Welcome to Shiyu Zhang's homepage

In Chinese, '世' means 'world' and '宇' means 'universe'. My name carries a nice meaning, though interestingly, it was chosen by my parents purely by coincidence. '世' indicates my generational position within the Zhang family, and since I was born on a rainy day, they picked '宇', which sounds the same in Chinese as rain.

The titie is 'Nonabelian Hogde correspondence over Kahler log terminal spaces' from my paper [7]. I love Japan Ramen and have it for every meal when staying at Japan... Unfortulately, I can't stand Sashime.

In this note, I present an improvement on a result by Chu, Lee, and Tam.

An example of a projective manifold with ample canonical bundle admits an embedding of P^k was claimed in a paper by Li, L. Ni, and Zhu, this was in fact a typographical error. Thank Lei Ni for kind clarification. 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 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 M. Iwai how to extend it to the klt case, I contributed this work.

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

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, which is a natural generalization of classical work.

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 difficult via 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 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].

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?