The Kobayashi-Hitchin correspondence is the one to one correspondence between slope stable sheaves and Hermitian-Einstein sheaves. Traditionally, this correspondence has been studied in the context of Kaehler classes. In this talk, I introduce the notions of slope stability and the Hermitian-Einstein metrics for big classes on normal analytic spaces. These notions exhibit a form of bimeromorphic invariance. As a consequence of the invariance, I will explain the Kobayashi-Hitchin correspondence on projective manifolds of general type.

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