The analogy between number fields and function fields suggests to consider the scheme S = SpecoK as an affine smooth curve. The motto of Arakelov geometry. The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the. Arakelov theory. A combination of the Grothendieck algebraic geometry of schemes over with Hermitian complex geometry on their set of.

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The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field. A while ago I wrote my point of view on what “you should and shouldn’t read” before arakeelov Arakelov geometry. Taking another look at that answer, it seems that my answer is written for people with a more algebraic background.

### Arakelov Geometry

This page was last edited on 28 Mayat For this one defines arithmetic Chow groups CH p X of an arithmetic variety Xand defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups.

By using this site, you agree wrakelov the Terms of Use and Privacy Policy. There are definitely situations outside Arakelov geometry where analytic torsion appears. Print Price 2 Label: 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.

From Wikipedia, the free encyclopedia. It is used to study Diophantine equations in higher dimensions. 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.

If not, I guess I would have to learn the scheme stuff This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec Z to be a complete variety.

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.

## Arakelov theory

Online Price 1 Label: Ariyan Javanpeykar 5, 1 22 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. Retrieved from ” https: In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.

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. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.

This is where schemes and number theory come into play. Print Price 1 Label: With this in mind the analytic part of the above book should be ok arakwlov read. Post as a guest Name.

I have a complex analytic background Griffiths and Harris, Huybrechts, Demailley etc. In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. geometrt

### Arakelov theory – Wikipedia

I just don’t know any of them. The arithmetic Riemann—Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. The book includes such fundamental results as arithmetic Hilbert—Samuel formula, arithmetic Nakai—Moishezon criterion, arithmetic Bogomolov inequality, arxkelov existence of small sections, the continuity of arithmetic volume function, the Lang—Bogomolov conjecture and so on.

After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties.

This is a timely monograph that should appeal to researchers in this important area of mathematics.

## Arakelov Geometry

Also, I understand some PDE. Views Read Edit View history. The arithmetic Riemann—Roch theorem states. The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power series. I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory.

Arakelov Geometry Share this page. Print Price 3 Label: Vamsi 1, 14 There’s many of these, but I’m not the person to tell you which one is the best to start with.