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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 |
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Geometry & Topology 17 (2013) 1165–1198
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Abstract |
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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. |
Keywords
Hilbert scheme, symmetric product, generating series, power
structure, Pontrjagin ring, motivic exponentiation,
characteristic classes
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Mathematical Subject Classification 2000
Primary: 14C05, 55S15, 20C30
Secondary: 13D15, 32S35
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References |
Publication
Received: 15 April 2012
Revised: 30 October 2012
Accepted: 9 February 2013
Published: 18 May 2013
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Ralph Cohen
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Authors |
| Sylvain Cappell | |
| Courant Institute New York University 251 Mercer Street New York, NY 10012 USA |
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| http://www.math.nyu.edu/faculty/cappell/ | |
| Laurentiu Maxim | |
| Department of Mathematics University of Wisconsin Madison, WI 53706-1388 USA |
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| http://www.math.wisc.edu/~maxim/ | |
| Toru Ohmoto | |
| Department of Mathematics Hokkaido University Kita 10 Nishi 8 Sapporo 060-0810 Japan |
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| http://www.math.sci.hokudai.ac.jp/~ohmoto/ | |
| Jörg Schürmann | |
| Mathematische Institut Universität Münster Einsteinstr. 62 48149 Münster Germany |
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| http://wwwmath.uni-muenster.de/u/jschuerm/ | |
| Shoji Yokura | |
| Math. & Comp. Sci. Kagoshima Uni. 21-35 Korimoto 1-chome Kagoshima 890-0065 Japan |
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