Letbe a polarized log variety with an effective holomorphic torus action, and be a closed positive torus invariant -current. For any smooth positive function defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci -solitons. We prove a version of the Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform -twisted -Ding-stability. When is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kähler-Ricci/Mabuchi solitons or Kähler-Einstein metrics.

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We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kähler surfaces. This is used to prove that for any scalar-flat Kähler ALE surface, all small deformations of complex structure also admit scalar-flat Kähler ALE metrics. A local moduli space of scalar-flat Kähler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kähler ALE surface which deforms to a minimal resolution of C 2 /Γ, where Γ is a finite subgroup of U(2) without complex reflections.as r → ∞, where Γ is a finite subgroup of U(2) containing no complex reflections, B denotes a ball centered at the origin, and g Euc denotes the Euclidean metric. The real number µ is called the order of g.Remark 1.2. In this paper, henceforth Γ will always be a finite subgroup of U(2) containing no complex reflections.We are interested in the class of scalar-flat Kähler ALE metrics. These are interesting since they are extremal in the sense of Calabi [Cal85], and they arise as "bubbles" in gluing constructions for extremal Kähler metrics [ALM15, ALM16, AP06, APS11, BR15, RS05, RS09]. In the case of scalar-flat Kähler ALE metrics, it is known that there exists an ALE coordinate system for which the order of such a metric is at least 2 [LM08].We note that for an ALE Kähler metric of order µ, there exist ALE coordinates for which2) for any multi-index I as r → ∞, where J Euc is the standard complex structure on Euclidean space [HL16]. There are many known examples of scalar-flat Kähler ALE metrics. In the case that Γ ⊂ SU(2), Kronheimer constructed and classified the hyperkähler ALE metrics [Kro89a, Kro89b]. Calderbank-Singer constructed a family of scalar-flat Kähler ALE metrics on the 1 Γ(T X) Γ(End a (T X)) Γ {Λ 0,2 ⊗ Θ ⊕ Λ 2,0 ⊗ Θ} R , ∂ Re ∂ Re Re Z →− 1 2 J•L Z J I → 1 4 J•N ′ J (I) where L Z J is the Lie derivative of J, End a (T X) = {I ∈ End(T X) : IJ = −JI}, (1.4) and N ′ J is the linearization of Nijenhuis tensor N(X, Y ) = 2{[JX, JY ] − [X, Y ] − J[X, JY ] − J[JX, Y ]} (1.5) at J. Each isomorphism Re is simply taking the real part of a section. If g is a Hermitian metric compatible with J, then let denote the∂-Laplacian ≡∂ * ∂ +∂∂ * , (1.6)where∂ * denotes the formal L 2 -adjoint. Each complex bundle in the diagram (1.3) admits a -Laplacian, and these correspond to real Laplacians on each real bundle in (1.3). We will use the same -notation for these real Laplacians.To state our most general result, we need the following definition. This is necessary because there is a gauge freedom of Euclidean motions in the definition of ALE coordinates. Definition 1.3. Let (X, g, J) be a Kähler ALE surface. For any bundle E in the diagram (1.3), and τ ∈ R, define H τ (X, E) = {θ ∈ Γ(X, E) : θ = 0, θ = O(r τ ) as r → ∞}. (1.7)

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.(1.7)Define global base spaces1.8) Case (a) follows easily from [HV16, Theorem 1.4]. Cases (b) and (c) are obtained by applying a generalization of a result of Biquard-Rollin to the ALE case [BR15]. For the precise statement, see Theorem 6.2 below.Recall that for integers p, q satisfying (p, q) = 1, the cyclic action 1 p (1, q) is that generated by (z 1 , z 2 ) → (ζ p z 1 , ζ q p z 2 ) where ζ p is a primitive pth root of unity. Corollary 1.12. Let Γ = 1 p (1, q) be any cyclic group with (p, q) = 1, and let J M k be any component of J M . Then for any J ∈ J M k (J is away from the central fiber if k > 0), there exists a scalar-flat Kähler metric ω J in some Kähler class.This is obtained by using the Calderbank-Singer construction from [CS04], together with Theorem 1.11. 1.2. Global existence results. We now turn our attention to existence of global moduli spaces of ALE SFK metrics for certain groups Γ. The following theorem is an application of Case (a) in Theorem 1.11 together with Corollary 1.7.

In this article, we give a survey of our construction of a local moduli space of scalar-flat Kähler ALE metrics in complex dimension 2. We also prove an explicit formula for the dimension of this moduli space on a scalar-flat Kähler ALE surface which deforms to the minimal resolution of C 2 /Γ, where Γ is a finite subgroup of U(2) without complex reflections, in terms of the embedding dimension of the singularity.