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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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    $\begingroup$ Have you seen: mathoverflow.net/questions/54603/… ? $\endgroup$ – jmc Jun 6 '13 at 8:15
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    $\begingroup$ 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… $\endgroup$ – jmc Jun 6 '13 at 8:21
  • $\begingroup$ You could have a look at C. Soulé book "Lectures on Arakelov Geometry" (Cambridge Univ. Press). $\endgroup$ – Damian Rössler Jun 7 '13 at 14:06
  • $\begingroup$ Damian Rössler: But that book already supposes that I have the neccessary background in analysis from what I have seen. $\endgroup$ – Dedalus Jun 7 '13 at 15:59
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    $\begingroup$ 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). $\endgroup$ – Damian Rössler Jun 7 '13 at 18:37
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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!

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  • $\begingroup$ Just as comment, Arakelov geometry is wast now. A person must know enough current geometry. For example we can introduce hermitian verctor bundle on arithmetric varieties and hence we can define Mumford setability and Hitchin Kobayashi correspondence . or Arithmitical version of Yau Tian Donaldson conjecture by using Gillet Soule Arithmetric intersection theory, ..ect $\endgroup$ – user21574 May 21 '17 at 4:14
  • $\begingroup$ @HassanJolany: I do not follow "is wasted now". Can you explain further? Here I meant research in Arakelov theory, not applying Arakelov theory techniques to other fields in arithemetic geometry. $\endgroup$ – Bombyx mori May 21 '17 at 4:22
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    $\begingroup$ Well, I am beginner also, As it recently proved by Tian CM stability is equivalent with K-stability for existence of Kahler-Einstein metric. So in Arakelov point of view by using Gillet Soule intersection theory, K-stability is noting just generalized version of Faltings hight. See mathoverflow.net/questions/231058/… $\endgroup$ – user21574 May 21 '17 at 4:27
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    $\begingroup$ For Mumford Stability point of view on reflexive sheaf or vector bundle, We can extend it on arithmetric variety by defining the degree of a vector bundle on arithmetric variety using Arakelov theory. and extend Hitchin Kobayashi correspondence or finding canonical meric on Arithmetric hermitian vector bundle $\endgroup$ – user21574 May 21 '17 at 4:30
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    $\begingroup$ Well, even in index theory the heat kernel and the K-theory approach are hardly united. In general, it is rather difficult to extract interesting information using soft-analysis tools even if the underlying space is well-behaved. So I am very impressed by people who did these things. Thanks for the comment, though. $\endgroup$ – Bombyx mori May 21 '17 at 4:43

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