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Exponential motives Fresán, Javier | CIRM H

Multi angle

y

Research talks

I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally generalizes the Mumford-Tate group of a Hodge structure and, over number fields, governs all algebraic relations between exponential periods. This is a joint work with Peter Jossen (ETH). I will sketch the construction - following ideas of Kontsevich and Nori - of a Tannakian category of exponential motives over a subfield of the complex numbers. It is a universal cohomology theory for pairs of varieties and regular functions, whose de Rham and Betti realizations are given by twisted de Rham and rapid decay cohomology respectively. The upshot is that one can attach to any such pair a motivic Galois group which conjecturally ...

11R58 ; 14G25 ; 11F80 ; 14C15 ; 11E72 ; 14D07 ; 11G35

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0
Z
ings held at tromso#June 27 - July 8" style="background-image:url('icon/BES/Ressource_BES_200082.jpg');">V

- 244 p.
ISBN 978-3-540-08954-4

Lecture notes in mathematics , 0687

Localisation : Collection 1er étage

anneaux d'équivalence rationnelle # courbe # cycle # famille # fondements de la géométrie algébrique # géométrie algébrique # méthode de la géométrie algébrique # module # sous schéma # surface

14A05 ; 14C15 ; 14H99 ; 14J10 ; 14N10

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Algebraic cycles and motives. Vol. 2 :
Aug. 30 - Sept. 3
Nagel, Jan ; Peters, Chris | Cambridge University Press 2007

Congrès

V

- 359 p.
ISBN 978-0-521-70175-9

London mathematical society lecture note series , 0344

Localisation : Collection 1er étage

géométrie algébrique # cycles algébriques # motifs mixtes # conjecture de Bloch

14C15 ; 14-06 ; 00B25 ; 14C25

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Subtle Stiefel-Whitney classes and the J-invariant of quadrics Vishik, Alexander | CIRM H

Multi angle

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Research Talks

I will discuss the new ?subtle? version of Stiefel-Whitney classes introduced by Alexander Smirnov and me. In contrast to the classical classes of Delzant and Milnor, our classes see the powers of the fundamental ideal, as well as the Arason invariant and its higher analogues, and permit to describe the motives of the torsor and the highest Grassmannian associated to a quadratic form. I will consider in more details the relation of these classes to the J-invariant of quadrics. This invariant defined in terms of rationality of the Chow group elements of the highest Grassmannian contains the most basic qualitative information on a quadric. I will discuss the new ?subtle? version of Stiefel-Whitney classes introduced by Alexander Smirnov and me. In contrast to the classical classes of Delzant and Milnor, our classes see the powers of the fundamental ideal, as well as the Arason invariant and its higher analogues, and permit to describe the motives of the torsor and the highest Grassmannian associated to a quadratic form. I will consider in more details the relation of these classes ...

14F42 ; 14C15 ; 11E04 ; 11E81

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Stable rationality - Lecture 3 Pirutka, Alena | CIRM H

Multi angle

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Research talks

Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.
Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational ...

14C15 ; 14C25 ; 14E08 ; 14H05 ; 14J70 ; 14M20

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On examples of varieties that are not stably rational Pirutka, Alena | CIRM H

Multi angle

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Research Talks

A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational and not even retract rational. The proofs involve studying the properties of Chow groups of zero-cycles and the diagonal decomposition. As concrete examples, we will discuss some quartic double solids (C. Voisin), quartic threefolds (a joint work with Colliot-Thélène), some hypersurfaces (Totaro) and others. A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational and not even retract rational. The proofs involve studying the properties of Chow groups of zero-cycles and the diagonal decomposition. As concrete examples, we will discuss some quartic ...

14C15 ; 14M20 ; 14E08

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&record=19276207124910944899"> Hodge theory.
Based on lectures delivered at the summer school on Hodge theory and related topics
Trieste # june 14 - july 2, 2010 Cattani, Eduardo ; El Zein, Fouad ; Griffiths, Phillip A. ; Le, Dung Trang | Princeton University Press 2014

Congrès

V

- xvii; 589 p.
ISBN 978-0-691-16134-1

Mathematical notes , 0049

Localisation : Colloque RdC (TRIE)

théorie de Hodge # variété Kählerienne # groupes de Chow # variété de Shimura

14-06 ; 32-06 ; 14C30 ; 14D07 ; 32G20 ; 32J25 ; 32S35 ; 32Q15 ; 14C15 ; 14G35 ; 14C25 ; 00B25

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Geometric methods in the algebraic theory of quadratic forms :
summer school#April 17
Izhboldin, O. T. ; Kahn, B. ; Karpenko, N. A. ; Tignol, Jean-Pierre | Springer 2004

Congrès

V

- 190 p.
ISBN 978-3-540-20728-3

Lecture notes in mathematics , 1835

Localisation : Collection 1er étage

théorie algèbrique des formes quadratiques # groupe de Chow # cohomologie des quadriques # u-invariant # motif # motif de quadrique # équivalence birationnelle stable

11E81 ; 14C15 ; 14F42

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