Algebraic cycles and Hodge theory: lectures given at the 2nd by Mark L. Green, Jacob P. Murre, Claire Voisin, Alberto

By Mark L. Green, Jacob P. Murre, Claire Voisin, Alberto Albano, Fabio Bardelli

The most objective of the CIME summer time college on "Algebraic Cycles and Hodge thought" has been to assemble the main energetic mathematicians during this zone to make the purpose at the current cutting-edge. hence the papers integrated within the complaints are surveys and notes at the most crucial issues of this quarter of analysis. They comprise infinitesimal equipment in Hodge concept; algebraic cycles and algebraic elements of cohomology and k-theory, transcendental equipment within the research of algebraic cycles.

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7 . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . 15 . . . . . . . . . . . 52 54 55 56 57 60 6 Some Applications . . . . . . . . . . . . . . . . . . . . 1 Higher Forms of the Motives of Quadrics . . . . . . . . . . . . 2 Dimensions of Anisotropic Forms in I n . . . . . . . . . . . . . 3 Motivic Decomposition and Stable Birational Equivalence of 7-dimensional Quadrics .

36 5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . -P. ): LNM 1835, pp. 25–101, 2004. 8 . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . .

Proof. Let 0 ≤ i < i1 (Q). Then the quadric Q|k(Q) has a projective subspace L of dimension i. Let A ⊂ Q×Q be the closure of L ⊂ Spec k(Q) ×Q ⊂ Q×Q. We have dim A = dim Q + i, so A defines a map α : M (Q)(i)[2i] → M (Q). Let now ρi : M (Q) → M (Q)(i)[2i] be the map defined by a plane section of codimension i embedded diagonally into Q × Q. It is easy to see that (ρi ◦ α)(i) = 1. 8, M (Q)(i)[2i] contains an indecomposable direct summand N1 , and M (Q) contains an indecomposable direct summand N2 such that N1 ∼ = N2 and Z(i)[2i] is a direct summand of N1 |k .

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