I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can define a hermitian structure on $L$ by,

$$h(t) = \frac{|\psi(t)|^{2}}{\sum |\psi(s_{i})|^{2}},$$

where $t$ is a point in the fiber $L(x)$ and $\psi$ is a local trivialization of $L$ around $x$.

I'm mostly okay with this, however later in the Chapter, he moves to an example of the tautological bundle $\mathcal{O}(1)$ on $\mathbb{P}^{n}$. We consider $z_{0}, \ldots, z_{n}$ as the globally generating sections, such that the hermitian structure takes the form,

$$h(t) = \frac{|\psi(t)|^{2}}{\sum |\psi(z_{i})|^{2}}.$$

Then working locally over a standard open set $U_{0} \subseteq \mathbb{P}^{n}$, he writes the local hermitian structure as a positive, real function


where I'm guessing $w_{i}=z_{i}/z_{0}$. I see obviously how similar this is to the general expression for $h(t)$, but how does one rigorously arrive at this scalar function $h$? I think the way you're technically supposed to view $h(t)$ is as an anti-linear isomorphic from $L \to L^{*}$, so I'm not seeing the connection.

  • 1
    $\begingroup$ Define $s_k[z_0,\cdots,z_n] = z_k$. Consider $U_0$ the open subset of $\mathbb{C}P^n$ defined by $z_0 \neq 0$. Restricted to $U_0$, the section $s_0 = z_0$ is nowhere vanishing, so it defines a local holomorphic trivialization of $\mathcal{O}(1)$ restricted to $U_0$. The $h$ that Prof. Huybrechts wrote down is basically $h(s_0[z_0,\cdots,z_n])$. Is this clear enough? In local affine coordinates on $U_0$, $s_0 = 1$ and so on. $\endgroup$ – Malkoun Feb 8 '16 at 15:17

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