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 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-Donaldson-Sun [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.

Abstract. Recently it was shown by H. Guenancia and M. Pȃun that a singular metric satisfying the conical Kähler-Einstein equation with a simple normal crossing divisor is equivalent to a conical metric along that divisor. In this note, we present an alternative proof of their theorem.

We give criterions for the existence of toric conical Kähler-Einstein and Kähler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kähler-Einstein metrics in the Gromov-Hausdorff topology.One can relate R BE (X) to the continuity method for solving the Kähler-Ricci soliton equation on X as introduced in [47] as analogue of R(X) and explicitly calculate the value of R BE (X) for toric Fano manifolds. In fact, we conjecture that R BE (X) = 1 for any Fano manifold X. However, in this paper, we are more interested in generalizing R(X) and R BE (X) for log Fano manifolds and more specifically, toric conical metrics on toric manifolds. We start with a few definitions. Let X be an n-dimensional toric manifold and L a Kähler class (or equivalently, an ample R divisor) on X. In [16,38], smooth toric conical Kähler metrics are defined and studied in detail and a brief review is given in section 2. We let K c (X) be the set of all smooth toric conical Kähler metrics with each cone angle in (0, 2π]. Definition 1.3. Let X be a toric manifold. Let ω ∈ K c (X) be a smooth toric conical Kähler metric on X. We say Ric(ω) > αω if there exists η ∈ K c (X) and an effective toric divisor D such thatIn fact, ω and η have the same cone angles and the divisor D can be explicitly calculated in terms of the cone angles of ω.Definition 1.4. A smooth toric conical Kähler metric ω ∈ K c (X) is called a conical Kähler-Ricci soliton metric if it satsifiesfor some holomorphic vector field ξ and effective toric divisor D. If ξ = 0, the metric is a smooth toric conical Kähler-Einstein metric.Associated to any toric Kähler class, we define the following geometric invariants R(X, L), R BE (X, L) and S(X, L).Definition 1.5. Let X be a toric manifold and L be a Kähler class on X. Let {D j } N j=1 be the set of all prime toric divisors on X. Then we define(1) R(X, L) = sup{α | Ric(ω) > αω for some ω ∈ c 1 (L) ∩ K c (X)},(2) R BE (X, L) = sup{α | Ric(ω) + L ξ ω > αω for a ω ∈ c 1 (L) ∩ K c (X) and a toric ξ ∈ H 0 (X, T X)},(3) S(X, L) = sup {α | there exists D = N j=1 a j D j ∼ −K X − αL with a j ∈ [0, 1)} . R(X, L) and R BE (X, L) are natural generalizations of R(X) and R BE (X) for log Fano manifolds with polarization L. S(X, L) characterizes when (X, D) is log Fano as by definition K X + D is klt and negative. In the special case that X is toric Fano and L = −K X , R(X, −K X ) is the usual greatest Ricci lower bound studied in [39] and S(X, −K X ) = 1. In fact, for any toric pair (X, L), R(X, L) and S(X, L) are both positive. In general, one can define R(X, L) and R BE (X, L) for any log Fano pair (X, L) by requiring Ric(ω) − αω ≥ 0 and Ric(ω) + L ξ ω − αω ≥ 0 in the current sense.Any toric manifold X is induced by an integral Delzant polytope P and P determines a Kähler class on X. Without loss of generality, we let (1.3) P = {x ∈ R n

In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampère equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regular set. As a consequence, the classical theorems of Myers and Bishop-Gromov extend almost verbatim to this singular setting.

We construct Kähler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kähler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.Remark 3.1. The above theorem is true as long as ω KE and ω ′ KE are complete. No other metric properties of ω KE , ω ′ KE are required. We prove some corollaries of Theorem 1.3.Corollary 3.1. Suppose we are in the setting of theorem 1.3 i.e, U admits a complete Kähler-Einstein metric ω KE with negative scalar curvature and V ol ω KE (U ) < ∞, then for any other complete Kähler-Einstein metric ω ′ KE with negative scalar curvature,Another simple application is a quick proof of uniqueness of complete Kähler-Einstein metric on a complex manifold without boundary.

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen's result is a degenerate concentration of mass result. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.

We prove that generalised Monge-Ampère equations (a family of PDEs which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if and only if certain numerical critera are satisfied. As a corollary, we prove a uniform version of a conjecture of Székelyhidi in the projective case. Our result also includes an equivariant version which can be used to recover existing results on manifolds with large symmetry such as projective toric manifolds.