Geometry of compact complex homogeneous spaces with vanishing first Chern class

We prove that any compact complex homogeneous space with vanishing first Chern class after an appropriate deformation of the complex structure admits a homogeneous Calabi-Yau with torsion structure, provided that it also has an invariant volume form. A description of such spaces among the homogeneous C-spaces is given as well as many examples and a classification in the 3-dimensional case. We calculate the cohomology ring of some of the examples and show that in dimension 14 there are infinitely many simply-connected spaces with the same Hodge numbers and torsional Chern classes admitting such structure. We provide also an example solving the Strominger's equations in heterotic string theory.