By James D. Lewis

This ebook offers an creation to a subject of significant curiosity in transcendental algebraic geometry: the Hodge conjecture. along with 15 lectures plus addenda and appendices, the quantity is predicated on a sequence of lectures introduced through Professor Lewis on the Centre de Recherches Mathematiques (CRM). The publication is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and similar subject matters. It comprises many examples, and so much effects are thoroughly confirmed or sketched. the incentive at the back of some of the effects and history fabric is supplied. This complete method of the publication supplies it a ``user-friendly'' variety. Readers needn't seek somewhere else for numerous effects. The ebook is appropriate to be used as a textual content for a subject matters direction in algebraic geometry; comprises an appendix via B. Brent Gordon.

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Pour N > 4, Coker η 4 s’injecte dans CH3 (X)tors . 4 ([20, 22, 23]). Pour toute quadrique X, le groupe CH3 (X)tors est d’ordre 1 ou 2. De plus, on a CH3 (X)tors = 0 pour N > 10. En particulier (cf. 5. Coker η 4 = 0 pour N > 10 et Coker η24 = 0 pour N > 14. 3 est N = 4. 2) : a) Voisine : d = 1, XE hyperbolique. b) Interm´ediaire : d = 1, XE isotrope, non hyperbolique. c) Albert : d = 1. d) Albert virtuelle : d = 1, XE anisotrope. 6 ([18]). Soit φ une forme quadratique d´eﬁnissant X (avec dim X = 4).

Voevodsky, V. : Triangulated categories of motives over a ﬁeld. In Cycles, transfers and motivic cohomology theories, Annals of Math. Studies 143, 2000 ` paraˆıtre aux Publ. 45. Voevodsky, V. : Motivic cohomology with Z/2 coeﬃcients. A ´ Math. IHES. 46. Voevodsky, V. : Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Notices 2002, 351–355 47. Wang, S. : On the commutator group of a simple algebra. Amer. J. Math. ru Introduction This text is the notes of my lectures at the mini-course “M´ethodes g´eom´etriques en th´eorie des formes quadratiques” at the Universit´e d’Artois, Lens, June 26–28, 2000.

71 76 92 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 1 Grothendieck Category of Chow Motives Let k be any ﬁeld, and SmProj(k) the category of smooth projective varieties over k. We deﬁne the category of correspondences C (k) in the following way: the set Ob C (k) is identiﬁed with the set Ob SmProj(k) (the object corresponding to X will be denoted by [X]), and if X = i Xi is the decomposition into a disjoint union of connected components, then CHdim Xi (Xi × Y ), HomC (k) ([X], [Y ]) := i where CHdim Xi (Xi × Y ) is the Chow group of dim Xi -dimensional cycles on Xi × Y .