# Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\mathcal L$ be an arithmetic line bundle over $\mathcal X$ and $h$ be a positive Hermitian metric on $\mathcal L$,

Then

1- How can we compute the intersection number

$$(\overline{\mathcal L}^h)^n.(n+1)(\overline{K_{\mathcal X/C}}^{Ric(h)})$$

2.What is the definition of intersection number of two arithmetic line bundles

$$\overline{\mathcal L_1}.\overline{\mathcal L_2}$$

3.If we take $\mathcal D$ be an arithmetic divisor on $\mathcal X$, then how can we introduce $$\overline{\mathcal L_1}.\mathcal D$$

• Have you tried to read Arakelov theory by Soulé, Abramovich, Burnol, Kramer? It's not that thick, and should give you the main ideas. – Sebastian Goette Feb 5 '16 at 15:10
• yahh, I have read them, But the fact is part 1 of my question is related to Mabuchi energy functional, and, and I am woundring to see it, – user21574 Feb 5 '16 at 15:14