# Vector Bundles

# Properties

::: proposition Proposition Associating to a holomorphic vector bundle its sheaf of sections defines a canonical bijection between the set of holomorphic vector bundles of rank rr and the set of locally free OX\mc{O}_{X}-modules of rank rr. :::

# Constructions of Vector Bundles

# Pullback Bundles

Let p:LYp : \mc{L} \rightarrow Y be a vector bundle with fibre FF, and let f:XYf : X \rightarrow Y be a continous map. The pullback bundle is defined to be

fL:={(x,l)(X,L):f(x)=p(l)}X×L,f^{\ast}\mc{L} := \{ (x,l) \in (X,\mc{L}) : f(x) = p(l) \} \subseteq X \times \mc{L},

which is equipped with the subspace topology and the projection map π:fLX\pi : f^{\ast}\mc{L} \rightarrow X, given by projecting onto the first factor.

Example

Let LXL \rightarrow X be a vector bundle over a topological space XX, and consider a point xXx \in X. The inclusion map i:{x}Xi: \{x\} ↪ X gives rise to the induced bundle iL{x}i^{\ast}L \rightarrow \{x\}, which is the same as the trivial bundle {x}×Lx\{x\} \times L_{x}, where LxiLL_{x} \cong i^{\ast}L is the fibre over the point xx.

More generally, if i:YXi: Y \hookrightarrow X is the inclusion map of some subspace YY into XX, then LY:=iLL|_{Y} := i^{\ast}L is the restriction of LL from XX to YY.

Example

Let O(1)Pn\mc{O}(1) \rightarrow \PP^{n} be a line bundle over nn-dimensional projective space Pn\PP^{n}, where the homogeneous coordinates x0,,xnx_{0},\ldots, x_{n} give rise to global sections x0,,xnH0(Pn,O(1))x_{0},\ldots, x_{n} \in H^{0}(\PP^{n}, \mc{O}(1)). Now let XX be an algebraic variety, and let φ:XPn\varphi : X \rightarrow \PP^{n} be a projective embedding. Then L=φO(1)\mc{L} = \varphi^{\ast}\mc{O}(1) is a line bundle on XX, and the global sections s0,,sns_{0},\ldots, s_{n}, where si=φxis_{i} = \varphi^{\ast}x_{i}, siH0(X,L)s_{i} \in H^{0}(X,\mc{L}), generate the line bundle L\mc{L}.

# Direct Sum of Bundles

Let L\mc{L} and M\mc{M} be two vector bundles over XX. The direct sum LM\mc{L} \oplus \mc{M} is the vector bundle over XX whose fibre (LM)x(\mc{L} \oplus \mc{M})_{x} for any xXx \in X is canonically isomorphic to LxMx\mc{L}_{x} \oplus \mc{M}_{x} as vector spaces.

# Tensor Product of Bundles

Let L\mc{L} and M\mc{M} be two vector bundles over XX. The tensor product LM\mc{L} \otimes \mc{M} is the vector bundle over XX whose fibre (LM)x(\mc{L} \otimes \mc{M})_{x} for any xXx \in X is canonically isomorphic to LxMx\mc{L}_{x} \otimes \mc{M}_{x} as vector spaces.

# Exterior and Symmetric Powers of Bundles

Let L\mc{L} be a vector bundle over XX. The ii-th exterior power iL⋀^{i}\mc{L} and the ii-th symmetric power SiLS^{i}\mc{L} are the vector bundles over XX whose fibres for any xXx \in X are canonically isomorphic to iLx⋀^{i}\mc{L}_{x} and SiLxS^{i}\mc{L}_{x}, respectively.

# External Tensor Products of Bundles

For two vector bundles p1:LXp_{1}: \mc{L} \rightarrow X and p2:MYp_{2}: \mc{M} \rightarrow Y, their external tensor product, denoted by

LMX×Y,\mc{L} \boxtimes \mc{M} \rightarrow X \times Y,

is the tensor product of vector bundles of the pullback bundles p1LXp2MX×Yp_{1}^{\ast}\mc{L} \otimes_{X} p_{2}^{\ast}\mc{M} \rightarrow X \times Y over the product space X×YX \times Y.

# Dual Bundles

Let L\mc{L} be a vector bundle over XX. The dual bundle L\mc{L}^{\ast} is the vector bundle over XX whose fibre (L)x(\mc{L}^{\ast})_{x} over xXx \in X is canonically isomorphic with the dual vector space (Lx)(\mc{L}_{x})^{\ast}.

# Determinant Bundles

Let L\mc{L} be a vector bundle over XX of rank rr. The determinant bundle of L\mc{L} is the line bundle det(L):=rL\det(\mc{L}) := ⋀^{r}\mc{L}.

# Projective Bundles

Let L\mc{L} be a vector bundle over XX, and let s:XLs : X \rightarrow \mc{L} be the zero section of L\mc{L}, that is any point xXx \in X is mapped to 0Lx0 \in \mc{L}_{x} by ss. The projective bundle associated to L\mc{L}, or the projectivisation of L\mc{L}, is defined as

P(L):=(Ls(X))/C,\PP(\mc{L}) := (\mc{L} - s(X))/\mathbb{C}^{\ast},

and it comes with a projective π:P(L)X\pi: \PP(\mc{L}) \rightarrow X whose fibre over xXx \in X is isomorphic to the projectivised vector space P(Lx)\PP(\mc{L}_{x}).