Euler characteristic of a metrized line bundle

Question

During my readings on the web, I found the Euler characteristic of a metrized line bundle, let $L$ a line bundle on a variety $X \to \operatorname{spec}(K)$. And let $\varphi$ a metric on $L$.

Someone has a reference to define this object: $$ \chi (L,\varphi) $$

I can't find anymore where I saw it. And if this object really exists, what's the link with Euler's characteristic $ \chi (L)$ ?

Thanks !

Answer 1

Check for example Lang, Introduction to Arakelov Geometry.