Reduced classes and curve counting on surfaces II: calculations

TL;DR

This paper computes the stable pair theory for projective surfaces, revealing that it depends solely on topological invariants and extending the G"ottsche conjecture to non-ample systems, with implications for Gromov-Witten theory.

Contribution

It provides explicit calculations of stable pair invariants on surfaces, extending known conjectures and relating to reduced Gromov-Witten invariants via the MNOP conjecture.

Findings

Stable pair invariants depend only on topological data.

Extension of G"ottsche conjecture to non-ample linear systems.

Conditions for calculating the full 3-fold reduced residue theory.

Abstract

We calculate the stable pair theory of a projective surface $S$. For fixed curve class $\beta\in H^2(S)$ the results are entirely topological, depending on $\beta^2$, $\beta.c_1(S)$, $c_1(S)^2$, $c_2(S)$, $b_1(S)$ \emph{and} invariants of the ring structure on $H^*(S)$ such as the Pfaffian of $\beta$ considered as an element of $\Lambda^2 H^1(S)^*$. Amongst other things, this proves an extension of the G\"ottsche 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.