Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds

Abstract

We propose the study of a Monge-Ampère-type equation in bidegree (n-1, n-1) rather than (1, 1) on a compact complex manifold X of dimension n for which we prove ellipticity and uniqueness of the solution subject to positivity and normalisation restrictions.Existence will hopefully be dealt with in future work.The aim is to construct a special Gauduchon metric uniquely associated with any Aeppli cohomology class of bidegree (n-1, n-1) lying in the Gauduchon cone of X that we hereby introduce as a subset of the real Aeppli cohomology group of type (n-1, n-1) and whose first properties we study.Two directions for applications of this new equation are envisaged : to moduli spaces of Calabi-Yau ∂ ∂-manifolds and to a further study of the deformation properties of the Gauduchon cone beyond those given in this paper.