I'm doing some independent reading in differential geometry, and the following is my attempt to work out the details of the construction of the tautological bundle on $\mathbb CP^2$ and the induced metric. I'm posting this as a sort of sanity check. Any corrections or nitpicks are very welcome.
Let me first consider $CP^2$ with the standard charts, just to fix notation. That is , $U_2=\{[x:y:z] : z\neq 0\}$ with the map $\phi_0([x:y:z])=(x/z, y/z)$ and similarly for $U_0$ and $U_1$. The transition map from $U_2$ to $U_1$, for example, is $(x,y) \rightarrow [x:y:1]\rightarrow (x/y, 1/y)$.
We let $E\subset \mathbb CP^2 \times \mathbb C^3$ be the set of points $([s], ws)$ with $s$ nonzero and $w\in \mathbb C$. I claim this is a line bundle. It suffices to give trivializations and transition functions. Over $U_2$, we give $([x:y:z], w(x,y,z))\rightarrow (x/y, y/z, w)$ and similarly for the other $U_i$.
Let us compute the transition function from part of the bundle over $U_2$ to the part over $U_1$.
$$(x,y, t)\rightarrow ([x:y:1], t(x,y,1))=([x/y:1:1/y], ty(x/y,1,1/y))\rightarrow (x/y, 1/y, ty)$$ We get similar transitions for the other two intersections.
If $([z], tz)$ and $([z], sz)$ are two elements of the tangent space over $[z]$, we can introduce the metric $\langle tz,sz\rangle $, where the brackets indicate the standard Hermitian metric on $\mathbb C^3$.
In coordinates over $U_2$, we look at the points $(x,y,t)$ and $(x,y,s)$ and define the inner product here to be $\langle t(x,y,1), s(x,y,1)\rangle$.
Is there anything else I should be checking here?