the section of $\mathcal O(E)$ of the exceptional divisor

Question

In Huybrechts's Complex Geometry PAGE 103 Exrcise 2.5.2:

Show that $\mathcal O(E)$ of the exceptional divisor $E=\mathbb P(\mathcal N_{Y/X})$ of a blow-up $Bl_Y(X)\rightarrow X$ of a compact manifold $X$ admits (up to scaling) only one section.

Can anyone give some advice?Thanks a lot.

Answer 1

I write the answer here in case someone feel confused about this problem.(Thanks to @ Tabes Bridges)

Notice that $\mathcal O(E)|_E=\mathcal N_{E/\hat X}$ has negative degree(When $Y$ just a point,we have $\mathcal O(E)|_E=\mathcal O(-1)$),hence has no global section.(A)

$E$ is a hypersurface, and as such is an effective divisor. We know that we can then construct a global section of $\mathcal O(E)$.While the section cutting out $E$ vanishes along $E$,hence lies in the kernel of the restriction map $\mathcal O(E)\rightarrow\mathcal O(E)|_E$. Then if we have another linearly independent section, it will restrict to a non-trivial section of the restriction,contradiction(with (A)).So,every section is equal to this section up to multiplication by a global holomorphic function on $\hat X$.

Since $X$ is compact,the map $\sigma:\hat X\rightarrow X$ is proper,we get $\hat X$ is compact.Then global holomorphic function on $\hat X$ is constant.So we're done.