Related papers: Obstructions to the Existence of Sasaki-Einstein M…

This article is a summary of some of the author's work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal…

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat K\"{a}hler metrics on quasi-projective varieties Y=X\D with \alpha[D]=c_1(X), \alpha >1. The requirement that D admit a…

A five dimensional Sasaki-Einstein (SE) manifold provides a AdS/CFT pair for four dimensional $\mathcal{N}=1$ SCFT, and those pairs are very useful in studying field theory and AdS/CFT correspondence. The space of known SE manifolds is…

In this paper we study compact Sasaki manifolds in view of transverse K\"ahler geometry and extend some results in K\"ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse…

We describe an infinite family of quiver gauge theories that are AdS/CFT dual to a corresponding class of explicit horizon Sasaki-Einstein manifolds. The quivers may be obtained from a family of orbifold theories by a simple iterative…

This paper concerns obstruction flatness of hypersurfaces $\Sigma$ that arise as unit sphere bundles $S(E)$ of Griffiths negative Hermitian vector bundles $(E, h)$ over K\"ahler manifolds $(M, g).$ We prove that if the curvature of $(E, h)$…

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with…

We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is…

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…

We show that K\"ahler-Einstein metrics with cone singularities along simple normal crossing (SNC) divisors define RCD spaces, both in the compact setting and in certain non-compact cases, thereby producing many examples of Einstein RCD…

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory…

Obstruction flatness of a strongly pseudoconvex hypersurface $\Sigma$ in a complex manifold refers to the property that any (local) K\"ahler-Einstein metric on the pseudoconvex side of $\Sigma$, complete up to $\Sigma$, has a potential…

Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi-Yau cones, which…

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently,…

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize…

We review our study of Sasakian geometry as an agent for proving the existence of Einstein metrics on odd dimensional manifolds. Particular emphasis is given to the Sasakian structures occuring on links of isolated hypersurface…

We show the existence of strictly almost-Kahler anti-self-dual metrics on certain 4-manifolds by deforming scalar-flat Kahler metrics. On the other hand, we prove the non-existence of such metrics on certain other 4-manifolds by means of…

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with $\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic singularities…

There is an obstruction to the existence of K\"ahler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to…

We obtain a residue formula for an obstruction to the existence of coupled K\"ahler-Einstein metrics described by Futaki-Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with…