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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Graduate students interested in Diophantine and Arakelov geometry. I also want to know if there are any applications of Analytic geomeyry outside Arakelov geometry.
For this one 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. From Wikipedia, the free encyclopedia.
Arakelov theory – Wikipedia
Learning Arakelov geometry Ask Question. Dual Price 1 Label: 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. Compared to the earlier books on Arakelov geometry, the current arkaelov is much more up-to-date, detailed, comprehensive, and self-contained.
What should I read before reading about Arakelov theory? Dear Vamsi, A while ago I wrote my point of view on what “you should and shouldn’t read” before studying Arakelov geometry.
Retrieved from ” https: Arakelov geometry studies a scheme X over the ring of integers Zby araielov Hermitian metrics on holomorphic vector bundles over X Cthe complex points of X. Print Price 3 Label: I would say Fulton’s book is not necessary since you anyway do intersection theory via K-theory. It is used to study Diophantine equations in higher dimensions. If not, I guess I would have to learn the scheme stuff Ariyan Javanpeykar 5, 1 22 I know almost nothing of schemes or of number theory.
Bruin’s master’s thesis written under the supervision of R. Sign up or log in Sign up using Google. 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. 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.
You should know about schemes in general, and a good deal about K-theory and intersection theory in particular Fulton’s book alone arakeoov not suffice. I have a complex analytic background Griffiths and Harris, Huybrechts, Demailley etc.
Kyoto University, Kyoto, Japan.
Mathematics > Algebraic Geometry
There are definitely situations outside Arakelov geometry where analytic torsion appears. Dual Price 2 Label: 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.
This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec Z to be a complete variety. With this in mind the analytic part of the above book should be ok to read. The arithmetic Riemann—Roch theorem is similar except that the Todd class gets multiplied by a certain power series.
Vamsi 1, 14 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. Home Questions Tags Users Unanswered. Algebraic geometry Diophantine 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. Post as a guest Name. In this context Gfometry obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. Print Price 1 Label: Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality.
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 addition, the author presents, with full details, the proof of Faltings’ Riemann—Roch theorem. This is where schemes and number theory come into play. Translations of Mathematical Monographs. Author s Product display: A while ago I wrote my point of view on what “you should and shouldn’t read” before arakeloov Arakelov geometry.