论文标题
Vaisman流形的calabi-yau定理
A Calabi-Yau theorem for Vaisman manifolds
论文作者
论文摘要
如果$ dw = w \ w \wedgeθ$，而与levi-civita的连接相对于紧凑的复杂复杂的赫尔米尼式歧管$（m，i，w）$，则称为vaisman vaisman。相对于向量字段$ x $ dual至$θ$（称为Lee Field）和向量字段$ i（x）$，称为{anti-Lee field}的vector field $ x $ dual $ x $ dual $ x $ dual $ x $ dual的动作是不变的。 $θ$（称为Lee类）的同一个同类班级在Kahler几何学中扮演着与Kahler类相同的角色。我们证明，Vaisman度量是由其体积形式和Lee类唯一决定的，相反，对于每个Lee类$ [θ] $以及每个Lee-和Anti-Lee-Invariant Volume Volume $ v $，都存在Vaisman结构，具有Vaisman结构，具有V $和Lee Class $ C [θ] $。这是Calabi-yau定理的类似物，声称Kahler形式是由其数量和协同学类别决定的。
A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge θ$ and the 1-form $θ$, called the Lee form, is parallel with respect to the Levi-Civita connection. The volume form of $M$ is invariant with respect to the action of the vector field $X$ dual to $θ$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $θ$, called the Lee class, plays the same role as the Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely determined by its volume form and the Lee class, and, conversely, for each Lee class $[θ]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[θ]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class.
