Null Sasaki eta-Einstein Structures in Five Manifolds

We study null Sasakian structures in dimension five. First, based on a result due to Koll'ar [Ko], we improve a result by Boyer, Galicki and Matzeu in [BGM] and prove that simply connected manifolds diffeomorphic to # k(S^2\times S^3) admit null Sasaki $\eta$-Einstein structures if and only if $k\in {3,…, 21}$. After this, we determine the moduli space of simply connected null Sasaki $\eta$-Einstein structures. This is accomplished using information on the moduli of lattice polarized K3 surfaces.