Rank correction, top-Chern criterion, and an affine-surface counterexample

Definitions and mathematical statement

The supplied reconstruction of Madhav Nori's Question 14 asks: if $X$ is a smooth affine variety of dimension $d$ and $\alpha\in F^{d-1}K_0(X)$, must there be a vector bundle $E$ of rank $d-1$ with $[E]=\alpha$? Here $K_0$ is the Grothendieck group of vector bundles; under the topological filtration convention, $F^iK_0$ is generated by coherent classes supported in codimension at least $i$, using smoothness to identify coherent and vector-bundle $K$-theory. This filtration convention is not defined in the damaged excerpt.

For $d\geq2$, every such $\alpha$ has rank zero, while $[E]$ has rank $d-1>0$. Thus the printed equality is rank-inconsistent, as the report emphasizes. Its proposed corrected question is explicitly different: \[ [E]-(d-1)[\mathcal O_X]=\alpha,\qquad\operatorname{rank}E=d-1. \] No assertion is made that this reconstruction was the intended printed statement.

The report says that a rank-$d$ representative $W$ of $d[\mathcal O_X]+\alpha$ always exists by stable realization on a smooth affine $d$-fold. It identifies corrected solvability with existence of some such representative splitting a free line: \[ \exists E\ \Longleftrightarrow\ \exists W\text{ of rank }d:\ [W]=d[\mathcal O_X]+\alpha, \quad W\cong\mathcal O_X\oplus E. \] This existential formulation does not assume cancellation or that an arbitrary chosen representative splits. Over an algebraically closed field, the report attributes to Serre stable realization and the Murthy--Krishna top-rank splitting theorem the criterion \[ \text{corrected solvability}\quad\Longleftrightarrow\quad c_d(\alpha)=0\quad\text{in }\operatorname{CH}^d(X). \] The precise dimension and characteristic hypotheses of those cited results are not supplied.

For a surface and a rank-zero class, the report defines the determinant defect \[ \delta_2(\alpha)=\alpha-\left([\det(\alpha)]-[\mathcal O_X]\right). \] It vanishes exactly when the corrected rank-one representation is given by the determinant line bundle. The report refutes the corrected universal assertion over ${\mathbb{C}}$ using the smooth affine surface \[ X=\{(x,y,z)\in{\mathbb{A}}^3_{{\mathbb{C}}}:1+x^4+y^4+z^4=0\}. \] It claims $\operatorname{CH}_0(X)\ne0$; for a nonzero zero-cycle class $\zeta$, the cycle class $\alpha=\operatorname{cyc}_K(\zeta)\in F^2K_0(X)$ satisfies $c_2(\alpha)=-\zeta\ne0$, and therefore is not $[L]-[\mathcal O_X]$ for any line bundle $L$. No explicit representative cycle or proof of the nonzero Chow group is supplied. These claims address $d\geq2$; the original excerpt's unspecified low-dimensional range needs separate interpretation.

Short English statement

The printed bundle-class equality has incompatible ranks in dimension at least two. The report separately studies a rank-corrected version, reducing it to splitting a free line from some rank-$d$ representative and claiming a top-Chern criterion over algebraically closed fields. A supplied affine-surface example reportedly disproves even that corrected universal statement.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000251$, source code catalogue entry; damaged original text, supplied reconstruction and complete research summary. Statement status reconstructed and unverified.
• AIM workshop "Projective modules and $A^1$-homotopy theory", Question 14, attributed to Madhav Nori: https://aimath.org/pastworkshops/projectiveA1problems.pdf. Canonical location aim-algebraic-geometry-notes.json, notes[250]; original literature field empty. No PDF was opened here.
• Supplied attributions: Serre stable realization and the Murthy--Krishna top-rank splitting theorem; no full bibliographic references, theorem statements or characteristic hypotheses supplied.
• Supplied machine-generated research attempt 1: counterexample; status refuted at the run, registry solved and classification SOLVED-IN-LITERATURE; rigor metadata "proved", novelty confidence low. The non-peer-reviewed report requires independent verification before citation.