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This may be more suitable as a long comment. I remember someone asking Soulé current open problems in Arakelov theory during a walk at the summer school (2017) in Grenoble. The adelic intersection the …

Here is a rather low-brow way of tracing through Arakelov's original ideas. Recall that the intersection of two ordinary divisors $D,E$ can be written as $$ (D.E)_{v}=\sum^{r}_{i=1}-\log \lVert (f| …

If I am not mistaken, you can find such an example in the following lecture notes by Akshay: http://virtualmath1.stanford.edu/~conrad/mordellsem/Notes/L01.pdf (to be precise, see the bottom part of pa …

There is a definition of $h^{1}(\overline{D})$ by many people, and a definition of $H^{1}(\overline{D})$ by Alexrander Borisov using the notion of ghost spaces of the second kind. But neither is the s …

I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says: If $D$ is a divisor on $X$, we would like to define a hermiti …

Here is another low brow way to look at it by tracing Arakelov's ideas in his paper. Let $X$ be a curve over $K$, where $K$ is a number field. Let $\infty$ denoting the archimedean valuations of $K$ …

A very short answer I learned from summer school - because they want to construct local-global correspondence. A Neron function correspond exactly to the local part and Riemann-Roch is a local-global …

Recall that Serre duality says that $H^{0}(X,F)\cong H^{1}(X, K\otimes F^{*})^{*}$. So the determinant of cohomology just follows from this. However we do need slightly more machinery than Serre duali …

I do not know if this is really the end of the story. You may be interested in the following paper by Jay Jorgenson. Basically he extended the work by Ray-Singer by fixing all unknown invariants, bu …

I apologize for answering late. I think the 1D case has been discussed multiple times in the forum already. The high dimensional case you suggested was first defined by Bost. See page 63 in below: Thé …

Let us consider the arithemetic surface case, which is already very difficult (see the recent work by Gerald Montplet, for example). In this case Faltings-Riemann-Roch established that $$ \chi(O_{X}(D …