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James Sparks
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This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
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Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f, mathbb{Z})$ is completely determined. Using the Boyer-Galicki construction of links $L_f$ over particular Kahler-Einstein orbifolds, we apply a valid case of Orliks conjecture to the links $L_f $ so that one is able to explicitly determine $H_{3}(L_f,mathbb{Z}).$ We give ten such new examples, all of which have the third Betti number satisfy $10leq b_{3}(L_{f})leq 20$.
We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result is to generalize T. Darvas theory on the geometric structure of the space of Kahler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kahler structure.
We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng cite{CC1, CC2, CC3} on existence of Kahler metrics with constant scalar curvature (csck) on compact Kahler manifolds. Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the $mathcal{K}$-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure. Technically, the proof mainly consists of two parts. The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Chengs results in cite{CC2, CC3} in Sasaki setting. The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on Kahler manifolds. There are notable, and indeed subtle differences in Sasaki setting (compared with Kahler setting) for both parts (PDE and pluripotential theory). The PDE part is an adaption of deep work of Chen-Cheng cite{CC1, CC2, CC3} to Sasaki setting with necessary modifications. While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in Kahler setting which is very intricate. We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper cite{HL} (joint work with Jun Li).
Given a Sasaki-Einstein manifold, M_7, there is the N=2 supersymmetric AdS_4 x M_7 Freund-Rubin solution of eleven-dimensional supergravity and the corresponding non-supersymmetric solutions: the perturbatively stable skew-whiffed solution, the perturbatively unstable Englert solution, and the Pope-Warner solution, which is known to be perturbatively unstable when M_7 is the seven-sphere or, more generally, a tri-Sasakian manifold. We show that similar perturbative instability of the Pope-Warner solution will arise for any Sasaki-Einstein manifold, M_7, admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the range between 2(9-4sqrt 3) and 2(9+4sqrt 3). Existence of such (1,1)-forms on all homogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kahler quotient construction or the standard harmonic expansion. The latter shows that the instability arises from the coupling between the Pope-Warner background and Kaluza-Klein scalar modes that at the supersymmetric point lie in a long Z-vector supermultiplet. We also verify that the instability persists for the orbifolds of homogeneous Sasaki-Einstein manifolds that have been discussed recently.
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kahler manifolds. We give an explicit non-compact example of an Einstein almost cokahler manifold that is not cokahler. We prove that compact Einstein almost cokahler manifolds with non-negative $*$-scalar curvature are cokahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cokahler. We prove similar bounds for almost Kahler Einstein manifolds that are not Kahler.
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