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 ...

The stable converse soul question (SCSQ) asks whether, given a real vector bundle $E$ over compact manifold, some stabilization $E \times \mathbb{R}^k$ admits metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all bundles any simply connected homogeneous manifold positive curvature, except possibly Berger space $B^{13}$...