2020/07/21 by Dinew, Sławomir, Popovici, Dan · 2 citations
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2007.10647
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 E2-cohomology class, that we call the E2-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.