiao Chen.