Duality and universality for stable pair invariants of surfaces

M. Kool

Published 2013-03-21, updated 2014-11-28Version 2

Let $\beta$ be a curve class on a surface $S$. The moduli space of stable pairs on $S$ with class $\beta$ carries a natural reduced virtual cycle. This cycle is defined when $h^2(L) = 0$ for any effective $L \in \mathrm{Pic}^\beta(S)$ (weak assumption). When $h^2(L) = 0$ for any $L \in \mathrm{Pic}^{\beta}(S)$ (strong assumption), the associated invariants are given by universal functions in $\beta^2$, $\beta.c_1(S)$, $c_1(S)^2$, $c_2(S)$, and certain invariants of the ring structure of $H^*(S,\mathbb{Z})$. In this paper, we show the following. (1) Universality no longer holds when just the weak assumption is satisfied. (2) For any $S,\beta$ (no conditions), the BPS spectrum of the non-reduced stable pair invariants of $S,\beta$ with maximal number of point insertions consists of a single number. This number is the Seiberg-Witten invariant of $S, \beta$. (3) The GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence in certain cases, e.g. when $\beta$ is irreducible. (4) When $p_g(S)=0$, the difference between the stable pair invariants in class $\beta$ and $K_S-\beta$ is universal. The results of this paper follow from relating certain stable pairs invariants of surfaces to the Poincar\'e invariants of M. D\"urr, A. Kabanov, and Ch. Okonek.