Vector bundles of rank four and A_3 = D_3

TL;DR

This paper characterizes when rank four vector bundles over schemes with 2 invertible have line sub or quotient bundles, linking it to symmetric bilinear forms and Witt-theoretic Euler classes.

Contribution

It establishes a new criterion connecting subbundles of rank four vector bundles to symmetric bilinear forms and Euler classes, using a novel isomorphism between moduli functors of certain homogeneous bundles.

Findings

Sub or quotient line bundles correspond to lagrangian subspaces in the exterior square.

Existence of line subbundles is equivalent to vanishing of the Witt-theoretic Euler class.

Provides explicit isomorphisms between moduli functors for types A_3 and D_3.

Abstract

Over a scheme with 2 invertible, we show that a vector bundle of rank four has a sub or quotient line bundle if and only if the canonical symmetric bilinear form on its exterior square has a lagrangian subspace. For this, we exploit a version of "Pascal's rule" for vector bundles that provides an explicit isomorphism between the moduli functors represented by projective homogeneous bundles for reductive group schemes of type A_3 and D_3. Under additional hypotheses on the scheme (e.g. proper over a field), we show that the existence of sub or quotient line bundles of a rank four vector bundle is equivalent to the vanishing of its Witt-theoretic Euler class.