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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On jets, extensions and characteristic classes II


Author: Helge Maakestad
Journal: Proc. Amer. Math. Soc. 141 (2013), 151-169
MSC (2010): Primary 14F10, 14F40
Published electronically: May 22, 2012
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Abstract: In this paper we define the generalized Atiyah classes $ c_{\mathcal {J}}(\mathcal {E})$ and $ c_{\mathcal {O}_X}(\mathcal {E})$ of a quasi-coherent sheaf $ \mathcal {E}$ with respect to a pair $ (\mathcal {I},d)$, where $ \mathcal {I}$ is a left and right $ \mathcal {O}_X$-module and $ d$ a derivation. We relate this class to the structure of left and right modules on the first order jet bundle $ \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})$. In the main result of the paper we show $ c_{\mathcal {O}_X}(\mathcal {E})=0$ if and only if there is an isomorphism $ \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{left} \cong \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{right}$ as $ \mathcal {O}_X$-modules. We also give explicit examples where $ c_{\mathcal {O}_X}(\mathcal {E})\neq 0$ using jet bundles of line bundles on the projective line. Hence the classes $ c_{\mathcal {J}}(\mathcal {E})$ and $ c_{\mathcal {O}_X}(\mathcal {E})$ are nontrivial. The classes we introduce generalize the classical Atiyah class.


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Additional Information

Helge Maakestad
Affiliation: Institut Fourier, 100 rue des maths, BP 74, 38402 St. Martin d’Hères cedex, France
Email: h{\textunderscore}maakestad@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11412-1
PII: S 0002-9939(2012)11412-1
Keywords: Atiyah sequence, jet bundle, characteristic class, generalized Atiyah class, square zero extension, lifting
Received by editor(s): January 11, 2011
Received by editor(s) in revised form: January 28, 2011, April 12, 2011, and June 13, 2011
Published electronically: May 22, 2012
Communicated by: Lei Ni
Article copyright: © Copyright 2012 American Mathematical Society