Polarisation of SKT Calabi-Yau ∂∂̄-manifolds by Aeppli classes

Abstract

Given a ∂ ∂ ¯ \partial \bar \partial -manifold X X with trivial canonical bundle and carrying a metric ω \omega such that ∂ ∂ ¯ ω = 0 \partial \bar \partial \omega =0 , we introduce the concept of small deformations of X X polarised by the Aeppli cohomology class [ ω ] A [\omega ]_A of a strong Kähler with torsion metric ω \omega . There is a correspondence between the manifolds polarised by [ ω ] A [\omega ]_A in the Kuranishi family of X X and the Bott-Chern classes that are primitive in a sense that we define. We also investigate the existence of a primitive element in an arbitrary Bott-Chern primitive class and compare the metrics on the base space of the subfamily of manifolds polarised by [ ω ] A [\omega ]_A within the Kuranishi family.