Existence and Compactness Theory for Ale Scalar-Flat Kähler Surfaces

Abstract

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.

On the ADM mass of Kähler scalar flat ALE metrics

“…The existence and geometric properties of Ricci and scalar flat metrics on ALE Kähler spaces with a given structure group at infinity have been subject of intense research in past few decades (see e.g. [8], [15], [14], [13], [7], [10], [16]). In this paper we study to which extent the machinery developed in [2] for constructing constant scalar curvature Kähler metrics on blow ups can be used and extended in this setting.…”