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OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

Part of: Theory of modules and ideals (Co)homology theory Fiber spaces and bundles Holomorphic fiber spaces

Published online by Cambridge University Press:  21 March 2019

A. ASOK ,
J. FASEL  and
M. J. HOPKINS
A. ASOK
Affiliation:
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA; asok@usc.edu
J. FASEL
Affiliation:
Institut Fourier - UMR 5582, Université Grenoble Alpes CS 40700, 38058 Grenoble Cedex 09, France; jean.fasel@gmail.com
M. J. HOPKINS
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA; mjh@math.harvard.edu

Abstract

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Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory
Secondary: 32L05: Holomorphic bundles and generalizations 55R25: Sphere bundles and vector bundles 13C10: Projective and free modules and ideals
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e6
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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