# 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.

## 46 Citations

Families of exotic affine 3-spheres

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We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically…

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LOCALLY NILPOTENT DERIVATIONS ON AFFINE SURFACES WITH A C -ACTION

- Mathematics
- 2004

We give a classification of normal affine surfaces admitting analgebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the…

Challenging problems on affine $n$-space

- Mathematics
- 1995

Complex affine n-space C^n, the basic object of algebraic geometry, offers a number of exciting and striking problems. The most famous one, the Jacobian Conjecture is the still unsolved. Others are…

Computations in A1-homotopy theory

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- 2020

We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine A1-contractible 3-folds.…

Smooth contractible threefolds with hyperbolic $\mathbb{G}_{m}$-actions via ps-divisors

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- 2017

The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the…

Vector bundles on contractible smooth schemes

- Mathematics
- 2007

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The…

A1 contractibility of affine modifications

- Mathematics
- 2018

We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine A1-contractible 3-folds.…

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© Institut Fourier – Université de Grenoble, 1993, tous droits réservés. L’accès aux archives de la collection « Cours de l’institut Fourier » implique l’accord avec les conditions générales…