A compact HKT manifold is a hyperhermitian manifold $(M,I,J,K,g)$ such that either $\partial (\omega_J+i\omega_K)=0$ (if endomorphisms act on the left on the tangent space and $\partial$ is taken with respect to $I$) or, equivalently, the three Bismut connections for hermitian structures $(I,g)$, $(J,g)$, $(K,g)$ are all the same.
My question is related to the situation when the canonical bundle (with respect to $I$) is non trivial$^1$ (holomorphically). Can it happen that $\mathcal{K}_M$ has many section or it always has no sections at all, like for example for Hopf surfaces?
The second question is whether the answer is know for example for homogeneous examples of hypercomplex manifolds due to Joyce, $SU(3)$ for instance?
$1.$ From the work of M. Verbitsky it is know that a compact HKT manifold has a trivial canonical bundle iff the holonomy of the Obata connection is reduced to $Sl_n(\mathbb{H})$.
