Sufficiently generic orthogonal Grassmannians (Q2376701)

Let \(F\) be a field, and let \(q\) be a non-zero non-degenerate quadratic form over \(F\) of dimension \(n\). For each \(0\leq i\leq\left\lfloor n/2\right\rfloor \) let \(Q_{i}\) denote the variety of \(i\)-dimensional totally isotropic subspaces of \(q\), and let \(M\left( Q_{i}\right) \) be the Chow motive of \(Q_{i}\) This work proves for ``sufficiently generic'' \(q\) that, with two exceptions, the \(M\left( Q_{i}\right) \) are indecomposable. It follows that each of the corresponding \(Q_{i}\) is \(2\)-incompressible, i.e. its \(2\)-dimension is precisely \(\dim Q_{i}.\) The notion of ``sufficient generic'' depends on the parity of \(n\) as well as the discriminant of \(q\). Associated to \(q\) is the \(J\)-invariant which also depends on the parity of \(n\). In the case \(n\) is odd, say \(n=2m+1,\) then \(q\) is sufficiently generic if \(J\left( q\right) \) is empty. If instead \(n=2m\), \(J\left( q\right) =\emptyset,\) and \(q\) has non-trivial discriminant, then \(q\) is sufficiently generic. Finally, if \(n=2m,\;q\) has trivial discriminant, and \(J\left( q\right) =0.\) In each of these cases, \(M\left( Q_{i}\right) \) is indecomposable. and \(Q_{i}\) is 2-incompressible. In both of these cases, \(Q_{i}\) is a connected variety. The remaining case to be considered is \(n=2m\) and \(q\) has trivial discriminant. Here, \(M\left( Q_{i}\right) \) is indecomposable. for \(0\leq i\leq m-2.\) On the other hand, \(Q_{m}\) is not connected: it decomposes into two isomorphic components. Also, \(M\left( Q_{m-1}\right) \) decomposes completely into \(m\) copies of the motive of a component of \(Q_{m}.\) A conjecture of Mathews states: if the degree of every closed point on \(Q_{i}\) is divisible by \(2^{i}\) and the Witt index of the quadratic form \(q_{F\left( Q_{i}\right) }\) is \(i\), then \(Q_{i}\) is 2-incompressible. This conjecture had been proved for various cases, namely for. \(i=1\) and for \(i=2\), \(n\) odd.\ Here, the author proves the conjecture for all \(i\).