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 H. We prove that R is non-trivial if the C× action on X0, the central fiber of X , is nontrivial. The ray R is obtained as limit of smooth geodesic rays Rk ⊆ Hk, where Hk ⊆ H is the subspace of Bergman metrics.