Some remarks on Calabi–Yau and hyper-Kähler foliations

We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol(∇)Hol(∇) is contained either in SU(n)SU(n) or in Sp(n)Sp(n). We prove a Weitzenböck formula involving complex basic forms on Kähler 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 Kähler foliation admits a transverse hyper-Kähler 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 [46]. 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.