A nontrivial rank-two bundle with trivial Chern classes on a smooth complex affine fourfold

By [F1], choose one smooth complex affine fourfold X and nontrivial rank-two E with an actual stable isomorphism $E\oplus {\mathcal{O}}_X^N\cong {\mathcal{O}}_X^{2+N}$ for some $N\ge 1$. This is an existence witness supplied by the explicitly imported established construction, rather than a newly specified system of equations. Its base field is exactly $\mathbb{C}$, its algebraic dimension is four, and it is affine.

For $A=\mathrm{\Gamma }(X,{\mathcal{O}}_X)$, [F2] constructs a right-invertible matrix $q:A^{2+N}\to A^N$ with E as its kernel bundle. The imported nontriviality means exactly that this q has no completion to an invertible matrix: a completion would give a global frame of the kernel. Thus the bundle is nontrivial as an algebraic bundle even though its finite free stabilization is trivial. The no-completion assertion follows from the imported existence and local equivalence; no independent calculation of matrix entries is claimed.

By [F3], this actual stable isomorphism gives $[E]=2[{\mathcal{O}}_X]$ in perfect spectral $K_0(X)$ and $\det E\cong {\mathcal{O}}_X$. Neither equality cancels a bundle summand; E remains nontrivial by step 2.1.

Apply [F4] independently to the ordinary Chow Chern operation and to the actual motivic operation fixed in [F5]. The stable isomorphism gives $c_t(E)c_t({\mathcal{O}}_X^N)=c_t({\mathcal{O}}_X^{2+N})$, so both trivial-bundle series being 1 implies $c_t(E)=1$. Every coefficient $c_i(E)$ for $i\ge 1$ is therefore zero in ordinary integral Chow groups, and likewise for the article's motivic Chern classes. This deduction uses the actual Whitney identity, not an identification of maps from their targets alone.

The pair (X,E) consequently satisfies all the smooth complex affine fourfold and rank-two hypotheses of the proposed criterion and has all its positive Chern classes zero, while $E\cong ̸{\mathcal{O}}_X^2$. It refutes that criterion and preserves the article's stronger stable-triviality assertion.