hyperplane line bundle

Let $\mathbb{P}^n = \mathbb{P}^n_k$ be the $n$-dimensional projective space over a field $k$, whose points are the equivalence classes $[z_0,\ldots,z_n]$ of $(n+1)$-tuples $(z_0,\ldots,z_n)\in k^{n+1}\backslash \{0\}$; the (co)domains of usual open charts in the sense of manifolds, which are Zariski-open subsets for general $k$, have $U_i = \{[z_0,\ldots,z_n] | z_i\neq 0\}$.

The **hyperplane line bundle** $\mathcal{O}(1)$ on $\mathbb{P}^n$ is a line bundle given by the transition functions $g_{ij}([z_0,\ldots,z_n])=z_j/z_i$ on $U_i\cap U_j$. Its dual bundle $\mathcal{O}(1)^*$ is the **tautological bundle** (or universal bundle) usually denoted by $\mathcal{O}(-1)$ and the tensor powers are $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}$, $\mathcal{O}(-n)=\mathcal{O}(-1)^{\otimes n}$ for $n\geq 0$. The total space of the tautological line bundle can be identified with $\mathbb{C}^{n+1}$ and the projection is exactly $(z_0,\ldots,z_m)\mapsto[z_0,\ldots,z_m]$, i.e. the fiber over $[z_0,\ldots,z_m]$ is the line $\{(\lambda z_0,\ldots,\lambda z_n), 0 \neq \lambda\in \mathbb{C}\}$. The **canonical line bundle** $K = \Lambda^n T^* \mathbb{P}^n$ equals $\mathcal{O}(-n-1)$.

The bundles $\mathcal{O}(n)$ are holomorphic if $k=\mathbb{C}$. The sheaves of (regular or holomorphic) sections are also denoted as $\mathcal{O}(n)$ and are said to be the *twists* of the structure sheaf $\mathcal{O}$; they restrict to the equally denoted sheaves on any projective subvariety and these restrictions up to an isomorphism do not depend on a particular embedding into a particular projective space.

Revised on August 8, 2012 18:17:06
by Andrew Stacey
(192.76.7.219)