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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The Witten top Chern class via $K$-theory

Author: Alessandro Chiodo
Journal: J. Algebraic Geom. 15 (2006), 681-707
Published electronically: May 2, 2006
MathSciNet review: 2237266
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Abstract | References | Additional Information

Abstract: The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In the paper by Polishchuk and Vaintrob (Contemp. Math., vol. 276, Amer. Math. Soc., 2001, pp. 229–249) an algebraic construction of such a class is provided. We present a more straightforward construction via $K$-theory. In this way we short-circuit the passage through bivariant intersection theory and the use of MacPherson’s graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the $K$-theory ring.

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Alessandro Chiodo
Affiliation: Laboratoire J.-A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France

Received by editor(s): February 25, 2005
Received by editor(s) in revised form: May 11, 2005, and December 1, 2005
Published electronically: May 2, 2006
Additional Notes: Supported by the Istituto Nazionale di Alta Matematica, and the Marie Curie Intra-European Fellowship within the sixth European Community Framework Programme, MEIF-CT-2003-501940.