Convergence of Bergman Geodesics on Cp 1 1

This article is concerned with geodesics in spaces of Hermitian metrics of positive curvature on an ample line bundle L → X over a Kähler manifold. Stimulated by a recent article of Phong-Sturm [PS], we study the convergence as N → ∞ of geodesics on the finite dimensional symmetric spaces HN of Bergman metrics of ‘height N ’ to Monge-Ampére geodesics on the full infinite dimensional symmetric space H of C∞ metrics of positive curvature. Our main result is C2 convergence of Bergman geodesics to Monge-Ampére geodesics in the case of toric (i.e. S1invariant) metrics on CP. Although such metrics constitute the simplest case of toric Kähler metrics, the CP case already exhibits much of the complexity of general toric varieties for the approximation problem studied here. The general case will be studied in [SoZ]. The convergence problem raised by Phong-Sturm [PS] and Arezzo-Tian [AT] belongs to the intensively studied program initiated by Yau [Y2] of relating the algebro-geometric issue of stability to the analytic issue existence of canonical metrics on holomorphic line bundles. In this program, metrics in HN have a simple description in terms of algebraic geometry, while metrics in H are ‘transcendental’. The approximation of transcendental objects inH by ‘rational’ objects in HN lies at the heart of this program. The reasons for studying Monge-Ampére geodesics were laid out by Donaldson in [D1] (see also Mabuchi [M] and Semmes [S2]). The existence, uniqueness and regularity of such geodesics is connected to existence and uniqueness of metrics of constant scalar curvature. Donaldson conjectured [D2] that there exist smooth Monge-Ampére geodesics between any pair of metrics h0, h1 ∈ H. The best general result, due to Chen [Ch] and Chen-Tian [CT], is the existence of a unique C1,1 geodesic ht joining h0 to h1. This result is sufficient to prove uniqueness of extremal metrics. In the case of toric varieties, the much stronger result is known that the geodesic between any two metrics is C∞ [G]. In fact, the Monge-Ampére equation can be linearized by the Legendre transform and the symmetric space is flat. But in general, solutions of the Monge-Ampére equation are difficult to analyze. It seems rather remarkable that one can study solutions of the homogeneous Monge-Ampère equation by means of ‘algebro-geometric approximations’. It has been proved by Phong-Sturm [PS] (see also [B]) that Bergman geodesics, which are orbits of one-parameter subgroups of GL(dN +1,C) between two Bergman metrics, converge uniformly to a given Monge-Ampére geodesic for a general ample line bundle over a Kähler manifold. The question we take up in this article and in [SoZ] is whether Bergman geodesics converge to Monge-Ampère geodesics in a stronger sense. Convergence in C2 is especially interesting since it implies that the curvatures and moments maps for the metrics along the Bergman geodesic converge to those along the Monge-Ampère geodesics. In this article and in the subsequent