Excellent connections in the motives of quadrics
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 1, p. 183-195

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.

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.

DOI : https://doi.org/10.24033/asens.2142
Classification:  11E04,  14C15,  14C25
Keywords: quadratic forms, motives, Chow groups, Steenrod operations
@article{ASENS_2011_4_44_1_183_0,
     author = {Vishik, Alexander},
     title = {Excellent connections in the motives of quadrics},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {1},
     year = {2011},
     pages = {183-195},
     doi = {10.24033/asens.2142},
     zbl = {1223.14005},
     mrnumber = {2760197},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2011_4_44_1_183_0}
}
Vishik, Alexander. Excellent connections in the motives of quadrics. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 1, pp. 183-195. doi : 10.24033/asens.2142. http://www.numdam.org/item/ASENS_2011_4_44_1_183_0/

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