几类非Kahler复流形的研究

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.

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。

复几何是一个非常丰富的研究领域。过去数十年，Kähler复几何取得了巨大的发展。最近，非Kähler复几何也逐渐受到人们的重视。众所周知，Kähler度量不是紧致复流形的双有理不变量，而 balanced度量是双有理不变量。目前，已知的满足dd\bar-引理的紧致复流形都具有balanced度量；并且满足dd\bar-引理的紧致复流形有着类似于紧致Kähler复流形的良好性质，例如：Hodge-to-de Rham谱序列在E_1处退化、Hodge对称和formality等等。本项目研究的主要问题是：紧致复流形上dd\bar-引理的双有理不变性问题。利用层上同调理论，本项目获得了紧致复流形上Bott-Chern上同调的一个blow-up公式，并结合弱分解定理和dd\bar-引理的上同调刻画，证明了dd\bar-引理数和dd\bar-引理都是三维紧致复流形的双有理不变量。同时，本项目获得了紧致复流形上 Dolbeault上同调、全纯向量丛值Dolbeault上同调和twisted de Rham上同调的blow-up公式，并证明了Hodge-to-de Rham谱序列的E_1-退化性是三维和四维紧致复流形的双有理不变量。另外，本项目拓展了相关研究目标，探索了特殊三次复4-流形的Hassett除子的相交性问题，证明了任意两个Hassett除子都相交非空，并且证明每个Hassett除子包含至少3个余维2的子簇参数化光滑有理三次复4-流形；研究了三次复3-流形的Kuznetsov分支的Bridgeland稳定性条件，给出了三次复3-流形上线的Hilbert scheme的Bridgeland稳定对象的模空间刻画，并证明了Kuznetsov分支的Serre函子保持这些稳定性条件。