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: The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand-Diki{\u{\i\/}}\kern.15emhierarchies 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.