Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollar's simply-connected example which is a Seifert bundle over the complex projective space CP2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.