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S. Kov\'acs proposed a conjecture on rigidity results induced by amplesubsheaves of some exterior power of tangent bundles for projective manifolds.We verify the conjecture in the case of second exterior products under a rankcondition. Besides, we prove a structure theorem satisfied by projectivemanifolds whose third exterior power of tangent bundle is nef. Additionally, weprove a weaker version of log Campana-Peternell conjecture for fourfolds.Finally, we give the structure of manifolds with a regular foliation whoseexterior powers are strictly nef.

Positivity of exterior products of tangent bundles and their subsheaves