On exotic algebraic structures on affine spaces
@article{Zaidenberg1995OnEA, title={On exotic algebraic structures on affine spaces}, author={Mikhail Zaidenberg}, journal={arXiv: Algebraic Geometry}, year={1995} }
By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on the subject, which emphasizes its analytic aspects and points out some open problems.
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References
SHOWING 1-10 OF 56 REFERENCES
Complex hypersurfaces diffeomorphic to affine spaces
- Mathematics
- 1994
PROBLEM A. Topological characterization of the complex affine space C. Let X be an w-dimensional smooth affine variety over C. Is it possible to impose some topological conditions on X which imply…
Exotic analytic structures and Eisenman intrinsic measures
- Mathematics
- 1994
Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onCn under which we understand smooth complex affine…
Isotrivial Families of Curves on Affine Surfaces and Characterization of the Affine Plane
- Mathematics
- 1988
The main result is a characterization of C2 as a smooth acyclic algebraic surface on which there exist simply connected algebraic curves (possibly singular and reducible) or isotrivial…
Smooth contractible hypersurfaces in Cn and exotic algebraic structures on C3
- Mathematics
- 1993
In this paper we study complex affine algebraic varieties that are diffeomorphic to R E" but not isomorphic to C". According to [-Z1 Z3] we shall refer to these varieties as to exotic algebraic…
A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces
- Mathematics
- 1977
Explicit construction for decomposition of algebraic hyper- surfaces into a connected sum of handles and a homological projective space is discussed. Also a connection is provided between Levine's…
A topological characterisation of the affine plane as an algebraic variety
- Mathematics
- 1971
This investigation arose from an attempt to answer the following question, raised by M. P. Murthy: If X is an affine algebraic variety over a field k such that X x A' e A3, where An denotes the…
Homology Planes an Announcement and Survey
- Mathematics
- 1989
We review recent advances dealing with homology planes, i.e., non singular affine acyclic surfaces over the complex numbers C. Here acyclic means vanishing integral reduced homology. From a…
Affine surfaces containing cylinderlike open sets
- Mathematics
- 1980
L e t k be a n algebraically closed field of characteristic z e ro , a n d le t X be a n a ff in e su rfa c e defined over k such that XxA A, where A,'; denotes the n-dimensional affine sp ace . A…
On Some Family of Contractible Hypersurfaces in $C^4$
- Mathematics
- 1993
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