The Jacobian Conjecture fails for pseudo-planes

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its kale endomorphisms are proper. We study the conjecture for Q-acyclic surfaces of negative logarithmic Kodaira dimension. We show that G-equivariant counterexamples for infinite group G exist if and only if G = C* and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected C*-surfaces of negative logarithmic Kodaira dimension which admit non-proper C*-equivariant kale endomorph isms. We prove also that for every integers r >= 1, k >= 2 the Q-acyclic rational hyperplane u(1 + u(r)v) = w(k), which has fundamental group Z(k) and negative logarithmic Kodaira dimension, admits families of non-proper kale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition. (C) 2018 Elsevier Inc. All rights reserved.