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

TL;DR

This paper investigates Riemannian foliations with special holonomy, establishing properties of transverse Calabi-Yau and hyper-K"ahler structures, and connecting these to known geometric frameworks like Sasakian manifolds and the Heisenberg group.

Contribution

It proves new results on the properties of transverse Calabi-Yau and hyper-K"ahler structures in foliations, extending classical theorems to the foliated setting.

Findings

Links provide examples of compact simply-connected contact Calabi-Yau manifolds.

A simply-connected compact manifold with a K"ahler foliation admits a transverse hyper-K"ahler structure iff it admits a transverse hyper-Hermitian structure.

A compact Sasakian manifold has trivial transverse holonomy iff it is a quotient of the Heisenberg Lie group.

Abstract

We study Riemannian foliations whose transverse Levi-Civita connection $\nabla$ has special holonomy. In particular, we focus on the case where $Hol(\nabla)$ 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.