scholarly journals Analytic Curves in Algebraic Varieties over Number Fields

2009 ◽  
pp. 69-124 ◽  
Author(s):  
Jean-Benoît Bost ◽  
Antoine Chambert-Loir
Keyword(s):  
Number Fields ◽  
Algebraic Varieties ◽  
Analytic Curves

scholarly journals Algebraic varieties are homeomorphic to varieties defined over number fields

2020 ◽  
Vol 95 (2) ◽  
pp. 339-359
Author(s):  
Adam Parusiński ◽  
Guillaume Rond
Keyword(s):  
Number Fields ◽  
Algebraic Varieties

Higher Order Tangents to Analytic Varieties along Curves. II

2008 ◽  
Vol 60 (1) ◽  
pp. 33-63 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Finite Number ◽  
Algebraic Variety ◽  
Higher Order ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Open Set ◽  
Local Case ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Curves

AbstractLet V be an analytic variety in some open set in ℂn. For a real analytic curve γ with γ(0) = 0 and d ≥ 1, define Vt = t−d(V − γ(t)). It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support Tγ,δV is an algebraic variety as t tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the Vt. As a corollary, it is shown that Tγ,δV is either inhomogeneous or coincides with Tγ,δV for all δ in some neighborhood of d. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragmén-Lindelöf conditions. Corresponding results for limit varieties Tσ,δW of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

scholarly journals Some remarks concerning the Grothendieck period conjecture

2016 ◽  
Vol 2016 (714) ◽  
Author(s):  
Jean-Benoît Bost ◽  
François Charles
Keyword(s):  
Number Fields ◽  
Algebraic Varieties ◽  
De Rham ◽  
New Evidence

AbstractWe discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham–Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich–Zagier concerning transcendence properties of the torsors of periods of varieties over number fields.LetWe notably establish that

scholarly journals Periodicity of hermitianK-groups

2011 ◽  
Vol 7 (3) ◽  
pp. 429-493 ◽  
Author(s):  
A. J. Berrick ◽  
M. Karoubi ◽  
P. A. Østvær
Keyword(s):  
Upper Bound ◽  
Number Fields ◽  
Continuous Functions ◽  
Local Fields ◽  
Symplectic Groups ◽  
Group Rings ◽  
Algebraic Varieties ◽  
Compact Spaces ◽  
Smooth Complex ◽  
Rings Of Integers

AbstractBott periodicity for the unitary and symplectic groups is fundamental to topologicalK-theory. Analogous to unitary topologicalK-theory, for algebraicK-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraicK-groups for any ring implies periodicity for the hermitianK-groups, analogous to orthogonal and symplectic topologicalK-theory.The proofs use in an essential way higherKSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitianK-groups in terms of higher algebraicK-groups.We also relate periodicity to étale hermitianK-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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