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

摘要

Given a compact complex $n$-fold $X$ satisfying the$partialarpartial$-lemma and supposed to have a trivial canonical bundle$K_X$ and to admit a balanced (=semi-K"ahler) Hermitian metric $omega$, weintroduce the concept of deformations of $X$ that are {f co-polarised} by thebalanced class $[omega^{n-1}]in H^{n-1,,n-1}(X,,C)subsetH^{2n-2}(X,,C)$ and show that the resulting theory of balanced co-polariseddeformations is a natural extension of the classical theory of K"ahlerpolarised deformations in the context of Calabi-Yau or even holomorphicsymplectic compact complex manifolds. The concept of Weil-Petersson metricstill makes sense in this strictly more general, possibly non-K"ahler context,while the Local Torelli Theorem still holds.