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 introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.

We show that on a Kähler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kähler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture in [18] in this case. We also strengthen existing results on more general inverse σ k equations on Kähler manifolds.

We prove that the non-Kähler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kähler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustaţȃ-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kähler-Ricci flow on compact Kähler manifolds always form along analytic subvarieties, thus answering a question of Feldman-Ilmanen-Knopf and Campana. We also extend the second author's results about noncollapsing degenerations of Ricci-flat Kähler metrics on Calabi-Yau manifolds to the nonalgebraic case.

Let (X, α) be a Kähler manifold of dimension n, and let [ω] ∈ H 1,1 (X, R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equationwhere λi are the eigenvalues of ω with respect to α. When h(x) is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori C 2,β estimates when |h| > (n − 2) π 2 and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.