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The goal of this short note is to point out that every Fano manifold with anef tangent bundle possesses an almost K{\"a}hler-Einstein metric, in a weaksense. The technique relies on a regularization theorem for closed positive (1,1)-currents. We also discuss related semistability questions and Cherninequalities.

Fano manifolds with nef tangent bundles are weakly almost K{\"a}hler-Einstein