Sasaki–Einstein metrics and K–stability

Abstract

We show that a polarized affine variety admits a Ricci flat Kähler cone metric if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.To date, many Sasaki-Einstein manifolds have been found by employing estimates for the α-invariant [76,34]. For example, the affine varieties Z BP (p, q) are a special case of the Brieskorn-Pham singularities, which have been thoroughly studied in the literature. Boyer-Galick-Kollár [15] used estimates for the α-invariant of Brieskorn-Pham singularities to produce 68 distinct Sasaki-Einstein metrics on S 5 , as well as SE metrics on all 28 oriented diffeomorphism types of S 7 , and the the standard and Kervaire spheres S 4m+1 . Note that previously infinitely many Einstein (not Sasakian) metrics on spheres in dimensions 5 to 9 were constructed by Böhm [11].Estimates for the α-invariant were also used by Boyer-Galicki [12, 13], Boyer-Nakamaye [17], Kollár-Johnson [54], Ghigi-Kollár [47], Kollár [58,56] and others to produce many infinite families of Sasaki-Einstein metrics in dimensions 5 and 7, and higher. For example, #k(S 2 × S 3 ) is known to admit infinite families of Sasaki-Einstein metrics for any k 1. We refer the reader to [14] for a thorough discussion of these results. We note that Kollár has classified the possible topologies of Sasaki-Einstein manifolds [56,57,59]. For example it is known that for affine varieties of complex dimension 3 with a 2-torus action, the only possible topologies of the links are S 5 and k#(S 2 × S 3 ) for any k 1 (see [14, Proposition 10.2.27]). Our techniques also produce new infinite families of distinct Sasaki-Einstein metrics on k#(S 2 × S 3 ) for all k 1, and hence cover all possible topologies that can occur with a 2-torus action.We expect that many more examples can be found along the same lines. A particularly interesting problem is to find Sasaki-Einstein metrics with irregular Reeb vector fields. Remarkably, the first examples of irregular Sasaki-Einstein metrics were discovered by Gauntlett-Martelli-Sparks-Waldram [45] by explicitly writing down the metric in coordinates. We expect K-stability to be particularly useful for finding irregular Sasaki-Einstein manifolds in real dimension 5, since if the cone X has dim C X = 3, and ξ is an irregular Reeb field, then X admits a complexity-one action of a 2-torus. In particular, using the methods of Ilten-Süß [52] we can effectively test whether (X, ξ) admits a Ricci flat Kähler cone metric.The overall strategy of our proof is the same as that of Chen-Donaldson-Sun [22, 23, 24], as adapted in [75,30] to the smooth continuity method. We will set up this continuity method in Section 2, where we also give the precise definition of K-stability based on our previous work [27], extending the definition of Ross-Thomas [67] from the quasi-regular case. The main technical results ar...

Toric geometry and the dual of c-extremization

“…We have thus proven the desired relation (4. 14), and conclude that we may write the supersymmetric action (4.13) as…”

“…The main object of interest in [7] was the volume of the Sasakian manifold Y 5 . By 14) where recall that the transverse Kähler form ω Sasakian is given by (3.5). In the following Vol in (3.14) will always refer to the Sasakian volume.…”

New 3d $$ \mathcal{N} $$ = 2 SCFT’s with N3/2 scaling

“…where Vol(B 5 ) is the volume of the base B 5 of the CY 3 . Notice that the ratio a 1 /a 0 is always a number independent of ξ which only depends on the dimensionality of the CY singularity [53,Prop. 6.4].…”

“…In the non-toric case (which is the one of interest in this paper), one is left with the task of testing against r T -equivariant configurations λ, which can easily be guessed (see e.g. [53,Sec. 8] for other examples).…”

Interaction Between Singularity Theory and the Minimal Model Program

“…Later, in [CS15], following [Tia97,Don01], the K-(semi)polystability was formulated for Fano cone singularities. It was a straightforward calculation from the definition to show that if (X, ξ 0 ) is K-semistable, then v ξ0 is a minimiser among all valuations of the form v ξ for ξ ∈ t + R .…”