本项目主要研究复几何中的形变Hermitian-Yang-Mills方程和复k-Hessian方程。这两类方程来源于超弦理论的研究和预定曲率问题，是当前复几何分析的重要研究内容。. 首先，我们讨论形变Hermitian-Yang-Mills方程，希望得到与其存在性相关的能量泛函逆紧性定理和Chern数不等式，并研究复空间中光滑解的刘维尔型定理。. 另一方面，我们讨论与复k-Hessian方程有关的多重位势理论，我们希望定义k-凸函数空间上面的Weil-Peterson度量、联络和测地线，并研究测地线的存在性和唯一性，进而研究k-凸函数空间在该Weil-Peterson度量下的完备化问题。. 本项目的研究内容是代数几何、多复变函数论、几何分析等几个数学分支的交叉，对超弦理论的研究有推动作用，具有很高的学术价值。

In this project, we mainly study deformed Hermitian-Yang-Mills equations and complex k-Hessian equations in the complex geometry. These PDEs are related to the superstring theory and prescribed curvature problems. They are also important parts in the modern complex geometric analysis.. Firstly, we consider the deformed Hermitian-Yang-Mills equations. We hope to get the relations betweent the existence of deformed Hermitian-Yang-Mills metrics and the properness of some energy functionals. We also consider the Chern number inequalities related to deformed Hermitian-Yang-Mills metrics. Furthermore, we will discuss the Liouville type theorem of deformed Hermitian-Yang-Mills equations and the theory of viscosity solutions about deformed Hermitian-Yang-Mills equations. . Secondly, we will discuss the geometry related to complex k-Hessian equations. More precisely, we consider the infinity dimensional space consisted of the k-convex functions on compact Kaehler manifolds. We hope to define a Weil-Peterson type metrics and consider the connections and geodesics related to it. At last, we consider the existence and uniqueness of such geodesics and also the completion of this function space under such Weil-Peterson metric.