厄米-辛群Aeppli上同调类的广义体积不变量
A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics
摘要：
研究了紧致复Hermitian-辛流形类$X$ .对于$X$上的每个Hermite-辛度量$ω$，我们引入一个作用于$ω$的Aeppli上同调类中度量的泛函，并证明当$X$是$3$ -维时，它的临界点（如果有的话）一定是Kähler.我们继续展示这些临界点作为最大化的量的度量在其Aeppli类，并提出了一个蒙格安培型方程来研究它们的存在。我们的功能是进一步利用定义一个数值不变的任何Aeppli上同调类的厄米辛度量，概括的量的Kähler类。我们得到了这个不变量的两个上同调解释。同时，我们构造了一个$E_2$ -上同调类形式的不变量，我们称之为$E_2$ -挠类，它对应于每一个厄米特-辛度量的Aeppli类，并证明了它的消失是给定厄米特-辛Aeppli类中Kähler度量存在的必要条件.
英文摘要：
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.
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