Some remarks on Calabi-Yau and hyper-K\"ahler foliations

We study Riemannian foliations whose transverse Levi-Civita connection $\mathrm{\nabla }$ has special holonomy. In particular, we focus on the case where $Hol(\mathrm{\nabla })$ is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on K"ahler foliations and we apply this formula for pointing out some properties of transverse Calabi-Yau structures. This allows us to prove that links provide examples of compact simply-connected contact Calabi-Yau manifolds. Moreover, we show that a simply-connected compact manifold with a K"ahler foliation admits a transverse hyper- K"ahler structure if and only if it admits a compatible transverse hyper-Hermitian structure. This latter result is the “foliated version” of a theorem proved by Verbitsky. In the last part of the paper we adapt our results to the Sasakian case, showing in addition that a compact Sasakian manifold has trivial transverse holonomy if and only if it is a compact quotient of the Heisenberg Lie group.