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.
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 want to learn Arakelov geomeyry atleast till the point I can “apply” computations of Bott-Chern forms and Analytic torsion to producing theorems of interest in Arakelov geometry. 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.
See our librarian page for additional eBook ordering options. Online Price 2 Label: Thanks for the answer. For this geomstry 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.
In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. Home Questions Tags Users Unanswered. I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory.
Arakelov theory – Wikipedia
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Learning Arakelov geometry Ask Question. 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. This extra Hermitian structure is applied as arakelpv substitute for the failure of the scheme Spec Z to be a complete variety.
Kyoto University, Kyoto, Japan. In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. In mathematicsArakelov theory or Arakelov geometry is an approach to Diophantine geometrynamed for Suren Arakelov. Ariyan Javanpeykar 5, 1 22 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 classical birational geometry, i. Print Price 1 Label: Taking another look at that answer, it seems that my answer is written for people with a more algebraic background.
Arakelov geometry in nLab
Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.
I have a complex analytic background Griffiths and Harris, Huybrechts, Demailley etc. Dear Vamsi, A while ago I wrote my point of view on gometry “you should and shouldn’t read” before studying Arakelov geometry. Bruin’s master’s thesis written under the supervision of R. 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.