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

TL;DR

This paper introduces a new volume-invariant functional for Aeppli cohomology classes of Hermitian-symplectic metrics on compact complex manifolds, linking critical points to Kähler metrics and proposing a Monge-Ampère-type equation for their existence.

Contribution

It defines a novel functional on Aeppli cohomology classes, characterizes its critical points, and introduces invariants generalizing volume, with cohomological interpretations and conditions for Kähler metrics.

Findings

Critical points of the functional are Kähler in 3D cases.

The functional's maximizers correspond to volume maximizers within Aeppli classes.

Introduces the $E_2$-torsion class as an invariant linked to Kähler metric existence.

Abstract

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.