Rank four vector bundles on four-folds

Abstract: Throughout the 20th century, the "accidental" equalities of Dynkin diagrams of small rank have been exploited in various ways to describe torsors for the corresponding linear algebraic groups over fields and rings. In this talk, we will present these descriptions in the context of torsors over algebraic varieties. In the case of A_3=D_3, we will show how this description can be used to give a "hermitian K-theoretic" obstruction to the existence of sections of projective three-space bundles (i.e. line subbundles of rank four vector bundles) over four folds.