Distinction or relation between tautological bundle and cotangent bundle

I want to know the distinction or relation between tautological bundle and cotangent bundle, i.e. I want to distinguish $\mathcal O(-n)$ over $\mathbb {CP}^{n-1}$ (tautological bundle of tensor weight $n$ over $\mathbb {CP}^{n-1}$) and $T^* \mathbb {CP}^{n-1}$ (cotangent bundle over $\mathbb {CP}^{n-1}$).

Another question (might be wrong, just from curiosity): is $\mathcal O(-1) \oplus \mathcal O(-1)$ over $\mathbb {CP}^1$ is equivalent to $T^* \mathbb {CP}^1$ ?

In the mateirals vakil's note 24 page 4, There is a statement \begin{align} \Omega^1_{P^1} \simeq \mathcal O(-2)_{P^1} \end{align} I think in terms of my notation

\begin{align} T^* CP^1 \simeq \mathcal O(-2) \quad \textrm{over } \quad CP^1 \end{align}

Is it valid approach?

• See en.m.wikipedia.org/wiki/Euler_sequence for a relation – Hoot Oct 29 '15 at 19:40
• @Hoot, i think what you mean from wiki, from the section "The canonical line bundle of projective spaces", is $\mathcal O(-(n+1))$ over $CP^n$ is $w_{CP^n}$ which is $T^* CP^n$, am i right? – phy_math Oct 30 '15 at 6:01
• @Hoot, i edit my question, from the vakil's note on algebraic geometry, there is similar statement. But i still don't know what is the difference between tautological bundle and cotangent bundle without considering dimension. – phy_math Oct 30 '15 at 6:04
• (1) $\omega_{\mathbb P^n}$ is the determinant of the cotangent bundle; if $n > 1$ then that's not the same (2) the dimension is property #1 of a locally free sheaf so I don't think it's such a tiny difference. I think the Euler sequence gives a nice geometric picture, and the Wiki article does talk about this. – Hoot Oct 30 '15 at 20:23

The dimension does not match in general. $\mathcal O(k)$ is by definition a line bundle for all $k\in \mathbb Z$, while $T^*\mathbb{CP}^{n-1}$ is the cotangent bundle, so it's a rank-$n-1$ vector bundle (the fibers are $n-1$ dimensional vector space).
Again, in the $\mathbb{CP}^1$ case, $\mathcal O(-1) \oplus \mathcal O(-1)$ is a rank two vector bundle, so it cannot be $T^*\mathbb{CP}^1$ as it's one dimensional.