Complex hypersurfaces diffeomorphic to affine spaces
Department or Administrative Unit
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.
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
Kodai Mathematical Journal
Copyright © 1994 Tokyo Institute of Technology, Department of Mathematics