STABLE PAIR INVARIANTS OF SURFACES AND SEIBERG-WITTEN INVARIANTS

The moduli space of stable pairs on a local surface X = K-S is in general non-compact. The action of 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 of X. Sometimes there are no other contributions, for example, when the curve class beta is irreducible. We relate these surface stable pair invariants to the Poincare invariants of Durr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of S by the work of Durr-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.