Excellent connections in the motives of quadrics

Vishik, Alexander

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 183-195.

Dans cet article, nous prouvons la conjecture qui dit que le motif d'une quadrique réelle est le « plus décomposable » parmi ceux des quadriques de la même dimension sur n'importe quel corps. Cela restreint sûrement les motifs possibles pour une quadrique anisotrope quelconque. Nous en tirons en corollaire une minoration du rang d'un facteur direct indécomposable du motif d'une quadrique en fonction de sa dimension, ce qui généralise le théorème bien connu du motif binaire. De plus, nous obtenons une description des motifs de Tate qui apparaissent, ce qui implique alors une nouvelle preuve du théorème de Karpenko sur les valeurs du premier indice de Witt. D'autres relations entre les indices de Witt supérieurs s'en suivent également.

In this article we prove the conjecture claiming that the motive of a real quadric is the “most decomposable” among anisotropic quadrics of given dimension over all fields. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalizes the well-known Binary Motive Theorem. Moreover, we have the description of the Tate motives involved. This, in turn, gives another proof of Karpenko's Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.

Classification : 11E04, 14C15, 14C25
Keywords: quadratic forms, motives, Chow groups, Steenrod operations
Mot clés : formes quadratiques, motifs, groupes de Chow, opérations de Steenrod

[1] P. Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003), 1869-1903. | MR | Zbl

[2] R. Elman, N. A. Karpenko & A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, Amer. Math. Soc., 2008. | MR | Zbl

[3] O. Haution, Lifting of coefficients for Chow motives of quadrics, in Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, Springer, 2010, 239-247. | MR | Zbl

[4] J. Hurrelbrink, N. A. Karpenko & U. Rehmann, The minimal height of quadratic forms of given dimension, Arch. Math. (Basel) 87 (2006), 522-529. | MR | Zbl

[5] J. Hurrelbrink & U. Rehmann, Splitting patterns of excellent quadratic forms, J. reine angew. Math. 444 (1993), 183-192. | EuDML | MR | Zbl

[6] O. Izhboldin & A. Vishik, Quadratic forms with absolutely maximal splitting, in Quadratic forms and their applications (Dublin, 1999), Contemp. Math. 272, Amer. Math. Soc., 2000, 103-125. | MR | Zbl

[7] N. A. Karpenko, On the first Witt index of quadratic forms, Invent. Math. 153 (2003), 455-462. | MR | Zbl

[8] M. Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. 33 (1976), 65-93. | MR | Zbl

[9] M. Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. 34 (1977), 1-31. | MR | Zbl

[10] M. Rost, Some new results on the Chow groups of quadrics, preprint http://www.mathematik.uni-bielefeld.de/~rost/data/chowqudr.pdf, 1990.

[11] M. Rost, The motive of a Pfister form, preprint http://www.math.uni-bielefeld.de/~rost/data/motive.pdf, 1998.

[12] A. Vishik, Motives of quadrics with applications to the theory of quadratic forms, in Geometric methods in the algebraic theory of quadratic forms, Lecture Notes in Math. 1835, Springer, 2004, 25-101. | Zbl

[13] A. Vishik, Symmetric operations, Tr. Mat. Inst. Steklova 246 (2004), 92-105, English translation: Proc. of the Steklov Institute of Math. 246 (2004), 79-92. | Zbl