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Characteristic classes of Hilbert schemes of points via symmetric products

Sylvain Cappell, Laurentiu Maxim, Toru Ohmoto, Jörg Schürmann and Shoji Yokura

Geometry & Topology 17 (2013) 1165–1198

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.

Hilbert scheme, symmetric product, generating series, power structure, Pontrjagin ring, motivic exponentiation, characteristic classes
Mathematical Subject Classification 2000
Primary: 14C05, 55S15, 20C30
Secondary: 13D15, 32S35
Received: 15 April 2012
Revised: 30 October 2012
Accepted: 9 February 2013
Published: 18 May 2013
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Ralph Cohen
Sylvain Cappell
Courant Institute
New York University
251 Mercer Street
New York, NY 10012
Laurentiu Maxim
Department of Mathematics
University of Wisconsin
Madison, WI 53706-1388
Toru Ohmoto
Department of Mathematics
Hokkaido University
Kita 10 Nishi 8
Sapporo 060-0810
Jörg Schürmann
Mathematische Institut
Universität Münster
Einsteinstr. 62
48149 Münster
Shoji Yokura
Math. & Comp. Sci.
Kagoshima Uni.
21-35 Korimoto 1-chome
Kagoshima 890-0065