Nikita A. Karpenko and Alexander S. Merkurjev: Hermitian forms over quaternion algebras

karpenko at, merkurev at

Submission: 2013, May 13

We study a hermitian form h over a quaternion division algebra Q over a field (h is supposed to be alternating if the characteristic of the field is 2). For generic h and Q, for any integer i in the interval [1, n/2], where n is the dimension of h over Q, we show that the variety of i-dimensional (over Q) totally isotropic right subspaces of h is 2-incompressible. The proof is based on a computation of the Chow ring for the classifying space of a certain parabolic subgroup in a split simple adjoint affine algebraic group of type C_n. As an application, we determine the smallest value of the J-invariant of a non-degenerate quadratic form divisible by a 2-fold Pfister form; we also determine the biggest values of the canonical dimensions of the orthogonal Grassmannians associated to such quadratic forms.

2010 Mathematics Subject Classification: 14L17; 14C25

Keywords and Phrases: Algebraic groups, classifying spaces, hermitian and quadratic forms, projective homogeneous varieties, Chow groups and motives.

Full text: dvi.gz 52 k, dvi 127 k, ps.gz 972 k, pdf.gz 244 k, pdf 273 k.

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