Reduced classes and curve counting on surfaces II: calculations

We calculate the stable pair theory of a projective surface $S$. For fixed curve class $β\in H^2(S)$ the results are entirely topological, depending on $β^2$, $β.c_1(S)$, $c_1(S)^2$, $c_2(S)$, $b_1(S)$ \emph{and} invariants of the ring structure on $H^*(S)$ such as the Pfaffian of $β$ considered as an element of $Λ^2 H^1(S)^*$. Amongst other things, this proves an extension of the Göttsche conjecture to non-ample linear systems. We also give conditions under which this calculates the full 3-fold reduced residue theory of $K_S$. This is related to the reduced residue Gromov-Witten theory of $S$ via the MNOP conjecture. When the surface has no holomorphic 2-forms this can be expressed as saying that certain Gromov-Witten invariants of $S$ are topological. Our method uses the results of \cite{KT1} to express the reduced virtual cycle in terms of Euler classes of bundles over a natural smooth ambient space.

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