The original version of the preprint [5] has been divided into two parts, and this is the second part, generalizing some results concerning Chern number inequalities to the Fujiki setting.

In this paper, I successfully constructed Kähler metrics with HSC > 0 on all rational surfaces. This gives a curvature characterization of rational surfaces, thus resolving a problem of Hitchin and Yau.

In this paper, we establish the nonabelian Hodge correspondence over compact Kähler klt spaces as well as their regular loci, thereby generalizing the previous result of Greb–Kebekus–Peternell–Taji for projective klt varieties.

As an interesting application, we prove that if a projective klt variety with big canonical divisor satisfies the orbifold Miyaoka–Yau type equality in the sense of the previous work [5] joint with M. Iwai and S. Jinnouchi, then its canonical model must be a singular quotient of the unit ball. See also Jinnouchi's recent preprint for K-stable klt varieties with big anti-canonical divisor.

The main result is establishing the Miyaoka–Yau inequality for projective klt varieties with big canonical divisor.

The basic idea is proving the Bogomolov–Gieseker inequality formulated via the non-pluripolar product. When I discussed with M. Iwai how to extend it to the klt case, I contributed to this work.

A basic question is characterizing the equality case of the Bogomolov–Gieseker inequality. It is subtle because we used a limiting process in the proof.

Our first main result provides a new differential-geometric criterion for rational connectedness. We prove that a compact Kähler manifold is rationally connected if and only if its tangent bundle is BC-p positive for every p ≥ 1. This notion of BC-p positivity is very weak and arises naturally from a Bochner-type formula associated with MRC fibrations (Proposition 3.3).

As a direct application, we confirm a conjecture of Prof. Lei Ni, showing that positive orthogonal Ricci curvature implies rational connectedness.

Our second result establishes structure theorems for manifolds with semi-positive curvature conditions, offering a natural generalization of several classical works.

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

On a personal note, the first version of this paper contained an error—specifically, a mistaken equality in a key integral inequality. The reviewer's careful reading pointed out this issue. In revising those details, I discovered something new and came to a deeper understanding: mathematics is not only about grand visions and directions, but also emerges from meticulous attention to detail. We must take full responsibility for what we write.

We extend the classical Miyaoka–Yau inequality to all minimal Kähler klt spaces.

One key is the existence of an orbifold modification by Kollár and Ou for klt spaces.

This paper provides a relatively general approach to considering Chern number inequalities and computing HN types of Higgs sheaves on Kähler klt varieties.

Notably, the framework developed here—based on L^p approximate Hermitian–Einstein metrics—yields an important byproduct: we obtain the semistability (respectively, generic nefness) of torsion-free sheaves under symmetric powers, exterior powers, and tensor products in the singular setting.

The first version contained an insufficient discussion of Harder–Narasimhan filtrations on singular varieties. That exposition was consequently difficult to follow; the present version supersedes it entirely.

The main result is the establishment of a pseudoeffective version of a 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 in the structure theorems for HSC ≥ 0 in the Kähler case obtained by Professor S. Matsumura.

Concerning nonnegative holomorphic sectional curvature, it remains open whether it implies some form of minimality in algebraic geometry.

The primary contribution to this work belongs to my collaborators. The most significant guidance throughout my doctoral studies came from my supervisor, yet I wish to express my deepest personal gratitude to my senior, Jiawei Liu. During my difficult third year of PhD, I was grappling with the Kähler–Ricci flow. Despite the distance and his own commitments, he offered patient and consistent guidance through our online correspondence. Through this process, I learned how to engage with a research problem, how to question, and how to begin.