### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1165-1198 |

Number of pages | 34 |

Journal | Geometry and Topology |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*17*(2), 1165-1198. https://doi.org/10.2140/gt.2013.17.1165

**Characteristic classes of hilbert schemes of points via symmetric products.** / Cappell, Sylvain; Maxim, Laurentiu; Ohmoto, Toru; Schürmann, Jörg; Yokura, Shoji.

Research output: Contribution to journal › Article

*Geometry and Topology*, vol. 17, no. 2, pp. 1165-1198. https://doi.org/10.2140/gt.2013.17.1165

}

TY - JOUR

T1 - Characteristic classes of hilbert schemes of points via symmetric products

AU - Cappell, Sylvain

AU - Maxim, Laurentiu

AU - Ohmoto, Toru

AU - Schürmann, Jörg

AU - Yokura, Shoji

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84878035128&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878035128&partnerID=8YFLogxK

U2 - 10.2140/gt.2013.17.1165

DO - 10.2140/gt.2013.17.1165

M3 - Article

VL - 17

SP - 1165

EP - 1198

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1364-0380

IS - 2

ER -