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

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Abstract

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 β n of a rational curve. We determine the two point extremal GW-invariants of X [n] with respect to the class d β n 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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, USA

    Jun Li

  2. Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong Kong

    Wei-Ping Li

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  1. Jun Li
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  2. Wei-Ping Li
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Correspondence to Jun Li.

Additional information

J. Li was partially supported by an NSF grant DMS-0601002 and W.-P. Li was partially supported by the grant CERG601905 and GRF601808.

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Li, J., Li, WP. Two point extremal Gromov–Witten invariants of Hilbert schemes of points on surfaces. Math. Ann. 349, 839–869 (2011). https://doi.org/10.1007/s00208-010-0542-2

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  • DOI: https://doi.org/10.1007/s00208-010-0542-2

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