  For compact complex manifolds with vanishing first Chern class that are
compact torus principal bundles over K\"ahler manifolds, we prove that all
holomorphic geometric structures on them, of affine type, are locally
homogeneous. For a compact simply connected complex manifold in Fujiki class
$\mathcal C$, whose dimension is strictly larger than the algebraic dimension,
we prove that it does not admit any holomorphic rigid geometric structure, and
also it does not admit any holomorphic Cartan geometry of algebraic type. We
prove that compact complex simply connected manifolds in Fujiki class $\mathcal
{C}$ and with vanishing first Chern class do not admit any holomorphic Cartan
geometry of algebraic type.
