Geometry & Topology

Tautological integrals on curvilinear Hilbert schemes

Gergely Bérczi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety X as a projective completion of the nonreductive quotient of holomorphic map germs from the complex line into X by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.

Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 2897-2944.

Dates
Received: 12 November 2015
Revised: 18 August 2016
Accepted: 11 November 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859277

Digital Object Identifier
doi:10.2140/gt.2017.21.2897

Mathematical Reviews number (MathSciNet)
MR3687110

Zentralblatt MATH identifier
06774936

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14N10: Enumerative problems (combinatorial problems) 55N91: Equivariant homology and cohomology [See also 19L47]

Keywords
Hilbert scheme of points curve counting Göttsche formula tautological integrals nonreductive quotients equivariant localisation iterated residue

Citation

Bérczi, Gergely. Tautological integrals on curvilinear Hilbert schemes. Geom. Topol. 21 (2017), no. 5, 2897--2944. doi:10.2140/gt.2017.21.2897. https://projecteuclid.org/euclid.gt/1510859277


Export citation

References

  • M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1–28
  • G Bérczi, Thom polynomials of Morin singularities and the Green–Griffiths–Lang conjecture, preprint (2010)
  • G Bérczi, B Doran, T Hawes, F Kirwan, Geometric invariant theory for graded unipotent groups and applications, preprint (2016)
  • G Bérczi, B Doran, T Hawes, F Kirwan, Projective completions of graded unipotent quotients, preprint (2016)
  • G Bérczi, F Kirwan, Graded unipotent groups and Grosshans theory, preprint (2015)
  • G Bérczi, A Szenes, Tautological integrals on Hilbert schemes and counting singular hypersurfaces, in preparation
  • G Bérczi, A Szenes, Thom polynomials of Morin singularities, Ann. of Math. 175 (2012) 567–629
  • N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, corrected 1st edition, Grundl. Math. Wissen. 298, Springer (2004)
  • N Berline, M Vergne, Zéros d'un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983) 539–549
  • R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982)
  • J-P Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, from “Algebraic geometry” (J Kollár, R Lazarsfeld, D R Morrison, editors), Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI (1997) 285–360
  • M Duflo, M Vergne, Orbites coadjointes et cohomologie équivariante, from “The orbit method in representation theory” (M Duflo, N V Pedersen, M Vergne, editors), Progr. Math. 82, Birkhäuser, Boston (1990) 11–60
  • D Edidin, W Graham, Characteristic classes in the Chow ring, J. Algebraic Geom. 6 (1997) 431–443
  • D Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer (1995)
  • G Ellingsrud, L Göttsche, M Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001) 81–100
  • L M Fehér, R Rimányi, Thom series of contact singularities, Ann. of Math. 176 (2012) 1381–1426
  • W Fulton, Intersection theory, Ergeb. Math. Grenzgeb. 2, Springer (1984)
  • W Fulton, Equivariant cohomology in algebraic geometry, Eilenberg lectures, Columbia University (2007) Available at \setbox0\makeatletter\@url https://people.math.osu.edu/anderson.2804/eilenberg/ {\unhbox0
  • T Gaffney, The Thom polynomial of $\overline{\Sigma \sp{1111}}$, from “Singularities, 1” (P Orlik, editor), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, RI (1983) 399–408
  • L Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998) 523–533
  • M Green, P Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, from “The Chern Symposium 1979” (H H Wu, S Kobayashi, I M Singer, A Weinstein, J Wolf, editors), Springer (1980) 41–74
  • M E Kazarian, Characteristic classes of singularity theory, from “The Arnold–Gelfand mathematical seminars” (V I Arnold, I M Gelfand, V S Retakh, M Smirnov, editors), Birkhäuser, Boston (1997) 325–340
  • M E Kazarian, Multisingularities, cobordisms, and enumerative geometry, Uspekhi Mat. Nauk 58 (2003) 29–88 In Russian; translated in Russian Math. Surveys 58 (2003) 665–724
  • S Kleiman, R Piene, Enumerating singular curves on surfaces, from “Algebraic geometry: Hirzebruch 70” (P Pragacz, M Szurek, J Wiśniewski, editors), Contemp. Math. 241, Amer. Math. Soc., Providence, RI (1999) 209–238
  • M Kool, V Shende, R P Thomas, A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011) 397–406
  • M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999) 157–207
  • J Li, Zero dimensional Donaldson–Thomas invariants of threefolds, Geom. Topol. 10 (2006) 2117–2171
  • A-K Liu, Family blowup formula, admissible graphs and the enumeration of singular curves, I, J. Differential Geom. 56 (2000) 381–579
  • A Marian, D Oprea, R Pandharipande, Segre classes and Hilbert schemes of points, preprint (2015)
  • V Mathai, D Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986) 85–110
  • E Miller, B Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer (2005)
  • H Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. 145 (1997) 379–388
  • J Rennemo, Universal polynomials for tautological integrals on Hilbert schemes, Geom. Topol. 21 (2017) 253–314
  • W Rossmann, Equivariant multiplicities on complex varieties, from “Orbites unipotentes et représentations, III”, Astérisque 173–174, Société Mathématique de France, Paris (1989) 313–330
  • A Szenes, Iterated residues and multiple Bernoulli polynomials, Internat. Math. Res. Notices (1998) 937–956
  • Y-J Tzeng, A proof of the Göttsche–Yau–Zaslow formula, J. Differential Geom. 90 (2012) 439–472
  • M Vergne, Polynômes de Joseph et représentation de Springer, Ann. Sci. École Norm. Sup. 23 (1990) 543–562