The titie of the talk is 'Nonabelian Hogde correspondence over Kahler log terminal spaces' from my paper [7]. This talk was invited by R. Goto and M. Iwai. I enjoy Japan Ramen.

In this paper, we establish the nonabelian Hodge correspondence over compact K\"ahler 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. Toghther, these results confirm the Miyaoka-Yau inequality formulated via non-pluripolar product developed in the paper [5] provide an effective criterion for projective klt varieties with big canonical divisor.

This is just a note, I present an improvement on a result by Chu, Lee, and Tam based on a use of Gauss-Codazzi equation.

An example of a projective manifold with negative k-Ricci curvature admits an embedding of P^k was claimed in a paper by Li--Ni--Zhu, this was in fact a typographical error. Thank Prof. Lei Ni for kind clarification.

I will not revise and submit this paper until I have made sure of the existence of this example. If someone find such an example, it will be nice because it means that the study on negative k-Ricci curvature indeed provide new perspectives than negative holomorphic sectional curvature. I am not good at construting an example.

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

The basic idea is proving 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 this work.

A basic question is characterizing the equality case of Bogomolov-Gieseker inequality. It's subtle because we used the 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 immediate curvature conditions, offering a natural generalization of several classical works.

It would be interesting whether BC-p quasi-positivity (p≥1) imply the rational connectedness and BC-2 quasi-positivity implies the projectivity. The latter is posed by L. Ni. The former is posed by me and indeed implies a conjecture of Xiaokui Yang that quasi-positive uniformly RC-positivity ensures the rational connectedness.

I am grateful to Prof. Lei Ni for his attention to this work, and for his generous assistance on numerous other occasions.

On a personal note, I must candidly share that the first version of this paper contained an error—specifically, a mistaken equality in a key integral inequality. It was the reviewer’s careful reading that patiently pointed out this issue. In the process of carefully 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 K\"ahler klt spaces.

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

This paper does provide a relatively general approach to considering Chern number inequalities and computing HN types of Higgs sheaves on K\"ahler KLT varieties.

On the other hand, our argument is new for the smooth case. After completing this paper, Iwai shared an alternative approach with me. Please see the paper [5].

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.

Finally, I should acknowledge that the first version of this paper contained an insufficient discussion of Harder–Narasimhan filtrations on singular varieties, due to my then-unfamiliarity with certain technical aspects. That exposition was consequently difficult to follow; the present version supersedes it entirely.

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.

Concerning the nonnegative holomorphic sectional curvature, it remains open whether it implies some minimality in AG?

The primary contribution to this work belongs to my collaborators. The most significant guidance throughout my doctoral studies undoubtedly 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. At that time, Jiawei had already graduated and was conducting his research in Germany. Despite the distance and his own commitments, he offered me patient and consistent guidance through our online correspondence. That period culminated in a painful personal realization: that this particular field might not be the right path for me. The sense of frustration and doubt was significant. Yet, looking back, I see that this very struggle—and the generous, remote mentorship he provided across continents—was fundamental. It was through this process that I truly learned how to engage with a research problem, how to question, and how to begin.