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

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 $\overline{\mathfrak M}_{g, r}(X^{[n]}, \beta)$ 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: $\langle 1 \rangle_{0, \beta_{K_{X}}-d\beta_2}^{X^{[2]}}$ with $K_{X}^{2} = 1$ and d ≤ 3, and $\langle 1 \rangle_{1, d\beta_2}^{X^{[2]}}$ with d ≥ 1. When $K_{X}^{2} = 1$, we...