Characteristic classes of Hilbert schemes of points via symmetric products

TL;DR

This paper derives formulas for the generating series of Hirzebruch homology characteristic classes of Hilbert schemes of points on smooth varieties, extending to virtual motives and relating to the MNOP conjecture.

Contribution

It introduces a geometric construction of motivic exponentiation and provides new generating series formulas for characteristic classes of Hilbert schemes and symmetric products.

Findings

Formulas for generating series of Hirzebruch classes of Hilbert schemes

Extension of methods to virtual motives of threefolds

Connections to the dimension zero MNOP conjecture for Calabi-Yau threefolds

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.