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What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?

I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function $\varphi$. Moreover the metric $h=e^{-\varphi}$ has non-negative curvature iff $\varphi$ is subharmonic.

Somebody can me explain this.

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1 Answer 1

The trivial line bundle $\newcommand{\bC}{\mathbb{C}}$ $\underline{\bC}_M:=\bC\times M\to M$ over a complex manifold $M$ admits a trivial metric $h_0$. This is defined by requiring that the trivial section $u_0$

$$ M\ni p\mapsto u_0(p)=1\in\bC$$

has pointwise length $1$. If $h$ is another metric on $\underline{\bC}_M$, then $h= w^2\cdot h_0$ where $w$ is the positive function $w(p)=|u_0(p)|_h$, $p\in M$. The function $w$ thus can be expressed as an exponential $w= e^{-\varphi/2}$,

$$\varphi=\log |u_0|_h^2. $$

As shown in many books on complex differential geometry (e.g. Griffiths and Harris) the curvature of this line bundle is

$$ \bar{\partial}\partial \log |u_0|_h^2= - \bar{\partial}\partial \varphi. $$

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