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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 $\Mbar_{g, r}(X^{[n]}, β)$ vanish except possibly when $β= d_0 β_{K_X} - d β_n$ where d is an integer, $d_0 \ge 0$ is a rational number, and $β_n$ and $β_{K_X}$ are defined in (3.2) and (3.3) respectively. When $n=2$, the exceptional cases can be further reduced to the invariants: $<1>_{0, β_{K_X} - dβ_2}^{X^{[2]}}$ with $K_X^2 = 1$ and $d \le 3$, and $<1>_{1, dβ_2}^{X^{[2]}}$ with $d \ge 1$. We show that when $K_X^2 = 1$, $$<1>_{0, β_{K_X} - 3 β_2}^{X^{[2]}} = (-1)^{χ(\mathcal O_X)}$$ 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 $$<1>_{1, dβ_2}^{X^{[2]}} = K_X^2/(12d).$$