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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Vamsi 1, 14 Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.
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Mathematics > Algebraic Geometry
I also want to know if there are any applications of Analytic torsion outside Arakelov geometry. 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.
soft question – Learning Arakelov geometry – MathOverflow
There are definitely situations outside Arakelov geometry where analytic torsion appears. Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. For this one defines arithmetic Chow groups CH p X of an arithmetic variety Xand defines Chern classes for Hermitian vector bundles over Aarkelov taking values in the arithmetic Chow groups.
This is explained very well in Chapter 1. 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. The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power arakeov. Online Price 3 Label: What should I read before reading about Arakelovv theory?
If you’re more comfortable with analysis than algebraic geometry, I think a good idea would be to start with wrakelov analytic part of Arakelov geometry. Taking another look at that answer, it seems that my answer is written for people with a more algebraic background.
This is where schemes and number theory come into play. 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. In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.
I have a complex analytic background Griffiths and Araielov, Huybrechts, Demailley etc. Peter Arndt 8, 3 41 This page was last gometry on 28 Mayat 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. Email Required, but never shown. Also, I understand some PDE.
I just don’t know any of them. 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. In mathematicsArakelov theory or Arakelov geometry is an approach to Diophantine geometrynamed for Suren Arakelov.
Arakelov geometry studies a scheme X over the ring of integers Zby putting Hermitian metrics on holomorphic vector bundles over X Cthe complex points of X.
Arakelov geometry in nLab
Publication Month and Year: Graduate students interested in Diophantine and Arakelov geometry.