Null Sasaki η-Einstein structures in 5-manifolds

We study null Sasaki structures in dimension five. As a consequence of the transverse version of Yau's theorem due to El Kacimi-Alaoui (cf. Compositio Math. 73(1):57-106, 1990) every null Sasakian structure can be deformed to a Sasaki η-Einstein structure which is transverse Calabi-Yau. One refers to these structures as null Sasaki η-Einstein. First, based on a result of Kollár (J Geom Anal 15:445-476, 2005), we improve a result of Boyer et al. (Commun Math Phys 262(1):177-208, 2006) and prove that simply connected manifolds diffeomorphic to # k(S2 × S3) admit null Sasaki η-Einstein structures for 3 ≤ k ≤ 21. We also determine the moduli space of simply connected null Sasaki η-Einstein metrics. This is accomplished using information on the moduli of lattice polarized K3 surfaces of the minimal resolutions of a K3 surface with at worst cyclic singularities. Then, applying the non-degeneracy of the quadratic form in the Sasakian manifold, naturally induced by basic cohomology, we give an explicit expression for the moduli space as a quadric in complex projective space. © 2013 Springer Science+Business Media Dordrecht.