The Mabuchi completion of the space of Kähler potentials
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THE MABUCHI COMPLETION OF THE SPACE OF KÄHLER POTENTIALS By TAMÁS DARVAS Dedicated to Andrea. Abstract. Suppose (X,ω) is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials H can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space (H,d). We prove that the metric completion of (H,d) can be identified with (E2(X,ω), ˜ d), and this latter space is a complete non-positively curved geodesic metric space. In obtaining this result, we will rely on envelope techniques which allow for a treatment in a very general context. Profiting from this, we will characterize the pairs of potentials in PSH(X,ω) that can be connected by weak geodesics and we will also give a characterization of E(X,ω) in this context. 1. Introduction. Given (Xn,ω), a connected compact Kähler manifold, the space of smooth Kähler potentials is the set H = {u ∈ C∞ (X) | ω +i∂ ¯ ∂u > 0}. Using the ∂ ¯ ∂-lemma of Hodge theory, this set can be identified (up to a constant) with the space of Kähler metrics that are cohomologous to ω. H is a Fréchet manifold , and as such one can endow it with different Riemannian structures that induce path length metric spaces on H. One such Riemannian structure was introduced by E. Calabi [Ca] in 1954 (see also [Cl]), who proposed to find the metric completion of H using his metric. This was eventually carried out by B. Clarke and Y. Rubinstein in [CR]. As pointed out in [CR], it would be desirable, although much harder, to find the metric completion of H with respect to the much more studied Mabuchi metric. The main theorem of this paper answers this question completely, confirming a conjecture of V. Guedj [G] in the process. 1.1. Background. To state our main result it is necessary to recall some facts about the Mabuchi geometry of H. For a more exhaustive survey we refer to [Bl3]. For v ∈ H one can identify TvH with C∞(X). After Mabuchi [M], we introduce the following Riemannian metric on H: ξ,ηv :=  X ξη(ω +i∂∂v)n , ξ,η ∈ TvH. Manuscript received March 4, 2015. Research supported by NSF grant DMS-1162070. American Journal of Mathematics 139 (2017), 1275–1313. c  2017 by Johns Hopkins University Press. 1275 1276 T. DARVAS As discovered independently by Semmes [Se] and Donaldson [Do], the geodesic equation of this metric can be written as a complex Monge-Ampère equation. Let (0,1)  t → ut ∈ H be a smooth curve, S = {s ∈ C : 0 −h}(ω +i∂ ¯ ∂vh)n are increasing for h > 0. By definition, v ∈ E(X,ω) if lim h→∞  X½{v>−h}(ω +i∂ ¯ ∂vh)n =  X ωn =: Vol(X). Suppose χ : R∪{−∞} → R is a continuous increasing function, with χ(0) = 0 and χ(−∞) = −∞. Such χ is referred to as a weight. The set of all weights is denoted by W. Given v ∈ E(X,ω) we have v ∈ Eχ(X,ω) if Eχ(v) = lim h→∞  {v>−h} χ(vh)(ω +i∂ ¯ ∂vh)n > −∞. We will be most interested in E2(X,ω), this being the finite energy class given by the weight χ(t) = −t2, t ≤ 0 (see Section 2.3). Given u0,u1 ∈ E2(X,ω) and decreasing approximating sequences uk 0,uk 1 ∈ H, we define ˜ d(u0,u1), as promised, 1278 T. DARVAS by the formula: ˜ d(u0,u1) = lim k→∞ d(uk 0,uk 1). (4) We will prove that this definition is independent of the choice of decreasing approximating sequences and ˜ d is a metric on E2(X,ω). Moreover, for t → ut as defined in (3), we have ut ∈ E2(X,ω), t ∈ (0,1) and the following theorem holds, which is our main result: THEOREM 1. (Theorem 6.1, Theorem 9.2) (E2(X,ω), ˜ d) is a complete nonpositively curved geodesic metric space, with geodesic segments joining u0,u1 ∈ E2(X,ω) given by (3). Furthermore, (E2(X,ω), ˜ d) can be identified with the metric completion of (H,d). This result was proved for toric Kähler manifolds and conjectured to hold in general by V. Guedj in a preliminary version of [G]. Let us recall that a geodesic metric space (M,ρ) is a metric space for which...