Stable pair invariants of surfaces and Seiberg-Witten invariants

Abstract

The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $\mathbb{C}^*$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section $S \subset X$. Sometimes there are no other contributions, e.g. when the curve class $\beta$ is irreducible. We relate these surface stable pair invariants to the Poincar\'e invariants of D\"urr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of $S$ by work of D\"urr-Kabanov-Okonek and Chang-Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence for $S$. (2) When $p_g(S) = 0$, the difference of surface stable pair invariants in class $\beta$ and $K_S - \beta$ is a universal topological expression.