Rank-two uniqueness on singular threefolds

PDF p. 3: Following Corollary D

Assume the Axiom of Choice and the suitable universes used by the supplied constructions. Let $k$ be an algebraically closed field and let $X$ be a normal singular affine threefold of finite type over $k$. For rank-two vector bundles $E$ and $F$ on $X$, do the equalities $c_1(E)=c_1(F)$ in $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2,1}(X,\mathbb{Z})$ and $c_2(E)=c_2(F)$ in $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{4,2}(X,\mathbb{Z})$ imply $E\cong F$?

Here an affine threefold of finite type over $k$ is a scheme $\mathrm{Spec}A$, with $A$ a finitely generated commutative $k$-algebra, of Krull dimension three. Normal means that every local ring is an integrally closed domain; singular means that at least one local ring is not regular. A rank-two vector bundle is an ${\mathcal{O}}_X$-module locally isomorphic to ${\mathcal{O}}_X^2$, and $E\cong F$ means an invertible ${\mathcal{O}}_X$-linear map. The supplied construction assigns to such a bundle its first and second integral motivic Chern classes in the indicated motivic cohomology groups, invariant under isomorphism. Equivalently, the question asks whether the map $(c_1,c_2)$ from the set of isomorphism classes of rank-two bundles to the product of these two groups is injective.

Isomorphism classes of vector bundles

Motivic Chern classes

The Axiom of Choice

Fasel and Yakerson state this uniqueness question as open immediately after Corollary D. At that location they state surjectivity of $(c_1,c_2)$ for normal affine threefolds over algebraically closed fields; surjectivity alone does not settle injectivity. This entry records the article's status as of its cited 2026 version, not an exhaustive assessment of subsequent work. No solution is asserted.

The Choice assumption is inherited from the supplied bundle-class and motivic Chern-class constructions used to formulate the question.