 Nontrivial hyperplane powers on a compact projective submanifold

Let $X$ be a nonempty positive-dimensional compact complex submanifold of Complex projective space. On a positive-dimensional connected component choose a hyperplane meeting, but not containing, that component. Its linear equation gives a holomorphic section of ${\mathcal{O}}_X(1)$ with a zero and with a nonzero value. For $m>0$, its $m$th power cannot be a section of a trivial holomorphic line bundle, since such sections are constant on the compact connected component. Negative powers are also nontrivial by duality. Every positive-dimensional compact projective component meets every hyperplane: otherwise its affine coordinate functions would all be constant.