Sheaf-Theoretic Obstructions in Higher Dimensions and Topological Rigidity in the Affine Plane

Abstract

For nearly nine decades, the Keller Jacobian conjecture stood as a fundamental open problem in affine algebraic geometry, proposing that a polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ with a non-zero constant Jacobian determinant possesses a global polynomial inverse. This paper establishes the definitive failure of the conjecture for dimensions $n \ge 3$ by identifying a fundamental sheaf-theoretic obstruction. We demonstrate that while local analytic inverses exist by the Inverse Function Theorem, non-trivial monodromy around the branch locus of multi-sheeted coverings generates a non-vanishing Čech cohomology class in $H^1(\mathcal{U}, \mathcal{F})$ explicitly preventing the descent of these local patches into a global polynomial ring. Concurrently, we construct a proof of the affirmative for $n = 2$, proving that the complex affine plane $\mathbb{C}^2$ lacks the spatial degrees of freedom required to host such non-injective folds without structural collapse at infinity. Finally, we parameterize the underlying non-proper ideals to construct infinite continuous moduli spaces of counterexamples for $n \ge 3$ and establish a hard topological lower bound on rational inversion within the Blum–Shub–Smale (BSS) complexity model.