Gromov-Witten theory of bicyclic pairs
Abstract
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) 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.