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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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. Online Price 1 Label: 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.

In mathematicsArakelov theory or Arakelov geometry is an approach to Diophantine geometrynamed for Suren Arakelov. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.

## Mathematics > Algebraic Geometry

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. Vamsi 1, 14 Author s Product display: Translations of Mathematical Monographs.

Arakelov Geometry Share this page. 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.

Taking another look at that answer, it seems that my answer is written for people with srakelov more algebraic background.

## Arakelov Geometry

With this in mind the analytic part of the above book should be ok to read. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec Z to be a complete variety.

Email Required, but never shown. By using this site, you agree to the Terms of Use and Privacy Policy. Bruin’s master’s thesis written under the supervision of R. This is a timely monograph that should appeal to researchers in this important area of mathematics.

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The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power series. I just don’t know any of them. From Wikipedia, the free encyclopedia. This is where schemes and number theory come into play. By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an excellent textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry.

### soft question – Learning Arakelov geometry – MathOverflow

The book includes such fundamental results as arithmetic Hilbert—Samuel formula, arithmetic Nakai—Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang—Bogomolov conjecture and so on. Join our email list. If not, I guess I would have to learn the scheme stuff In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. 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.

### Arakelov geometry in nLab

Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. MathOverflow works best with JavaScript enabled. Dual Price 2 Label: Print Price 1 Label: The arithmetic Riemann—Roch theorem then describes how the Chern class behaves under pushforward of vector bundles araielov a proper map of arithmetic varieties.