1,720,992 research outputs found
1-motivic sheaves and the Albanese functor
Full text linkUsing sheaf theoretic methods, we define functors View the MathML source and View the MathML source. The functor View the MathML source extends the one in [L. Barbieri-Viale, B. Kahn, On the derived category of 1-motives, I. Prépublication Mathématique de l’IHÉS (M/07/22), June 2007, 144 pages] to non-necessarily geometric motives. These functors are then used to define higher Néron–Severi groups and higher Albanese sheaves
T-motives
No full textConsidering a (co)homology theory T on a base category C as a fragment of a first-order logical theory we here construct an abelian category A[T] which is universal with respect to models of T in abelian categories. Under mild conditions on the base category C, e.g. for the category of algebraic schemes, we get a functor from C to Ch(Ind(A[T])) the category of chain complexes of ind-objects of A[T]. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from D(Ind(A[T])) to Voevodsky's motivic complexes
Formal Hodge Theory
No full textWe introduce {\em formal}\, (mixed) Hodge structures (of level ) in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures (of level ) providing an equivalence of categories
A Pamphlet on Motivic Cohomology
No full textThis is an “elementary” introduction to the conjectural theory of motives along the lines indicated by Grothendieck. We further quote recent developments, also presenting some advances due to Voevodsky, and applications to the study of algebraic cycles and differential forms
Nori n-motives
No full textI introduced n-motives by showing that Nori’s construction of mixed motives can be applied to varieties of dimension ≤ n. I then have linked Nori 0- motives to Artin motives and Nori 1-motives to Deligne 1-motives (with torsion)
Che cos'è un numero. Una introduzione all'algebra
No full textChe cosa sono i numeri e come si giustifica la loro esistenza? L’approccio all’edificio matematico adottato in questo libro non richiede conoscenze matematiche preliminari. Il libro si sviluppa idealmente in tre fili del discorso che s’intrecciano tra loro: il primo affronta i fondamenti e il linguaggio della matematica come appare nella teoria degli insiemi e nell’algebra categoriale, il secondo sviluppa le problematiche elementari che scaturiscono dalla semplice esistenza dei numeri come si sono presentate nella storia, con l’aritmetica e la teoria dei numeri, e il terzo conduce direttamente all’algebra astratta presentando le sue principali strutture, alcuni risultati e soprattutto i metodi. Sono proprio le strutture algebriche che permettono di caratterizzare e classificare i numeri.
Il libro è una proposta originale per il giovane matematico, una sfida per il filosofo e un’avventura per chi si accosta per puro diletto al potente richiamo dell’algebra
Formal Hodge Structures
Full text linkThe aim of this work is to develop the program proposed by S. Bloch, L. Barbieri-Viale and V. Srinivas of generalizing Deligne mixed Hodge structures providing a new cohomology theory for complex algebraic varieties. In other words to construct and study cohomological invariants, of (proper) complex algebraic schemes, which are finer than the associated mixed Hodge structures in the case of singular spaces
On the theory of 1-motives
No full textThis is an overview and a preview of the theory of "mixed motives of level 1" explaining some results, projects, ideas and indicating a bunch of problems
On the derived category of 1-motives
No full textWe embed the derived category of Deligne 1-motives over a perfect field into the étale version of Voevodsky's triangulated category of geometric motives, after inverting the exponential characteristic. We then show that this full embedding ``almost'' has a left adjoint LAlb. Applying LAlb to the motive of a variety we get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. Among applications, we give motivic proofs of Roĭtman type theorems and new cases of Deligne's conjectures on 1-motives.Nous plongeons la catégorie dérivée des 1-motifs de Deligne sur un corps parfait dans la version étale de la catégorie triangulée des motifs géométriques de Voevodsky, après avoir inversé l'exposant caractéristique. Nous montrons ensuite que ce plongement a «presque» un adjoint à gauche LAlb. En appliquant LAlb au motif d'une variété, on obtient un complexe de 1-motifs, que nous calculons entièrement dans le cas des variétés lisses et partiellement dans le cas des variétés singulières. Parmi les applications, nous donnons des preuves motiviques de théorèmes de type Roĭtman, et établissons de nouveaux cas des conjectures de Deligne sur les 1-motifs
On the Deligne-Beilinson cohomology sheaves
No full textWe prove that the Deligne–Beilinson cohomology sheaves Hq+1(Z(q)D) are torsion-free as a consequence of the Bloch–Kato conjectures as proven by Rost and Voevodsky. This implies that H0(X,Hq+1(Z(q)D))=0 if X is unirational. For a surface X with pg=0 we show that the Albanese kernel, identified with H0(X,H3(Z(2)D)), can be characterized using the integral part of the sheaves associated to the Hodge filtration
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