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Suppose we have a complex holomorphic line bundle $L$ over a real manifold $M$.

We define a hermitian structure $h$ on $L \to M$ as a hermitian product $h_x$ on each fibre $E(x)$ which depends differentiable on $x$.

Suppose that $L$ is generated by the global sections $s_1, \dots, s_n$. Then according to Huybrecht's

$$ h(t) = \frac{|\psi(t)|^2}{\Sigma | \psi(s_i) |^2}$$

where $t$ is a point in the fibre $L(x) \cong \mathbb{C}$.

Do we want to view $L(x)$ a vector space of dimension $2$ over $\mathbb{R}$?

This is the only thing that makes sense to me since out input is single point $t=a+bi$. In that case $h: U \to \mathbb{R}^*$.

Furthermore, is $\psi(s_i)$ actually $\psi(s_i(x))$?

user7090
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1 Answers1

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Recall that we're dealing with a line bundle $L$, so its fiber over a point $x$ is just a one-dimensional vector space $L_x \cong \mathbb C$. A hermitian inner product on a one-dimensional vector space is given by a positive-definite $1 \times 1$ matrix $H$ such that ${}^t \overline H = H$, that is to say, a positive real number.

We now have our line bundle that we assume to be generated by global sections $s_1, \ldots, s_N$. We'd like to use them to define a hermitian metric on $L$, so we have to somehow define a positive real number $h_x$ for every point $x$.

Take a local trivialization $U$ of $L$, so that there is a bundle isomorphism $\psi : L|_U \cong U \times \mathbb C$. Since the sections $s_j$ globally generate $L$, at least one of them is nonzero at any point. If $\sigma$ is a local section of $L$, we can define a hermitian metric on $L$ by assigning to it a real number $|\sigma(x)|^2$ at any point $x$. We do this by setting $$ |\sigma(x)|^2 = \frac{|\psi(\sigma(x))|^2}{\sum_{j=1}^{N} |\psi(s_j(x))|^2 }. $$ This looks a bit weird and unmotivated, but one can check that this definition respects changes of local trivializations and is thus a global one.

To make this look less weird and unmotivated, we can write $|L| := H^0(X,L)$ for the space of global sections of $L$. We then get a holomorphic map from $X$ to a Grassmannian of hyperplanes $$ \phi : X \to \operatorname{Gr}(H^0(X,L), \dim H^0(X,L) - 1), \quad x \mapsto \{ \sigma \in H^0(X,L) \mid \sigma(x) = 0 \}. $$ This map is well-defined because $L$ is generated by global sections, so there is always a section that is not zero at a given point $x$.

A Grassmannian of hyperplanes is just a projective space of lines in the dual space, so this is a map to some projective space $\mathbb P(V)$. A projective space has the tautological line bundle $\mathcal O(1)$ along with a hermitian metric $h$ (whose curvature form is the Fubini-Study metric), and by construction we have $L = \phi^* \mathcal O(1)$. The metric we define on $L$ is thus the pullback metric $\phi^* h$.

Once you unpack all the abstractness above, you'll find that this is how one arrives at the original definition. Once you start questioning if we really need $L$ to be generated by global sections for us to get a metric and curvature form this way, you'll invent singular hermitian metrics on line bundles.

Gunnar Þór Magnússon
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