Hyperplane line bundle (${\mathcal{O}}_{{\mathbb{P}}^n}(1)$)

The dual of the complex tautological line bundle. On the complex projective line the induced frames satisfy $e_1=w{e}_0$ on the overlap of its standard affine charts.

The First Chern class of the hyperplane line bundle. On Complex projective space it generates degree-two integral cohomology, and on ${\mathbb{CP}}^n$ its top power evaluates as one on the complex-oriented fundamental class.

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.

For positive $k$ take the $k$th tensor power of the hyperplane line bundle; for negative $k$ take the corresponding power of its dual; for $k=0$ take the trivial line bundle. On the complex projective line, global sections for $k\ge 0$ correspond to polynomials of degree at most $k$ and form a space of dimension $k+1$.