Gromov–Witten theory of bicyclic pairs

A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov–Witten theory of bicyclic pairs. We establish correspondences with local Gromov–Witten theory and open Gromov–Witten theory in all genera, a correspondence with orbifold Gromov–Witten theory in genus zero, and correspondences between all-genus refined Gopakumar–Vafa invariants and refined quiver Donaldson–Thomas invariants. For self-nodal curves in P ( 1 , 1 , r ) P(1,1,r) we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments.