## Geometry & Topology

### Characteristic classes of Hilbert schemes of points via symmetric products

#### 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
Revised: 30 October 2012
Accepted: 9 February 2013
First available in Project Euclid: 20 December 2017

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

#### 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

#### References

• P Baum, W Fulton, R MacPherson, Riemann–Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. (1975) 101–145
• K Behrend, Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. 170 (2009) 1307–1338
• K Behrend, J Bryan, B Szendrői, Motivic degree zero Donaldson–Thomas invariants, Invent. Math. 192 (2013) 111–160
• K Behrend, B Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008) 313–345
• A A Beĭlinson, J Bernstein, P Deligne, Faisceaux pervers, from: “Analysis and topology on singular spaces I (Luminy, 1981)”, Astérisque 100, Soc. Math. France, Paris (1982) 5–171
• S Boissière, M A Nieper-Wisskirchen, Generating series in the cohomology of Hilbert schemes of points on surfaces, LMS J. Comput. Math. 10 (2007) 254–270
• L Borisov, A Libgober, Elliptic genera of singular varieties, Duke Math. J. 116 (2003) 319–351
• J-P Brasselet, J Schürmann, S Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010) 1–55
• S E Cappell, L Maxim, J Schürmann, J L Shaneson, Characteristic classes of complex hypersurfaces, Adv. Math. 225 (2010) 2616–2647
• S E Cappell, L G Maxim, J Schürmann, J L Shaneson, Equivariant characteristic classes of singular complex algebraic varieties, Comm. Pure Appl. Math. 65 (2012) 1722–1769
• S E Cappell, J L Shaneson, Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991) 521–551
• J Cheah, On the cohomology of Hilbert schemes of points, J. Algebraic Geom. 5 (1996) 479–511
• G Ellingsrud, S A Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987) 343–352
• W Fulton, S Lang, Riemann–Roch algebra, Grundl. Math. Wissen. 277, Springer, New York (1985)
• E Getzler, Mixed Hodge structures of configuration spaces
• E Gorsky, Adams operations and power structures, Mosc. Math. J. 9 (2009) 305–323, back matter
• L G öttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990) 193–207
• L G öttsche, W Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993) 235–245
• S M Gusein-Zade, I Luengo, A Melle-Hernández, A power structure over the Grothendieck ring of varieties, Math. Res. Lett. 11 (2004) 49–57
• S M Gusein-Zade, I Luengo, A Melle-Hernández, Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points, Michigan Math. J. 54 (2006) 353–359
• F Hirzebruch, Topological methods in algebraic geometry, 3rd edition, Grundl. Math. Wissen. 131, Springer New York (1966) Appendix and trans. from 2nd German edition by R L E Schwarzenberger, additional section by A Borel.
• M Kapranov, The elliptic curve in the $S$–duality theory and Eisenstein series for Kac–Moody groups
• E Looijenga, Motivic measures, from: “Séminaire Bourbaki 1999/2000”, Astérisque 276 (2002) 267–297
• R D MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974) 423–432
• R MacPherson, Characteristic classes for singular varieties, from: “Proceedings of the Ninth Brazilian Mathematical Colloquium (Poços de Caldas, 1973), Vol II (Portuguese)”, Inst. Mat. Pura Apl., São Paulo (1977) 321–327
• D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory I, Compos. Math. 142 (2006) 1263–1285
• B Moonen, Das Lefschetz–Riemann–Roch–Theorem für singuläre Varietäten, Bonner Mathematische Schriften 106, Universität Bonn Mathematisches Institut (1978) Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn
• M A Nieper-Wisskirchen, Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces, JP J. Geom. Topol. 8 (2008) 7–21
• T Ohmoto, Generating functions of orbifold Chern classes I: Symmetric products, Math. Proc. Cambridge Philos. Soc. 144 (2008) 423–438
• J Schürmann, Nearby cycles and characteristic classes of singular spaces, from: “Singularities in Geometry and Topology (Strasbourg 2009)”, (V Blanlœil, T Ohmoto, editors), IRMA Lectures in Mathematics and Theoretical Physics 20, European Math. Soc. (2012) 181–205
• S Yokura, A singular Riemann–Roch for Hirzebruch characteristics, from: “Singularities Symposium–-Łojasiewicz 70 (Kraków, 1996; Warsaw, 1996)”, Banach Center Publ. 44, Polish Acad. Sci., Warsaw (1998) 257–268