The Gromov-Witten invariants of the Hilbert schemes of points on surfaces with $p_g>0$

In this paper, we study the Gromov-Witten theory of the Hilbert schemes X^{[n]} of points on smooth projective surfaces X with positive geometric genus p_g. Using cosection localization technique due to Y. Kiem and J. Li [KL1, KL2], 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 $\Mbar_{g, r}(X^{[n]}, \beta)$vanishexceptpossiblywhen$\beta = d_0 \beta_{K_X} - d \beta_n$ where d is an integer, $d_0\ge 0$ is a rational number, and ${\beta }_n$ and ${\beta }_{K_X}$ are defined in (3.2) and (3.3) respectively. When $n=2$, the exceptional cases can be further reduced to the invariants: $<1>{0, \beta{K_X} - d\beta_2}^{X^{[2]}}$with$K_X^2 = 1$and$d \le 3$,and$<1>_{1, d\beta_2}^{X^{[2]}}$with$d \ge 1$\mathrm{.}Weshowthatwhen$K_X^2 = 1$,$$<1{>}_{0,{\beta }_{K_X}-3{\beta }_2}^{X^{[2]}}=(-1{)}^{\chi ({\mathcal{O}}_X)}$$ which is consistent with a well-known formula of Taubes [Tau]. In addition, for an arbitrary smooth projective surface X and $d\ge 1$, we verify that $$<1{>}_{1,d{\beta }_2}^{X^{[2]}}=K_X^2\mathrm{/}(12d)\mathrm{.}$$