Witt groups of Grassmann varieties (Q2913218)

The authors compute the Witt groups of split Grassmann varieties, over any regular base \(X\). They prove that the total Witt group of the Grassmannian is a free module over the total Witt ring of \(X\). Remark that also the Chow group, or the Grothendieck group, are free over \(X\) with a basis indexed by all Young diagrams.NEWLINENEWLINEThe authors provide an explicit basis of the total Witt group indexed by a class of Young diagrams which they call even Young diagrams. Recall that the total Witt group is a sum of all the Witt groups depending on a shift \(i\in \mathbb{Z}/4\) and a twist \(L\in \mathrm{Pic} (X)/2\). The cited basis consists of homogeneous elements; moreover the shift and the twist can be read on the corresponding Young diagram. In particular, this fact allows the author to describe the unshifted and untwisted Witt group. The elements of the basis of the total Witt group are defined as push-forwards of the unit form of certain desingularized Schubert varieties. Remark that pushing the unit form is not always possible. The condition for a Young diagram to be even implies the existence of such a push-forward, but it is not necessary.NEWLINENEWLINEThe computation of Witt groups of a Grassmann variety is harder than the computation of cohomology groups or Chow groups because of the following fact. The classical computation proceeds by induction, using the closed embedding of a smaller Grassmannian \(\mathrm{Gr}_X(d,n-1)\) inside \(\mathrm{Gr}_X(d,n)\), whose open complement \(U\) is an affine bundle over another smaller Grassmannian \(\mathrm{Gr}_X(d-1,n)\). Moreover the restriction morphism from the big Grassmannian \(\mathrm{Gr}_X(d,n)\) to the open \(U\) is split surjective. For Witt groups the restriction morphism is not even surjective. In other words, the connecting homomorphism in the localization long exact sequence in not zero in general.