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?
Best Answer
I think the amount of analysis you need to know is fairly modest. If you know distribution theory and some introductory material on pseudo-differential operators (say first half of Shubin), you should be right to go. In particular there is no "hard analysis" being involved in the literature I went through. While "soft analysis" is by no means easy, it has an algebraic flavor somewhat similar to algebraic geometry.
If you are reading papers, it might make more sense to learn material (like Sobolev embedding) and prove things yourself on the spot than reading through big books to have a decent understanding of the subject. For index theory: I think Faltings-Zhang's book has a nice section on the application of local index theorem to Arakelov theory. Since you asked the question as an undergraduate I imagine you might be very familiar with the background already. Good luck!