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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In mathematicsArakelov theory or Arakelov geometry is an approach to Diophantine geometrynamed for Suren Arakelov. Join our email list. I just don’t know any of them. Dual Price 1 Label: Ordering on the AMS Bookstore is limited to individuals for personal use only.
Post as a guest Name. 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.
Thanks for the answer. A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality.
Bruin’s master’s thesis written under the supervision of R. Dear Vamsi, A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. I don’t how much of these is needed to learn this stuff.
For this one defines arithmetic Chow groups CH p X geomerry an arithmetic variety Xand defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups.
Also, I understand some PDE. Print Price 3 Label: I have a complex analytic background Griffiths and Harris, Huybrechts, Demailley etc.
In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. What should I read before reading about Arakelov theory?
Print Price 2 Label: See What should I read before reading about Arakelov theory? Dual Price 2 Label: With this in mind the analytic part of the above book should be ok to read.
Publication Month and Year: 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. Online Price 3 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.
If not, I guess I would have to learn the scheme stuff 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. 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.
There’s many of these, but I’m not the person to tell you which one is the best to geomegry with. Compared to the earlier raakelov on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. Print Price 1 Label: Retrieved from ” https: Sign up using Email and Password. You should know about schemes in general, and a good deal about K-theory and intersection theory in arakelof Fulton’s book alone will not suffice.
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. Peter Arndt 8, 3 41 Home Questions Tags Users Unanswered. This page was last edited on 28 Mayat Sign up or log in Sign up using Google. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec Z to be a complete variety.