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

Over complex projective space, its fibre at a line $\ell \subseteq {\mathbb{C}}^{n+1}$ is $\ell$ itself. It is a holomorphic line bundle contained in the trivial bundle. Its dual is the hyperplane line bundle.

For any complex line bundle $L$, a nonzero dual vector $\varphi \in L_b^{\ast }$ determines the unique $v\in L_b$ with $\varphi (v)=1$. In local coordinates this is fiber inversion $a\mapsto a^{-1}$, giving a smooth projection-preserving diffeomorphism between the complements of the zero sections of $L^{\ast }$ and $L$. It is not a fiber-linear bundle isomorphism. In particular this condition cannot distinguish the tautological line on the complex projective line from its dual.

For $\zeta =z_2/z_1$ and $\eta =1/\zeta$, the tautological frames $(1,\zeta )$ and $(\eta ,1)$ differ by $1/\zeta$. Normalize their ambient Hermitian form norms; the resulting frames differ by the displayed phase. The unit circle total space is $S^3\subset {\mathbb{C}}^2$ and its projection to the complex projective line is the Hopf fibration. The transition of the dual line is the reciprocal phase.

The complex tautological line bundle is a holomorphic subbundle of the trivial bundle with fibre ${\mathbb{C}}^{n+1}$. A global holomorphic section therefore gives $n+1$ global holomorphic functions on compact connected Complex projective space. The maximum modulus principle makes them constant. The constant vector must lie in every projective line through the origin, whose intersection is zero for $n\ge 1$. For $n=0$, the base is one point and the section space is instead $\mathbb{C}$.

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 $k+1$.