Quasi-positive curvature and projectivity

In this paper, we first prove that a compact Kähler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive $S_2^{\perp },S_2^{+},Ri{c}_3^{\perp },Ri{c}_3^{+}$ or $2$-quasi-positive $Ri{c}_k$. Subsequently, we prove that a compact Kähler manifold with a restricted holonomy group is both projective and rationally conected if it satisfies some non-negative curvature condition, including non-negative $S_2^{\perp },S_2^{+},Ri{c}_3^{\perp },Ri{c}_3^{+}$ or $2$-non-negative $Ri{c}_k$.