4
$\begingroup$

Let $X$ be a compact complex manifold. According to Fulton and Lazarsfeld, a vector bundle $E$ on $X$ is called ample if the Serre line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ on the projectivized bundle $\mathbb{P}(E)$ is ample.

This notion should be a generalization of ampleness of line bundles, but I don't quite understand. Assume that $E$ is a line bundle, then its projectivization is isomorphic to $X$. In this case how can we conclude that $\mathcal{O}_{\mathbb{P}(E)}(1)\cong E$?

  • 1
    Isn't this just the relative form of the tautological isomorphism $\mathscr{O}_{\operatorname{Proj} S}(1) \cong S_1$?2012-09-09
  • 1
    @ZhenLin, you have to take the global section on the LHS (if $S_1$ denotes elements of degree $1$ of $S$), and this holds for certain $S$ (e.g. polynomial rings).2012-09-09

1 Answers 1