#### Title

Complex hypersurfaces diffeomorphic to affine spaces

#### Document Type

Article

#### Department or Administrative Unit

Mathematics

#### Publication Date

1994

#### Abstract

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

PROBLEM A. Topological characterization of the complex affine space C^{n}. 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 C^{n} as algebraic varieties? Such isomorphic varieties are denoted in this note by X≅C^{n}.

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

#### Recommended Citation

Choudary, A. D. R., & Dimca, A. (1994). Complex hypersurfaces diffeomorphic to affine spaces. *Kodai Mathematical Journal, 17*(2). https://doi.org/10.2996/kmj/1138039958

#### Journal

Kodai Mathematical Journal

#### Copyright

Copyright © 1994 Tokyo Institute of Technology, Department of Mathematics

## Comments

This article was originally published in

Kodai Mathematical Journal. The full-text article from the publisher can be found here.