Space H of Kähler metrics in the class

1 Overview 1. This talk is about geodesics in the infinite dimensional symmetric space of Kähler metrics in a fixed Kähler classá la Donaldson-Semmes. 2. These geodesics are solutions of a homogeneous complex Monge-Ampère equation in 'space-time'. One would like to know existence, regularity... 3. Phong-Sturm proved that one can construct weak solutions by special polynomial approximations. The purpose of this talk is to study the geodesics and the polynomial approximation on a toric variety. 2 Space H of Kähler metrics in the class [ω] Let L → M be an ample holomorphic line bundle over a compact Kähler manifold (M, ω 0) with 1 2π ω 0 ∈ H (1,1) (M, Z) and with c 1 (L) = [ω 0 ], the class of ω 0. Put m = dim M. Let h 0 be the unique Hermitian metric on L with curvature (1, 1) form ω 0. Any hermitian metric h with curvature in [ω 0 ] may be written h ϕ = e −ϕ h 0 , with ϕ in the space H = {ϕ ∈ C ∞ (M) : ω ϕ = ω 0 + √ −1 2 ∂ ¯ ∂ϕ > 0 }. 3 H is a symmetric space We endow H with a Riemannian metric: Identify the tangent space T ϕ H at ϕ ∈ H with C ∞ (M), let ψ ∈ T ϕ H H C ∞ (M) and define ||ψ|| 2 ϕ = M |ψ| 2 ω ϕ k .