Test configurations for K-stability and geodesic rays

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 ...

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...

On the Regularity of Geodesic Rays Associated to Test Configurations

Geodesic rays of class C1,1 are constructed for any test configuration of a positive line bundle L → X, using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampère equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor.