The Gromov–Witten invariants of the Hilbert schemes of points on surfaces with pg > 0

Abstract

In this paper, we study the Gromov–Witten theory of the Hilbert schemes X[n] of points on a smooth projective surface X with positive geometric genus pg. For fixed distinct points x1, …, xn-1 ∈ X, let βn be the homology class of the curve {ξ + x2 + ⋯ + xn-1 ∈ X[n] | Supp (ξ) = {x1}}, and let βKX be the homology class of {x + x1 + ⋯ + xn-1 ∈ X[n] | x ∈ KX}. Using cosection localization technique due to Y. Kiem and J. Li, we prove that if X is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor, then all the Gromov–Witten invariants of X[n] defined via the moduli space [Formula: see text] of stable maps vanish except possibly when β is a linear combination of βn and βKX. When n = 2, the exceptional cases can be further reduced to the Gromov–Witten invariants: [Formula: see text] with [Formula: see text] and d ≤ 3, and [Formula: see text] with d ≥ 1. When [Formula: see text], we show that [Formula: see text] which is consistent with a well-known formula of C. Taubes. In addition, for an arbitrary surface X and d ≥ 1, we verify that [Formula: see text].