The complete solution to Bass Generalized Jacobian Conjecture

TL;DR

This paper provides a complete solution to Bass's Generalized Jacobian Conjecture, demonstrating its validity only in dimension one, by analyzing unramified morphisms from complex affine varieties.

Contribution

It clarifies the conjecture's scope and proves it holds only in one-dimensional cases, resolving a long-standing open problem.

Findings

The conjecture is true only in dimension one.

Counterexamples exist in higher dimensions.

The paper connects Bass's conjecture with Kulikov's construction.

Abstract

The Classical Jacobian Conjecture claims that any unramified endomorphism of a complex affine space is an automorphism. In order to embed this conjecture in a geometric environment, where one could enjoy the beauty and the richness of tools of algebraic geometry and algebraic D-modules, as his paper [6] proves it, Hyman Bass proposed 25 years ago in [6], page 80 the following statement as the Generalized Jacobian Conjecture: "Any unramified morphism from a complex irreducible affine and unirational variety whose invertible regular functions are all constant to a complex affine space of the same dimension is an isomorphism". On the other hand, without any explicit connection with Bass conjecture, Victor Kulikov published in 1993 (see [18]) a non trivial construction of a complex irreducible rational and simply connected surface and an unramified morphism of geometric degree 3 (and hence which is not an isomorphism) from this surface to the complex affine space, without specifying if this surface is affine or not, or if its invertible regular functions are all constant or not. The main aim of this paper is to bring this precision and thanks to this to expose the complete solution to Bass Generalized Jacobian Conjecture which turned to be true only in dimension one (see Theorem 1 below).