On the Jacobian conjecture

We show that the Jacobian conjecture of the two dimensional case is true.

💡 Research Summary

The manuscript claims to prove the two‑dimensional Jacobian conjecture, namely that any polynomial map (T=(P,Q):\mathbb{C}^2\to\mathbb{C}^2) with constant Jacobian determinant equal to one is an automorphism of (\mathbb{C}^2). The author introduces a condition (K): the Jacobian (J(P,Q)=1) and every level curve ({P=\alpha}) is irreducible and nonsingular. Under this hypothesis the paper proceeds through three main parts.

First, the author reviews known topological facts about level curves. Propositions 1.1–1.4 assert that for a generic value (\alpha) the set (E_\alpha={(x,y)\mid P(x,y)\in U(\alpha)}) is a topologically trivial fiber bundle over a small disk (U(\alpha)) with fiber a Riemann surface (R_0). A global holomorphic section (L) of the projection (P:E_\alpha\to U(\alpha)) is constructed, and it is claimed that the image of (T) is Zariski‑open, with complement consisting of finitely many points.

Second, the core analytic argument is the linear partial differential equation \