Holomorphic Deformations of Balanced Calabi-Yau $\partial \bar{\partial }$-Manifolds

Abstract

Given a compact complex $n$-fold $X$ satisfying the $\partial \bar{\partial }$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $\omega$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1}(X,\,\C)\subset H^{2n-2}(X,\,\C)$ 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 even 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.

To appear in the Annales de l'Institut Fourier

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