Norming Sets on a Compact Complex Manifold

We describe the norming sets for the space of global holomorphic sections to a k -power of a positive holomorphic line bundle on a compact complex manifold X . We characterize in metric terms the sequence of measurable subsets {G_k}_k of X such that there is a constant C > 0 where ‖ s‖ ^2≤ C ∫ _G_k |s(z)|^2 dV(z) for every s∈ H^0(X,𝒪(L^⊗ k)) and for all k∈ℕ .