On motivic obstructions to Witt cancellation for quadratic forms over schemes

TL;DR

This paper computes specific $A^1$-homotopy sheaves of orthogonal Stiefel varieties to identify obstructions to Witt cancellation, advancing the classification of quadratic forms over schemes.

Contribution

It provides the first explicit obstructions for rationally trivial quadratic forms to split off hyperbolic planes using $A^1$-homotopy theory, refining previous invariants.

Findings

Computed the first non-vanishing $A^1$-homotopy sheaves of orthogonal Stiefel varieties.

Identified obstructions to splitting quadratic forms, refining the Euler class.

Demonstrated cases where obstructions are nontrivial, improving splitting results.

Abstract

The paper provides computations of the first non-vanishing $\mathbb{A}^1$-homotopy sheaves of the orthogonal Stiefel varieties which are relevant for the unstable isometry classification of quadratic forms over smooth affine schemes over perfect fields of characteristic $\neq 2$. Together with the $\mathbb{A}^1$-representability for quadratic forms, this provides the first obstructions for rationally trivial quadratic forms to split off a hyperbolic plane. For even-rank quadratic forms, this first obstruction is a refinement of the Euler class of Edidin and Graham. A couple of consequences are discussed, such as improved splitting results over algebraically closed base fields as well as examples where the obstructions are nontrivial.