Resolutions of Cones over Einstein-Sasaki Spaces

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups $H_{\bar\partial}^{(p,0)}(k)$ graded by their charge under the Reeb vector. We then introduce a new cohomology, $\eta$-cohomology, which is defined by a CR structure and a holomorphic function $f$ with non-vanishing $\eta\equiv \mathrm{d}f$. It is the natural cohomology associated to a class of...

In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by T^2-invariant special Lagrangians, where the generic leaf is topologically T^2 X R. We also obtain a family of SO(3)-invariant special Lagrangians. These special Lagrangian families in both the deformed and the resolved conifold approach asymptotic...

The 2d (0,2) supersymmetric gauge theories corresponding to the classes of Y^{p,k}(CP^1 x CP^1) and Y^{p,k}(CP^2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest classification of 2d (0,2) supersym...

We construct new complete, compact, inhomogeneous Einstein metrics on S^{m+2} sphere bundles over 2n-dimensional Einstein-Kahler spaces K_{2n}, for all n \ge 1 and all m \ge 1. We also obtain complete, compact, inhomogeneous Einstein metrics on warped products of S^m with S^2 bundles over K_{2n}, for m>1. Additionally, we construct new complete, non-compact Ricci-flat metrics with topologies S^m times R^2 bundles over K_{2n} that generalise the higher-dimensional Taub-BOLT me...

We consider a generalization of Einstein-Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure, and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain...

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $\eta$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhibit infinite families of compact Sasakian $\eta$--Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show...

Roughly speaking, an ALF metric of real dimension $4n$ should be a metric such that its asymptotic cone is $4n-1$ dimensional, the volume growth of this metric is of order $4n-1$ and its sectional curvature tends to 0 at infinity. In this paper, I will first show that the Taub-NUT deformation of a hyperk\"ahler cone with respect to a locally free $\mathbb{S}^1-$symmetry is ALF hyperk\"ahler. Modelled on this metric at infinity, I will show the existence of ALF Calabi-Yau me...

In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark ...

Using a variational approach, we establish the equivalence between a weighted volume minimization principle and the existence of a conical Calabi-Yau structure on horospherical cones with mild singularities. This allows us to do explicit computations on the examples arising from rank-two symmetric spaces, showing the existence of many irregular horospherical cones.

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.