On algebraic vector bundles of rank 2 over smooth affine fourfolds

2025/07/28 by Brazelton, Thomas, Opie, Morgan, Syed, Tariq
#13C10 #14C17 #14C35 #14F42 #14F45 #19A13 #55S35 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2507.21029

Abstract

The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the situation in lower dimensions. Given a smooth affine fourfold over an algebraically closed field of characteristic not equal to 2 or 3, we study cohomological criteria for finiteness of the fibers of the Chern class map for rank 2 bundles. As a consequence, we give a cohomological classification of such bundles in a number of cases. For example, if d≤ 4, there are precisely d² non-isomorphic algebraic vector bundles over the complement of a smooth hypersurface of degree d in \mathbb P⁴_{\mathbb C}.

Citations

• Strongly A¹-invariant sheaves (after F. Morel)
• Enumerating stably trivial vector bundles with higher real K-theory
• Higher Chow groups with finite coefficients and refined unramified cohomology
• Metastable complex vector bundles over complex projective spaces
• Suslin's cancellation conjecture in the smooth case
• Vector bundles on algebraic varieties
• Symplectic orbits of unimodular rows
• Refined unramified cohomology of schemes
• Topological classification of complex vector bundles over 8-dimensional spin^c manifolds
• Localization and nilpotent spaces in A¹-homotopy theory
• A¹-homotopy theory and contractible varieties: a survey
• A primer for unstable motivic homotopy theory
• Obstructions to algebraizing topological vector bundles
• Algebraic vs. topological vector bundles on spheres
• The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

Related

• Topological classification of complex vector bundles over 8-dimensional spin^c manifolds
• Secondary characteristic classes and the Euler class
• Vector bundles on Fano 3-folds without intermediate cohomology
• New families of rank 2 bundles on projective space
• Chern classes of rank two globally generated vector bundles on P²