Some new examples in the theory of quadratic forms

Abstract

We construct a 6-dimensional anisotropic quadratic form \(\phi\) and a 4-dimensional quadratic form \(\psi\) over some fieldF such that \(\phi\) becomes isotropic over the function field \(F(\psi)\) but every proper subform of \(\phi\) is still anisotropic over \(F(\psi)\). It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form \(\phi\) with trivial determinant such that the index of the Clifford invariant of \(\phi\) is 4 but \(\phi\) can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest.