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Hermitian forms over quaternion algebras

Part of: Algebraic groups Cycles and subschemes

Published online by Cambridge University Press:  15 September 2014

Nikita A. Karpenko  and
Alexander S. Merkurjev
Nikita A. Karpenko
Affiliation:
Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada email karpenko@ualberta.ca
Alexander S. Merkurjev
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA, USA email merkurev@math.ucla.edu
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Abstract

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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 two). For generic $h$ and $Q$, for any integer $i\in [1,\;n/2]$, where $n:=\dim _{Q}h$, 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.

Keywords

algebraic groupsclassifying spacesHermitian and quadratic formsprojective homogeneous varietiesChow groups and motives

MSC classification

Primary: 14C25: Algebraic cycles 14L17: Affine algebraic groups, hyperalgebra constructions
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2073 - 2094
Copyright
© The Author(s) 2014 

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