论文标题
Hermitian-symphectic指标的AEPPLI共同学类别的一般体积不变
A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics
论文作者
论文摘要
我们调查了紧凑的复杂的Hermitian-symplectic歧管$ x $的类别。对于$ x $上的每个Hermitian-Splectic Metric $ω$，我们在AEPPLI共同体$ω$的aeppli共同体中介绍了一个功能性，并证明当$ x $是$ 3 $ $ 3 $时，其关键点（如果有）必须是kähler。我们继续表现出这些关键点，作为其AEPPLI类中度量标准量的最大值，并提出了一个Monge-ampère-type方程来研究其存在。我们的功能进一步用于定义一个数值不变的遗传学 - 合成指标的AEPPLI同胞类别，该类别概括了Kähler类的体积。我们获得了这一不变的两个同谋解释。同时，我们以$ e_2 $ - 亚物种学课程的形式构建了一个不变性，我们称之为$ e_2 $ - torsion类，与每一个Aeppli类别的Hermitian-symplectic指标相关联，并表明其消失是在给定的Hermitian-Hermitian-Plectic appplectic appplectic appplectic class中存在的必要条件。
We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $ω$ on $X$, we introduce a functional acting on the metrics in the Aeppli cohomology class of $ω$ and prove that its critical points (if any) must be Kähler when $X$ is $3$-dimensional. We go on to exhibit these critical points as maximisers of the volume of the metric in its Aeppli class and propose a Monge-Ampère-type equation to study their existence. Our functional is further utilised to define a numerical invariant for any Aeppli cohomology class of Hermitian-symplectic metrics that generalises the volume of a Kähler class. We obtain two cohomological interpretations of this invariant. Meanwhile, we construct an invariant in the form of an $E_2$-cohomology class, that we call the $E_2$-torsion class, associated with every Aeppli class of Hermitian-symplectic metrics and show that its vanishing is a necessary condition for the existence of a Kähler metric in the given Hermitian-symplectic Aeppli class.
