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

TL;DR

This paper investigates the Gromov-Witten invariants of Hilbert schemes of points on surfaces with positive geometric genus, proving vanishing results and explicit formulas for certain cases using cosection localization techniques.

Contribution

It provides new vanishing theorems and explicit calculations for Gromov-Witten invariants of Hilbert schemes on surfaces with p_g > 0, extending previous understanding.

Findings

Most Gromov-Witten invariants vanish except for specific classes.

Explicit formula for invariants when K_X^2 = 1 and d=3.

Verification of a known Taubes formula for certain invariants.

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 $\Mbar_{g, r}(X^{[n]}, \beta)$ 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 $$<1>_{1, d\beta_2}^{X^{[2]}} = K_X^2/(12d).$$