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

Abstract

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 $\beta =d_0{\beta }_{K_X}-d{\beta }_n$ 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 $d\ge 1$. We show that when $K_X^2=1$, $$<1{>}_{0,{\beta }_{K_X}-3{\beta }_2}^{X^{[2]}}=(-1{)}^{\chi ({\mathcal{O}}_X)}$$ 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 $$<1{>}_{1,d{\beta }_2}^{X^{[2]}}=K_X^2/(12d).$$

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