Mathematische Annalen

, Volume 349, Issue 4, pp 839–869

Two point extremal Gromov–Witten invariants of Hilbert schemes of points on surfaces


DOI: 10.1007/s00208-010-0542-2

Cite this article as:
Li, J. & Li, WP. Math. Ann. (2011) 349: 839. doi:10.1007/s00208-010-0542-2


Given an algebraic surface X, the Hilbert scheme X[n] of n-points on X admits a contraction morphism to the n-fold symmetric product X(n) with the extremal ray generated by a class βn of a rational curve. We determine the two point extremal GW-invariants of X[n] with respect to the class dβn for a simply-connected projective surface X and the extremal quantum first Chern class operator of the tautological bundle on X[n]. The methods used are vertex algebraic description of H*(X[n]), the localization technique applied to \({X=\mathbb P^2}\) and the results on quantum cohomology of X[n] when \({X=\mathbb C^2}\) in Okounkov and Pandharipande (Invent. Math. 179(3):523–557, 2010), and a generalization of the reduction theorem of Kiem-J. Li to the case of meromorphic 2-forms.

Mathematics Subject Classification (2000)

Primary 14C05 Secondary 14F43 14N35 17B69 

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsHKUSTKowloonHong Kong