Welcome to Shiyu's homepage

In this note, I present an improvement on a result by Chu, Lee, and Tam. Specifically, I prove that a compact Kahler manifold with quasi-negative k-Ricci curvature has ample canonical bundle.

Furthermore, in the negative case, we can prove that any subvariety of dimension at least k is of general type. I haven't written down this result, becase I cannot address a fundamental open question, which arose from a private communication with Lei Ni, is to construct an example of a compact Kahler manifold X with negative k-Ricci curvature that admits a rational curve (1＜k＜dim X). Although such an example was claimed in a paper by Li, Ni, and Zhu, this was in fact a typographical error. The construction of such an example is very important for this topic because it means that the study on negative k-Ricci curvature provide new perspectives than negative holomorphic sectional curvature. I believe that such an example exists.

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 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.

A basic question is characterizing the equality case. It's quite subtle because we used the limiting process.

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. BC-p positviity is vert weak and naturally arises in a Bochner-type formula associated with MRC fibrations (Proposition 3.3). As an application, we confirm a conjecture of Ni 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. 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 not sure whether these two questions are true. And it seems to be very difficult by using Bochner techniques.

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.