Study of Quadratic Forms — Some Connections with Geometry

Abstract

Let X be an algebraic variety over a field k of characteristic not 2. A quadratic space on X is a locally free sheaf£ on X together with a self-dual isomor-phism q : £-> £V. In this article we outline some recent developments concerning the stable and nonstable study of quadratic spaces over algebraic varieties. Although this study borrows tools from algebra and geometry, it yields in return new insights into certain seemingly unrelated questions in algebra and geometry. 1 Quadratic spaces over the affine plane The nonstable study of quadratic spaces acquired an impetus with the solution of Serre's conjecture on the triviality of algebraic vector bundles on the affine space by Quillen and Suslin. In [OS], Ojanguren and Sridharan constructed nonfree, rank one projective modules over D[X, Y], where D is any noncommutative division ring. There was a classification [PS1] of nonfree projective left ideals of lHI[X, Y], where lHI denotes the algebra of real quaternions, in terms of certain 2 x 2 hermitian matrices, modulo "hermitian" equivalence. This led to the construction [P1] of an explicit family of indecomposable rank 4 quadratic spaces over JR[X, Y], thereby giving a negative answer to an analogue of Serre's conjecture for orthogonal bundles over A~. Given a quadratic space (c,q) over Ak, there is a quadratic space q0 over k such that the form on the fibre of (c, q) at any point of Ak is isometric to qo. We call qo the form on the fibre of (c, q). We say that a space over Ak is isotropic if the form on the fibre is isotropic. This is equivalent to the form being isotropic generically. The indecomposable spaces over JR[X, Y] mentioned above are anisotropic; indeed they have (1, 1, 1, 1) as the form on the fibre. It was shown by Ojanguren [OJ and independently by Kopeiko and Suslin [KS] that any isotropic quadratic space over Ak is extended from k. Thus, the obstruction to the quadratic spaces on Ak being extended from k lies in the existence of anisotropic quadratic spaces over k. That this is the precise obstruction to the extendibility question follows from the theorem [P2] below.