The Witten top Chern class via \(K\)-theory. (English) Zbl 1117.14008

In the formulation of the E. Witten conjecture [in: Topological methods in modern mathematics. Proc. symp. hon. John Milnor’s sixtieth birthday, Stony Brook, USA 1991, 235–269 (1993; Zbl 0812.14017)] concerning the relation of higher KdV hierarchies to higher spin curves, the Witten top Chern class plays a crucial role.
In the paper the author provides a straightforward construction of the Witten top Chern class using \(K\)-theory. A. Polishchuk and A. Vaintrob [in: Advances in algebraic geometry motivated by physics. Proc. AMS spec. sess. Univ. Massa., Lowell, MA, USA 2000. Contemp. Math. 276, 229–249 (2001; Zbl 1051.14007)] gave an algebraic construction of this class. However the construction presented in the paper is simpler and allows one to avoid usage of bivariant intersection theory and Mac-Pherson graph construction. The paper is an extended version of the previous note by the author [in: Gromov-Witten theory of spin curves and orbifolds. AMS spec. sess., San Francisco, USA 2003. Contemp. Math. 403, 21–29 (2006; Zbl 1105.14032)].


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14H10 Families, moduli of curves (algebraic)
Full Text: DOI


[1] Dan Abramovich and Tyler J. Jarvis, Moduli of twisted spin curves, Proc. Amer. Math. Soc. 131 (2003), no. 3, 685 – 699. · Zbl 1037.14008
[2] Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27 – 75. · Zbl 0991.14007
[3] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101 – 145. · Zbl 0332.14003
[4] Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97 – 136 (French). · Zbl 0091.33004
[5] A. Chiodo, Stable twisted curves and their \( r\)-spin structures. Preprint: math.AG/ 0603687. · Zbl 1179.14028
[6] -, Higher spin curves and Witten’s top Chern class, Ph.D. thesis, University of Cambridge, 2003.
[7] Pierre Deligne, Cohomologie à supports propres, Théorie des topos et cohomologie étale des schémas. Tome 3, exposé XVII, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305.
[8] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[9] Mark L. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), no. 1, 155 – 159. · Zbl 0674.14005
[10] Mark L. Green, Koszul cohomology and geometry, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 177 – 200. · Zbl 0800.14004
[11] Tyler J. Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), no. 5, 637 – 663. · Zbl 1094.14504
[12] 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. · Zbl 1015.14028
[13] Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1 – 23. · Zbl 0756.35081
[14] John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. · Zbl 0959.55001
[15] Martin Olsson, On (log) twisted curves, Preprint: http://www.math.ias.edu/  molsson/Logcurves.pdf. · Zbl 1138.14017
[16] Alexander Polishchuk, Witten’s top Chern class on the moduli space of higher spin curves, Frobenius manifolds, Aspects Math., E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 253 – 264. · Zbl 1105.14010
[17] Alexander Polishchuk and Arkady Vaintrob, Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 229 – 249. · Zbl 1051.14007
[18] 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. · Zbl 0757.53049
[19] Edward Witten, 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. · Zbl 0812.14017
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