### Some new examples in the theory of quadratic forms, by Oleg T. Izhboldin and Nikita A. Karpenko

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 $K_0$ 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.

Oleg T. Izhboldin <oleg@izh.urs.pu.ru>
Nikita A. Karpenko <karpenk@math.uni-muenster.de>