Norming sets on a compact complex manifold

Abstract

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 $$\Vert s{\Vert }^2\le C{\int }_{G_k}|s(z){|}^{2}dV(z)$$ for every $s\in H^0(L^k)$ and for all $k\in \mathbb{N}$.