A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics

We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $\omega$ on $X$, we introduce a functional acting on the metrics in the Aeppli cohomology class of $\omega$ and prove that its critical points (if any) must be K"ahler 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-Amp`ere-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 K"ahler 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 K"ahler metric in the given Hermitian-symplectic Aeppli class.