The groups \(H^3(F(X)/F)\) and \(\text{CH}^2(X)\) for generic splitting varieties of quadratic forms (Q5938992)

Let \(F\) be a field of characteristic \(\neq 2\) and \(\phi\) be a quadratic form over \(F\). A projective homogeneous variety \(X\) over \(F\) is said to be a generic splitting variety for \(\phi\) if, for some \(m<{1\over 2}\dim\phi\), the function field \(F(X)\) is place-equivalent over \(F\) to the function field of \(X_{\phi,m}\), the variety of totally isotropic subspaces of dimension \(m\) of the underlying vector space \(V\) of \(\phi\). Such a function field of a generic splitting variety is then place-equivalent to some field in the generic splitting tower of \(\phi\) as defined by Knebusch. The aim of this paper is to study and determine completely for such generic splitting varieties \(X\) the torsion part of \(\text{CH}^2(X)\) and the kernel \(H^3(F(X)/F)=\ker (H^3F\to H^3F(\phi))\), where \(H^3F\) denotes the third Galois cohomology group of \(F\) with coefficients in \({\mathbb Z}/2{\mathbb Z}\). \(\text{TorsCH}^2(X)\) and \(H^3(F(X)/F)\) are closely related, in fact, in many cases they are isomorphic, and one always has \(\text{TorsCH}^2(X)=0\) or \(\cong {\mathbb Z}/2{\mathbb Z}\). The different occuring cases are treated in detail and the list of results is comprehensive. Needed in this context and of interest in its own right is the determination of \(H^3(F(C_0(\phi))/F)\), where \(F(C_0(\phi))\) is some generic splitting field of the even part of the Clifford algebra of \(\phi\). As an application, the author proves the following result. Let \(\phi\) and \(\psi\) be forms of dimension \(8\) and suppose that in the case \(\text{ind}(C_0(\phi))\) we have that the division part of \(C_0(\phi)\) is defined over \(F\). Then \(\phi\) and \(\psi\) are motivically equivalent (i.e., the motives of the corresponding quadrics are isomorphic) if and only if \(\phi\) and \(\psi\) are half-neighbors (i.e., there exists \(a\in F^*\) such that \(\phi\perp a\psi\) is similar to a \(4\)-fold Pfister form). By a result due to Vishik in characteristic \(0\) and later \textit{N. A. Karpenko} in arbitrary characteristic \(\neq 2\) [Math. Ann. 317, No. 3, 585-611 (2000; Zbl 0965.11015)], it is known that two forms are motivically equivalent if and only if they are of the same dimension and their Witt indices are equal over all extension fields.