nLab
hyperplane line bundle

Let n= k n\mathbb{P}^n = \mathbb{P}^n_k be the nn-dimensional projective space over a field kk, whose points are the equivalence classes [z 0,,z n][z_0,\ldots,z_n] of (n+1)(n+1)-tuples (z 0,,z n)k n+1\{0}(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 kk, have U i={[z 0,,z n]z i0}U_i = \{[z_0,\ldots,z_n] | z_i\neq 0\}.

The hyperplane line bundle 𝒪(1)\mathcal{O}(1) on n\mathbb{P}^n is a line bundle given by the transition functions g ij([z 0,,z n])=z j/z ig_{ij}([z_0,\ldots,z_n])=z_j/z_i on U iU jU_i\cap U_j. Its dual bundle 𝒪(1) *\mathcal{O}(1)^* is the tautological bundle (or universal bundle) usually denoted by 𝒪(1)\mathcal{O}(-1) and the tensor powers are 𝒪(n)=𝒪(1) n\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}, 𝒪(n)=𝒪(1) n\mathcal{O}(-n)=\mathcal{O}(-1)^{\otimes n} for n0n\geq 0. The total space of the tautological line bundle can be identified with n+1\mathbb{C}^{n+1} and the projection is exactly (z 0,,z m)[z 0,,z m](z_0,\ldots,z_m)\mapsto[z_0,\ldots,z_m], i.e. the fiber over [z 0,,z m][z_0,\ldots,z_m] is the line {(λz 0,,λz n),0λ}\{(\lambda z_0,\ldots,\lambda z_n), 0 \neq \lambda\in \mathbb{C}\}. The canonical line bundle K=Λ nT * nK = \Lambda^n T^* \mathbb{P}^n equals 𝒪(n1)\mathcal{O}(-n-1).

The bundles 𝒪(n)\mathcal{O}(n) are holomorphic if k=k=\mathbb{C}. The sheaves of (regular or holomorphic) sections are also denoted as 𝒪(n)\mathcal{O}(n) and are said to be the twists of the structure sheaf 𝒪\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)