In this paper, we study smooth complex projective varieties X such that some exterior power r TX of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If TX is strictly nef, then X isomorphic to the projective space P n . If 2 TX is strictly nef and if X has dimension at least 3, then X is either isomorphic to P n or a quadric Q n .

In this article we show that if (X, ∆) is a log canonical compact Kähler threefold pair such that K X + ∆ is nef and the numerical dimension ν(X, K X + ∆) = 2, then K X + ∆ is semi-ample. This result combined with our previous work in [DO23] shows that the log abundance holds for log canonical compact Kähler threefold pairs.

We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.