Complex hypersurfaces diffeomorphic to affine spaces

Department or Administrative Unit

Mathematics

Document Type

Article

Author Copyright

Copyright © 1994 Tokyo Institute of Technology, Department of Mathematics

Publication Date

1994

Journal

Kodai Mathematical Journal

Abstract

This note is related to the following two basic problems in Algebraic Geometry.

PROBLEM A. Topological characterization of the complex affine space Cn. Let X be an n-dimensional smooth affine variety over C. Is it possible to impose some topological conditions on X which imply that X is isomorphic to the affine space Cn as algebraic varieties? Such isomorphic varieties are denoted in this note by X≅Cn.

PROBLEM B. Existence of exotic embeddings of Cn into Cn+1 (The Abhyankar-Sathaye Conjecture). Assume that X: f = 0 is a smooth hypersurface in Cn+1 such that X≅Cn. Does there exist an algebraic automorphism h : Cn+1→Cn+1 such that f◦h is a linear form? This can be restated as whether any embedding of Cn into Cn+1 is equivalent to a linear embedding.

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This article was originally published in Kodai Mathematical Journal. The full-text article from the publisher can be found here.

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