Incompressibility of orthogonal Grassmannians of rank 2
Let \(p\) be a prime or zero, and \(X\) a smooth complete variety over a field \(F.\) Then the canonical \(p\)-dimension cdim\(_{p}\left( X\right) \) of \(X\) is the minimal dimension of the image of a morphism \(X^{\prime}\rightarrow X\) for \(X^{\prime}\) an \(F\)-variety with a dominant morphism \(X^{\prime}\rightarrow X\) with \(F\left( X^{\prime}\right) /F\left( X\right) \) finite, degree prime to \(p\). We write cdim\(\left( X\right) \) for cdim\(_{0}\left( X\right) .\) It is clear that cdim\(_{p}\left( X\right) \leq\)cdim\(\left( X\right) \leq \dim\left( X\right) ;\) when these are all equalities we say \(X\) is \(p\)-incompressible.NEWLINENEWLINELet \(V\) be an \(F\)-vector space of odd dimension, say \(2n+1,\) and let \(\varphi\) be a nondegenerate quadratic form on \(V\). For \(1\leq d\leq n\) let \(X_{d}\) be the (smooth, projective) variety of \(d\)-dimensional totally isotropic subspaces of \(V\). \ The main result of this paper states that if the degree of the Chow ring of \(X_{2}\) is a multiple of \(4\) and the second relative higher Witt index of \(\varphi\) is \(1\), then cdim\(_{2}\left( X_{2}\right) =\dim\left( X_{2}\right) =4n-5\) in the case \(n\geq3\) -- the cases \(n=1\) and \(n=2\) having already been addressed by Karpenko and Merkurjev. This result suggests a more general conjecture, namely that \(X_{d}\) is \(2\)-incompressible if \(2^{d}\) divides the degree of the Chow ring of \(X_{d}\) and the \(d^{\text{th}}\) relative higher Witt index of \(\varphi\) is \(1.\)
scientific article; zbMATH DE number 6037582