On the Witt ring of a Grassmann variety, by Marek Szyjewski

A {+,-}\times Pic(X)/2 - indexed system of maps, with values in proper subquotients of Grothendieck group K_0(X) detects nontrivial (non-extended) bilinear bundles over a variety X, and other objects, considered in connection with bilinear spaces, e. g. Ranicki formations.

For any line bundle L on a variety X, the Hom_{O_X}(-,L) is exact (co)functor on vector bundles and induces an involution ^L on the Grothendieck group K_0(X). Tate cohomology groups of the group {id, ^L} with values in K_0(X), denoted here E^{+}(X,L), E^{-}(X,L), can be effectively computed in many cases. Moreover, the map e^0: W(X) --> E^{+}(X,1) is an epimorphism in many cases. For example, if L is trivial then any symmetric or skew-symmetric bilinear space produces a class in Tate cohomology \hat{H}^1({id,^L}, K_0(X)). Moreover, Witt equivalent spaces produce the same class.

Functoriality with respect to inverse image map and nice covariant properties yield effective computation of Herbrand index for a Grassmann variety X. This yields a lower bound for order of E^{+}(X,L). Moreover, for a Grassmann variety of planes in n-dimensional vector space X = Gr(n,2) each element of E^{+}(X,1) is a value of e^0. So there exist non-extended Witt classes on Grassmann varieties.

Marek Szyjewski <szyjewski@gate.math.us.edu.pl>