Let X$X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on X$X$. In order to define the height of an integral closed subset Y$Y$ we define it on closed points to be $h_{\bar{L}}(Y) = log \#(k(Y))$.$$h_{\bar{L}}(Y) = \log \#\bigl(k(Y)\bigr).$$
If $dim Y > 0$$\dim Y > 0$ we define the height recursively. Let s$s$ be a nontrivial rational section of L$L$ over $Y$. If $div_Y(s)=\sum_\alpha n_\alpha Y_\alpha$,$$\text{div}_Y(s)=\sum_\alpha n_\alpha Y_\alpha,$$ then $h_{\bar{L}}(Y) = \sum_\alpha n_\alpha h_{\bar{L}}(Y_\alpha) - \int_{Y(\Bbb{C})}log||s||c_1(\bar{L})^{dim Y(\Bbb{C})}$.$$h_{\bar{L}}(Y) = \sum_\alpha n_\alpha h_{\bar{L}}(Y_\alpha) - \int_{Y(\Bbb{C})}\log\|s\|c_1(\bar{L})^{\dim Y(\Bbb{C})}.$$
We notice that at the infinite fibre the order of vanishing $-log||s||$$-\log\|s\|$ occurs just as in the finite places. Continuing the analogy $c_1(\bar{L})^{dim Y(\Bbb{C})}$$c_1(\bar{L})^{\dim Y(\Bbb{C})}$ should be analogous to the height of the infinite fibre.
What is the philosophical reason that $c_1(\bar{L})^{dim Y(\Bbb{C})}$$c_1(\bar{L})^{\dim Y(\Bbb{C})}$ shows up as the height of the infinite fibre? In particular, when Y$Y$ is a curve: Does $c_1(\bar{L})$ in some sense describe the degree of closed points lying over infinity?
