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Null Sasaki \(\eta \)-Einstein structures in 5-manifolds

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Abstract

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 \(\eta \)-Einstein structure which is transverse Calabi–Yau. One refers to these structures as null Sasaki \(\eta \)-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 \(\# k(S^2\times S^3)\) admit null Sasaki \(\eta \)-Einstein structures for \(3\leqslant k \leqslant 21\). We also determine the moduli space of simply connected null Sasaki \(\eta \)-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.

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Acknowledgments

Some of these results were part of the author’s dissertation under the supervision of Charles Boyer. I thank him and Krzysztof Galicki for their invaluable help. I also would like to thank the support that I received, as a postdoctoral fellow, from McMaster University in Ontario Canada, specially from McKenzie Wang. Lastly, I thank the reviewer for very helpful comments and suggestions that improved the clarity of the article.

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  1. Departmento de Ciencias, Sección Matemáticas, Pontificia Universidad Católica del Perú, Apartado 1761, Lima, 100, Peru

    Jaime Cuadros Valle

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Correspondence to Jaime Cuadros Valle.

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Cuadros Valle, J. Null Sasaki \(\eta \)-Einstein structures in 5-manifolds. Geom Dedicata 169, 343–359 (2014). https://doi.org/10.1007/s10711-013-9859-9

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