在复几何中，如果紧致复流形具有特殊Hermitian度量则它的拓扑会有很强的限制；经典例子是紧致Kahler流形。过去几十年，数学家们从复几何和代数几何的角度研究了紧致Kahler流形，并取得了非常多的重要结果。相比而言，关于紧致非Kahler复流形的研究工作却不是很多。本项目将主要从双有理几何的观点来探索几类紧致非Kahler复流形，其内容包括以下三方面的问题：(1) Hermitian-symplectic度量的扩张定理以及tamed symplectic cone在blow-up下的稳定性问题；(2)紧致复流形上dd\bar-引理和dd\bar-引理数在blow-up下的双有理不变性问题；(3)一定条件下具有balanced度量和k-Gauduchon度量的紧致复流形是否是Kahler流形的问题。本项目的研究将有助于深化人们对非Kahler复几何的理解和认识。

In complex geometry, if a compact complex manifold admits a special Hermitian metric, then there exist many topological restrictions on it. The classical example is a compact Kahler manifold. In the past decades, mathematicians devoted many efforts to study compact Kahler manifolds from the complex and algebraic geometry points of view and obtained a lot of important results. Comparing with the extensive study of Kahler geometry, limited works have been done in the study of complex non-Kahler geometry. In this project, we will study some compact non-Kahler complex manifolds from the viewpoint of birational geometry, and we shall focus on the following problems: (1) the extension theorem of Hermitian-symplectic metrics and the stability properties of tamed symplectic cones under the blow-ups; (2) the birational invariance of the dd\bar-lemma and dd\bar-lemma numbers under the blow-ups; (3) the problem whether a compact complex manifold admit both balanced and k-Gauduchon metrics is a Kahler manifold under some additional conditions. This project is helpful to strengthen the understanding of complex non-Kahler geometry.