On the Regularity of Geodesic Rays Associated to Test Configurations

4 A ug 2 00 9 REGULARITY OF GEODESIC RAYS AND MONGE - AMPERE EQUATIONS 1

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class C1,α, 0 < α < 1. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampère equation on a Kähler manifold which is compact.

Test Configurations, Large Deviations and Geodesic Rays on Toric Varieties

This article contains a detailed study in the case of a toric variety of the geodesic rays φt defined by Phong-Sturm corresponding to test configurations T in the sense of Donaldson. We show that the ‘Bergman approximations’ φk(t, z) of Phong-Sturm converge in C to the geodesic ray φt, and that the geodesic ray itself is C 1,1 and no better. In particular, the Kähler metrics ωt = ω0 + i∂∂̄φt ass...

Test Configurations and Geodesic Rays

3 0 Ju n 20 06 TEST CONFIGURATIONS FOR K - STABILITY AND GEODESIC RAYS

Let X be a compact complex manifold, L → X an ample line bundle over X, and H the space of all positively curved metrics on L. We show that a pair (h0, T ) consisting of a point h0 ∈ H and a test configuration T = (L → X → C), canonically determines a weak geodesic ray R(h0, T ) in H which emanates from h0. Thus a test configuration behaves like a vector field on the space of Kähler potentials ...

m at h . D G ] 1 7 Ju n 20 06 TEST CONFIGURATIONS FOR K - STABILITY AND GEODESIC RAYS

Let X be a compact complex manifold, L → X an ample line bundle over X, and H the space of all positively curved metrics on L. We show that a pair (h0, T ) consisting of a point h0 ∈ H and a test configuration T = (L → X → C), canonically determines a weak geodesic ray R(h0, T ) in H which emanates from h0. Thus a test configuration behaves like a vector field on the space of Kähler potentials ...