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tautological bundle

Definition

Over ${\mathbf{CP}}^n$ , given by ${\mathcal{O}}(-1)$ sending each point $x\in {\mathbf{CP}}^n$ to the line it spans in ${\mathbf{C}}^{n+1}$ .
Set ${\mathcal{O}}(1) \coloneqq{\mathcal{O}}(-1) {}^{ \vee }$
Set ${\mathcal{O}}(n) \coloneqq{\mathcal{O}}(1)^{\otimes n}$

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examples of K theory rings

Letting $L$ be the class of the tautological bundle and $1$ the trivial bundle, $\begin{align*}{\mathsf{K}}\left(\mathbb{C} P^{n}\right) \simeq \mathbb{Z}[L] /(1-L)^{n+1}\end{align*}$
canonical bundle

tautological bundle
900_Changelog

2022-09-14 at 22h16 · tautological bundle