本项目主要研究具有非正或非负曲率的Hermitian流形。满足Kobayashi双曲条件的紧致复流形一定有负的第一陈类，这是Kobayashi的一个著名猜测。而具有负全纯截面曲率的Hermitian流形一定是Kobayashi双曲的。最近，吴大旻-丘成桐证明了具有负全纯截面曲率的射影流形有负的第一陈类。这是一项重要的突破。本项目将利用Hermitian曲率流和Pluriclosed流的方法来研究具有负全纯截面曲率的Hermitian流形，并为解决Kobayashi猜想提供理论基础。对于具有正的全纯截面曲率的紧致Kähler流形，丘成桐猜测：它一定是射影流形。在2018年，杨晓奎解决了这个猜想。本项目将利用Bochner公式以及Hermitian曲率流，把这个结果推广到Hermitian流形。最后，我们将研究Hermitian流形之间的全纯映射，以此考虑不同曲率条件下的Schwarz估计。

This project will focus on Hermitian manifolds with nonpositive or nonnegative curvature. Any compact complex manifold which is Kobayashi hyperbolic must have an ample canonical line bundle, or equivalently, must have negative first Chern class, which is a famous conjecture from Kobayashi. Any Hermitian manifold with holomorphic sectional curvature bounded from above by a negative constant is always Kobayashi hyperbolic. In their recent breakthrough, Wu-Yau proved that projective manifolds with negative holomorphic section curvature have negative first Chen class. This project will use Hermitian curvature flow and Pluriclosed flow to study Hermitian manifolds with negative holomorphic section curvature, and provide a theoretical basis for solving the Kobayashi’s conjecture. For compact Kähler manfolds with positive holomorphic sectional curvature, it was conjectured by Yau, and recently proved by X. Yang that such manifolds are all projective. This project will use the Bochner formula and Hermitian curvature flow to generalize this result to Hermitian manifolds. Finally, we will study holomorphic mapping between Hermitian manifolds to consider Schwarz estimates under different curvature conditions.