The Witten top Chern class via $K$-theory
Author:
Alessandro Chiodo
Journal:
J. Algebraic Geom. 15 (2006), 681-707
DOI:
https://doi.org/10.1090/S1056-3911-06-00444-9
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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[AV02]AV Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75 (electronic).
[BFM75]BFM Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 101–145.
[BS58]BS Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136.
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[Gre89]GrKos ---, Koszul cohomology and geometry, Lectures on Riemann surfaces (Trieste, 1987), World Sci. Publishing, Teaneck, NJ, 1989, pp. 177–200.
[Jar00]Jageom Tyler J. Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), no. 5, 637–663.
[JKV01]JKV Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies, Compositio Math. 126 (2001), no. 2, 157–212.
[Kon92]Ko Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23.
[McC01]Mc01 John McCleary, A user’s guide to spectral sequences, second ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001.
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[PV01]PV A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (Providence, RI), Contemp. Math., vol. 276, Amer. Math. Soc., 2001, pp. 229–249.
[Wit91]Wi2 Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310.
[Wit93]Wir ---, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235–269.
Additional Information
Alessandro Chiodo
Affiliation:
Laboratoire J.-A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France
Email:
chiodo@math.unice.fr
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.
