Positivity of exterior products of tangent bundles and their subsheaves

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