The main result is establishing the Miyaoka-Yau inequality for projective klt varieties with big canonical divisor. The idea of using the non-pluripolar product of ^{n-2} and providing a proof in the smooth case comes from Iwai and Jinnouchi. When I discussed with Prof. Iwai how to extend it to the klt case, I contributed this work. I am always grateful for their kind invitation to collaborate with me. And I have learned so much from the discussions with them.

The first result is providing a new differential geometric criterion for rational connectedness by proving that a compact K\"ahler manifold is rationally connected if and only if its tangent bundle is BC-p positive. As an application, we confirm a conjecture of Ni-Wang-Zheng that positive orthogonal Ricci curvature implies the rational connectedness. The second result is establishing the structure theorems for semi-positive immediate curvature conditions.

It would be interesting whether BC-p quasi-positivity (p≥1) imply the rational connectedness and BC-2 quasi-positivity implies the projectivity.

We extend the classical Miyaoka-Yau inequality to all minimal K\"ahler klt spaces. One key is the existence of orbifold modification by Ou for klt spaces. This paper appears to be quite basic, it does provide a relatively general approach to considering Chern number inequalities on K\"ahler KLT varieties, building on more advanced results. On the other hand, our argument is new for the smooth case. After completing this paper, Iwai shared an alternative and simplier approach with me. Please see the paper [5].

The main result is the establishment of a "pseudoeffective" version of Bochner-type theorem, which leads to a splitting of the tangent bundle. The innovation point is an elegant integral inequality. This is a key step of the structure theorems of HSC≥0 for the Kahler case obtained by Professor S. Matsumura recently.

The main contributions come from collaborators, and I didn't contribute much to this work.