Holomorphic Deformations of Balanced Calabi-Yau ∂∂ ¯-Manifolds

Abstract

Given a compact complex n-fold X satisfying the ∂∂ ¯-lemma and supposed to have a trivial canonical bundle K X and to admit a balanced (=semi-Kähler) Hermitian metric ω, we introduce the concept of deformations of X that are co-polarised by the balanced class [ω n-1 ]∈H n-1,n-1 (X,ℂ)⊂H 2n-2 (X,ℂ) and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi–Yau or holomorphic symplectic compact complex manifolds. The concept of Weil–Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.