The Kobayashi-Hitchin correspondence states that the existence of Hermitian-Einstein metrics on a holomorphic vector bundle is equivalent to an algebro-geometric stability condition, and was proved by Donaldson and Uhlenbeck-Yau. Their original proofs were based on technical and sophisticated applications of nonlinear PDE theory. We present some results that clarify variational aspects of the Kobayashi--Hitchin correspondence for smooth projective varieties; they are based on the theory of Quot-schemes in algebraic geometry and rely much less on analysis. Joint work with Julien Keller.