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We classify compact K\"ahler manifolds with semi-positive holomorphicbisectional and big tangent bundles. We also classify compact complex surfaceswith semi-positive tangent bundles and compact complex $3$-folds of the form$P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is aFano manifold such that $P(T^*X)$ has nef tangent bundle, then $X\cong P^n$.

Big vector bundles and compact complex manifolds with semi-positive tangent bundles