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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On exotic affine $3$-spheres

Authors: Adrien Dubouloz and David R. Finston
Journal: J. Algebraic Geom. 23 (2014), 445-469
Published electronically: January 31, 2014
MathSciNet review: 3205588
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Abstract | References | Additional Information

Abstract: Every $\mathbb {A}^{1}$-bundle over $\mathbb {A}_{\ast }^{2},$ the complex affine plane punctured at the origin, is trivial in the differentiable category, but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine $3$-sphere $\mathbb {S}_{\mathbb {C}}^{3},$ given by $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=1,$ admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb {A}^{1}$-bundles over $\mathbb {A}_{\ast }^{2}$ are classified up to $\mathbb {G}_{m}$-equivariant algebraic isomorphism, and a criterion for nonisomorphy is given. In fact $\mathbb {S}_{\mathbb {C}}^{3}$ is not isomorphic as an abstract variety to the total space of any $\mathbb {A}^{1}$-bundle over $\mathbb {A}_{\ast }^{2}$ of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a byproduct, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.

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Additional Information

Adrien Dubouloz
Affiliation: Adrien Dubouloz, Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary - BP 47870, 21078 Dijon cedex, France

David R. Finston
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Address at time of publication: Department of Mathematics, Brooklyn College, City University of New York, 2900 Bedford Avenue, Brooklyn, New York 11210

Received by editor(s): April 27, 2011
Received by editor(s) in revised form: October 26, 2011, and November 15, 2011
Published electronically: January 31, 2014
Article copyright: © Copyright 2014 University Press, Inc.