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Generalized and local Jacobian problems. (English. Russian original) Zbl 0796.14008

Russ. Acad. Sci., Izv., Math. 41, No. 2, 351-365 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 5, 1086-1103 (1992).
The Jacobian problem asks whether a polynomial endomorphism of complex affine \(n\)-space with non-vanishing Jacobian determinant is an isomorphism. Such a morphism is étale and surjective modulo codimension two. The generalized Jacobian problem asks whether an étale morphism from a simply connected complex variety of dimension \(n\) to complex \(n\)- space which is surjective modulo codimension two is an isomorphism. If the generalized problem had an affirmative answer, so would the original one. In this paper, the author constructs a counterexample to the generalized problem. His method is to show the equivalence between the generalized problem and a condition on fundamental groups of complex affine plane curve complements, namely whether such can have proper subgroups of finite index generated by geometric generators. He shows that if \(\overline D\) is a degree 4 projective plane curve with three cuspidal singularities and \(L\) is a projective line transversally intersecting \(\overline D\) in four points then the fundamental group of the plane curve \(D-L\) has a geometrically generated proper finite index subgroup.
He also considers the related local question, namely whether the fundamental group of the complement of a germ of an analytic curve in a two dimensional complex ball can have a proper finite index subgroup generated by geometric generators, and shows that this is equivalent to the question of whether such a ball can be the image by the germ of an étale surjective holomorphic map of degree more than one from a simply connected analytic surface. He shows that this second question has a negative answer, which thus settles the first negatively in the analytic case also.
The paper also establishes two additional equivalent formulations of the Jacobian problem: first, that injective Lie endomorphisms of the set of derivations of the polynomial ring are automorphisms, and the second in terms of properties of the differential equations associated to sets of certain derivations of complex \(n+1\) space.
Reviewer: A.R.Magid (Norman)

MSC:

14H37 Automorphisms of curves
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
32B10 Germs of analytic sets, local parametrization
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