Holonomy of non Kahler Hermitian manifold

Question

For an Hermitian manifold it is equivalent being Kahler and having holonomy in $U(n)$. I was wondering if someone can show me examples of non-Kahler Hermitian that illustrate what holonomy groups can I get (note that I don't assume the manifold to be irreducible or non-symmetric, so also examples with holonomies different than $SO(2m)$ would be cool).

Answer 1

This is hardly a classification, but a Calabi-Eckmann manifold is diffeomorphic to a product of odd-dimensional spheres $S^{2k-1} \times S^{2\ell-1}$. Picking an arbitrary Riemannian metric $g_{0}$ and averaging over the complex structure if necessary, i.e., defining $$ g(v, w) = \tfrac{1}{2}[g(v, w) + g(Jv, Jw)], $$ gives an Hermitian metric. We can therefore arrange holonomy contained in $SO(2k-2) \times SO(2\ell-2)$ for arbitrary positive integers $k$ and $\ell$. (Taking products naturally gives more possibilities.)