Characteristic classes of Hilbert schemes of points via symmetric products

Cappell, Sylvain¹ ; Maxim, Laurentiu² ; Ohmoto, Toru³ ; Schürmann, Jörg⁴ ; Yokura, Shoji⁵

¹ Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA
² Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA
³ Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Sapporo 060-0810, Japan
⁴ Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
⁵ Math. & Comp. Sci., Kagoshima Uni., 21-35 Korimoto 1-chome, Kagoshima 890-0065, Japan

Geometry & topology, Tome 17 (2013) no. 2, pp. 1165-1198.

Résumé

We obtain a formula for the generating series of (the push-forward under the Hilbert–Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of “virtual motives” of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi–Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture.

Détail

DOI : 10.2140/gt.2013.17.1165

Keywords: Hilbert scheme, symmetric product, generating series, power structure, Pontrjagin ring, motivic exponentiation, characteristic classes

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