= Nontrivial hyperplane powers on a compact projective submanifold
{title2=$\mathcal O_X(m)\cong\mathcal O_X\Longleftrightarrow m=0$}
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.
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