# What analysis should I know for studying Arakelov Theory?

Hi!

I have a fairly good background in Algebraic Geometry (say at the level of Hartshorne's book and some Intersection Theory from Fulton) and since I think working over $\text{Spec } \mathbb{Z}$ is fun, I would like to learn some Arakelov Theory.

My background in differential geometry and analysis is not that good, though - I know basic definitions in both fields and have taken some courses, but I have forgot a lot, and what more, I seem to need complex differential geometry, which I have never studied. From what I understand, residuce currents is important in Arakelov Theory.

So my question is:

What books, what articles should I read to get a good analytical / complex differential geometric background (covering for example, residue currents) sufficient to study Arakelov Theory?

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Have you seen: mathoverflow.net/questions/54603/… ? –  jmc Jun 6 '13 at 8:15
By the way, that answer does not really answer your question. But it does tell how to get started with Arakelov theory, so that is why I shared the link… –  jmc Jun 6 '13 at 8:21
You could have a look at C. Soulé book "Lectures on Arakelov Geometry" (Cambridge Univ. Press). –  Damian Rössler Jun 7 '13 at 14:06
Damian Rössler: But that book already supposes that I have the neccessary background in analysis from what I have seen. –  Dedalus Jun 7 '13 at 15:59
Global Arakelov theory requires a lot of index theory. Some of that material is covered in the book Heat kernels and Dirac operators, by N. Berline, E. Getzler and M. Vergne, Grundlehren Math. Wiss., vol. 298, Springer-Verlag, New York, 1992. The difficult part of local index theory, which is needed in the proof of the arithmetic Riemann-Roch theorem is only described in the articles of Bismmut and his coworkers (the most self-contained one is his article with Lebeau). –  Damian Rössler Jun 7 '13 at 18:37