
Mathematics 2013
Holomorphic Deformations of Balanced CalabiYau $\partial\bar\partial$ManifoldsAbstract: 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 (=semiK\"ahler) Hermitian metric $\omega$, we introduce the concept of deformations of $X$ that are {\bf copolarised} by the balanced class $[\omega^{n1}]\in H^{n1,\,n1}(X,\,\C)\subset H^{2n2}(X,\,\C)$ and show that the resulting theory of balanced copolarised deformations is a natural extension of the classical theory of K\"ahler polarised deformations in the context of CalabiYau or even holomorphic symplectic compact complex manifolds. The concept of WeilPetersson metric still makes sense in this strictly more general, possibly nonK\"ahler context, while the Local Torelli Theorem still holds.
