On the Chow groups of quadratic Grassmannians
Let \(k\) be a field of characteristic not \(2\), and let \(Q\) be a \(d\)-dimensional smooth projective quadric over \(k\) given by the equation \(q=0\) for a nondegenerate quadratic form \(q\) over \(k\). For any integer \(m\) with \(1\leq m\leq [d/2]+1\), let \(G(m,Q)\) denote the Grassmannian of \((m-1)\)-dimensional projective subspaces of \(Q\). The author defines the generic discrete invariant \(\text{GDI}(Q)\) of \(Q\) as the collection of the images \(C^*(G(m,Q))\) of the maps \(\text{CH}^*(G(m,Q))/2\to \text{CH}^*(G(m,Q)| _{\overline{k}})/2\) where \(\overline{k}\) denotes an algebraic closure of \(k\). This generic discrete invariant encodes many very important properties of the quadric \(Q\), for example the decomposition type of the Chow motive of \(Q\) as well as the splitting pattern of \(Q\). In the present paper, the author computes \(C^*(G(m,Q))\) for the Grassmannian of middle-dimensional planes, i.e. for the highest possible value \(m=[d/2]+1\) (where he assumes \(d\) to be odd as the even case can be deduced from the odd case). He shows that as an algebra it is generated by certain elementary classes \(\overline{z_i}\) contained in it, where \(i\) is taken from a certain subset \(J(Q)\subset\{ 1,\ldots ,m\}\). The use of Steenrod operations in the proofs implies a set of restrictions on the values that occur in \(J(Q)\), and the author asks if to any subset \(J\) of \(\{ 1,\ldots ,m\}\) satisfying these restrictions, there exists a quadric \(Q\) with \(J(Q)=J\), which would essentially imply that there are no other restrictions in general. The author has checked that this is the case at least up to \(m=4\). As an application, the author shows that for \(m=[d/2]+1\), \(G(m,Q)\) is \(2\)-incompressible provided \(Q\) is generic (where ``generic'' means that any splitting field of \(Q\) has degree divisible by \(2^m\)), and he also ventures a conjecture relating the canonical dimension of \(Q\) to the values in \(J(Q)\), where the canonical dimension of \(Q\) is defined in the sense of \textit{G. Berhuy} and \textit{Z. Reichstein} [Adv. Math. 198, No. 1, 128--171 (2005; Zbl 1097.11018)] as the smallest dimension a smooth variety \(Y\) can have such that there exist rational maps from \(G(m,Q)\) to \(Y\) and vice versa.
scientific article; zbMATH DE number 2146819