Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$

Harvey, Curtis R.; Karpenko, Nikita A.

Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1539-1543

For $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the corresponding quadratic forms up to odd degree base field extensions and get this way a strong upper bound on their essential $2$-dimension. As well, we compute the reduced Chow group of the maximal orthogonal grassmannian of the quadratic form and conclude that its canonical $2$-dimension is $2^n+2^{n-2}-2$.

Pour $n\ge 3$, en confirmant une version faible d’une conjecture de Hoffmann, on montre que toute forme quadratique anisotrope de dimension $2^n+2^{n-1}$ dans $I^n$ se déploie sur une extension finie du corps de base d’un degré qui n’est pas divisible par $4$. Le premier nouveau cas est celui de $n=4$, où l’on obtient une classification des formes quadratiques correspondantes à une extension de degré impair près ce qui donne une forte borne supérieure pour leur $2$-dimension essentielle. De plus, on détermine le groupe de Chow réduit de la grassmannienne orthogonale maximale de la forme quadratique et on en déduit que sa dimension $2$-canonique est égale à $2^n+2^{n-2}-2$.

Classification : 11E04, 14C25
Keywords: Quadratic forms over fields, projective homogeneous varieties, Chow rings
Mots-clés : Formes quadratiques sur des corps, variétés projectives homogènes, anneaux de Chow.

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