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Contents
Vector Bundles on P1
February 2, 2023
What are all the vector bundles on \(\PP ^1\)? There are several related questions of this sort. One can ask for a complete classification of vector bundles, or one can ask for something easier such as the \(K\)-theory. As for what one can mean by vector bundles, they can be topological or analytic/algebraic. One would also like to know something about the structure on vector bundles: for example what does the tensor product do to isomorphism classes?
\(\PP ^1\) is the easiest nontrivial example of a space for which you can ask these questions, for which there is a nice answer.
Aside from the trivial bundle, the basic example of a vector bundle on \(\PP ^1\) is \(\cO (-1)\), called the tautological bundle. If \(\PP ^1\) is the space of lines in a \(2\)-dimensional vector space, the fibre at a point for \(\cO (-1)\) is (tautologically) that line. The notation is meant to tell you that it is the dual of a bundle called \(\cO (1)\), called the hyperplane bundle, where the global (algebraic) sections correspond to hyperplanes (i.e points) in \(\PP ^1\). More generally \(\cO (n) = \cO (1)^{\otimes n}\).