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Let $M=\mathbb{CP}^n$ be our manifold, and $U_j$ be the standard coordinate charts, i.e. $$U_j=\{[z_0:z_1:\cdots:z_n] : z_j\neq 0\}$$ with coordinates $w^{(j)}_i=z_i/z_j$ for $i\neq j.$

Consider the section $s$ of $\mathcal{O}(1)$ which is $s=z_0$ in homogeneous coordinates. In local cordinates: over $U_1$ is equal to $w_1$.

Now consider the section $s^*$ of $\mathcal{O}(2)$ which is $s=z_0z_1$ in homogeneous coordinates. In local cordinates: over $U_1$ is equal to $w_1$.

Is that correct? The two sections $s$ and $s^*$ are equal on $U_1$?

Similarly, what is the local expression in $U_1$ for $t=z_0 z_1 z_2 \cdots z_n$ (as a section of $\mathcal{O}(n+1)$) ?

I'm a bit confused. I would honestly appreciate any help.

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In a pointwise setting, the confusion is the following: You have two vector spaces $V, W$ of the same dimension. Let $\{v_1, \cdots, v_n\}$ and $\{ w_1, \cdots, w_n\}$ be basis of $V, W$ respectively. If

$$ v = a_1 v_1+ \cdots+a_n v_n, \ \ \ w = a_1 w_1+\cdots +a_n w_n.$$

(that is, $v, w$ are represented as the same column vector $(a_1, \cdots, a_n)$ under different basis). Can we say that $v = w$? The answer is of course no. They are just elements in different vector spaces.

In your situation, you have two line bundles $O(1), O(2)$ where locally at an open set $U_1$, you have trivializing basis $e_1$ and $e_1\otimes e_1$. The local section $ e_1 $ of $O(1)$ and $e_1 \otimes e_1$ of $O(2)$ both have the same "matrix representation", which is $1$. But they are just not the same thing.

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  • $\begingroup$ I see..I still have a question though. The sections of positive line bundles can be viewed as homogeneous polynomials (in the homogeneous coordinates). Suppose I want to find the integral $\int_M |t|_h \omega^n.$ What do I use as a local expression for $t$ in terms of the $w_i$'s? In a way, all sections of $\mathcal{O}(k)$ for $k>n$ will locally look like sections of $\mathcal{O}(n)$? $\endgroup$ – Joe Aaron Feb 18 '17 at 1:13

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