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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Online Price 3 Label: From Wikipedia, the free encyclopedia. In this context Bost arakelvo an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. Thanks for the answer. There’s many of these, but I’m not the person to tell you which one is the best to start with. Online Price 2 Label: Translations of Mathematical Monographs.
In addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. Sign up using Email and Password. I just don’t know any of them. I want to learn Arakelov geometry atleast till the point I can “apply” computations of Bott-Chern gemoetry 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.
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.
The arithmetic Riemann—Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. Libraries and resellers, please contact cust-serv ams. I also want to know if there are any applications of Analytic torsion outside Arakelov geometry.
Now, I think after reading the relevant parts in the above references, you could start reading papers about arakelvo torsion assuming you’re already familiar with what this is. Sign up using Facebook. 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. Graduate students interested in Diophantine and Arakelov geometry. 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.
Author s Product display: 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.
This extra Hermitian structure geomehry applied as a substitute for the failure of the scheme Spec Z to be a complete variety. Arakelov geometry studies a scheme X over geomtery ring of integers Zby putting Hermitian metrics on holomorphic vector bundles over X Cthe complex points of X.
I know almost nothing of schemes or of number theory.
Publication Month and Year: If not, I guess I would have to learn the scheme stuff Email Required, but never shown. Many geomtery results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality.
Algebraic geometry Diophantine geometry. Also, I gdometry some PDE. Ariyan Javanpeykar 5, 1 22 Views Read Edit View history. I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory.
Learning Arakelov geometry Ask Question. This is where schemes and number theory come into play. What should I read before reading about Arakelov theory?
After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of geomefry surfaces and higher-dimensional varieties.
A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry. Bruin’s master’s thesis written under the supervision of R.