# Vector Bundles
::: 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 and the set of locally free -modules of rank . :::
# Constructions of Vector Bundles
# Pullback Bundles
Let be a vector bundle with fibre , and let be a continous map. The pullback bundle is defined to be
which is equipped with the subspace topology and the projection map , given by projecting onto the first factor.
Let be a vector bundle over a topological space , and consider a point . The inclusion map gives rise to the induced bundle , which is the same as the trivial bundle , where is the fibre over the point .
More generally, if is the inclusion map of some subspace into , then is the restriction of from to .
Let be a line bundle over -dimensional projective space , where the homogeneous coordinates give rise to global sections . Now let be an algebraic variety, and let be a projective embedding. Then is a line bundle on , and the global sections , where , , generate the line bundle .
# Direct Sum of Bundles
Let and be two vector bundles over . The direct sum is the vector bundle over whose fibre for any is canonically isomorphic to as vector spaces.
# Tensor Product of Bundles
Let and be two vector bundles over . The tensor product is the vector bundle over whose fibre for any is canonically isomorphic to as vector spaces.
# Exterior and Symmetric Powers of Bundles
Let be a vector bundle over . The -th exterior power and the -th symmetric power are the vector bundles over whose fibres for any are canonically isomorphic to and , respectively.
# External Tensor Products of Bundles
For two vector bundles and , their external tensor product, denoted by
is the tensor product of vector bundles of the pullback bundles over the product space .
# Dual Bundles
Let be a vector bundle over . The dual bundle is the vector bundle over whose fibre over is canonically isomorphic with the dual vector space .
# Determinant Bundles
Let be a vector bundle over of rank . The determinant bundle of is the line bundle .
# Projective Bundles
Let be a vector bundle over , and let be the zero section of , that is any point is mapped to by . The projective bundle associated to , or the projectivisation of , is defined as
and it comes with a projective whose fibre over is isomorphic to the projectivised vector space .