Geometry & Topology

Characteristic classes of Hilbert schemes of points via symmetric products

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

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 17, Number 2 (2013), 1165-1198.

Dates
Received: 15 April 2012
Revised: 30 October 2012
Accepted: 9 February 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732574

Digital Object Identifier
doi:10.2140/gt.2013.17.1165

Mathematical Reviews number (MathSciNet)
MR3070522

Zentralblatt MATH identifier
1318.14008

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 55S15: Symmetric products, cyclic products 20C30: Representations of finite symmetric groups
Secondary: 13D15: Grothendieck groups, $K$-theory [See also 14C35, 18F30, 19Axx, 19D50] 32S35: Mixed Hodge theory of singular varieties [See also 14C30, 14D07]

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

Citation

Cappell, Sylvain; Maxim, Laurentiu; Ohmoto, Toru; Schürmann, Jörg; Yokura, Shoji. Characteristic classes of Hilbert schemes of points via symmetric products. Geom. Topol. 17 (2013), no. 2, 1165--1198. doi:10.2140/gt.2013.17.1165. https://projecteuclid.org/euclid.gt/1513732574


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