We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson, which relate to the case of Kähler–Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

Abstract. We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-DonaldsonSun [17], and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.

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...

We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.