# Learning Arakelov geometry

I have a complex analytic background (Griffiths and Harris, Huybrechts, Demailley etc). Also, I understand some PDE. I want to learn Arakelov geometry (atleast till the point I can "apply" computations of Bott-Chern forms and Analytic torsion to producing theorems of interest in Arakelov geometry). I know almost nothing of schemes or of number theory. I don't how much of these is needed to learn this stuff. I'd be grateful if any good references/suggestions are pointed out.

• You should know about schemes in general, and a good deal about K-theory and intersection theory in particular (Fulton's book alone will not suffice). I suggest you have a look at "Lectures on Arakelov Geometry" by Soulé, Abramovich etc..., and a check the references given there. Oct 18, 2011 at 15:27
• I would say Fulton's book is not necessary since you anyway do intersection theory via K-theory. Oct 19, 2011 at 0:17

Dear Vamsi,

A while ago I wrote my point of view on what "you should and shouldn't read" before studying Arakelov geometry. See

Taking another look at that answer, it seems that my answer is written for people with a more algebraic background. I think the "road to Arakelov geometry" for someone from analysis is a bit different, but I'm convinced that the following is a good way to start for everyone.

If you're more comfortable with analysis than algebraic geometry, I think a good idea would be to start with the analytic part of Arakelov geometry. This is explained very well in Chapter 1.1 of R. de Jong's thesis

http://www.math.leidenuniv.nl/~rdejong/publications/

and P. Bruin's master's thesis (written under the supervision of R. de Jong and B. Edixhoven)

http://www.math.leidenuniv.nl/~pbruin/

These two explain very well what Faltings and Arakelov did in their articles.

Since you don't want to apply the analysis to do intersection theory on an arithmetic surface, you don't have to go into this, I believe. (This is where schemes and number theory come into play.)

Now, I think after reading the relevant parts in the above references, you could start reading papers about analytic torsion (assuming you're already familiar with what this is). There's many of these, but I'm not the person to tell you which one is the best to start with.

Good luck!

• Thanks for the answer. I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. If not, I guess I would have to learn the scheme stuff.... Oct 18, 2011 at 19:41
• There are definitely situations outside Arakelov geometry where analytic torsion appears. I just don't know any of them. I only know that analytic torsion appears in Arakelov geometry when one wants to define the Quillen metric on the determinant of cohomology of a hermitian line bundle. See the bourbaki talk by Jean-Benoit Bost in the early 90's or Soule's book on Arakelov geometry. Oct 18, 2011 at 22:00

I second the suggestion of the book "Lectures on Arakelov Geometry" by Soulé, Abramovich, Burnol and Kramer. There is this nice text by Demailly which motivates the treatment of intersection theory on the infinite fibers and probably suits you with your background. With this in mind the analytic part of the above book should be ok to read.