Stable maps to Looijenga pairs

Abstract: Start with a rational surface $Y$ admitting a decomposition of its anticanonical divisor into at least 2 smooth nef components. We associate 5 curve counting theories to this Looijenga pair: 1) all genus stable log maps with maximal tangency to each boundary component; 2) genus 0 stable maps to the local Calabi-Yau surface obtained by twisting $Y$ by the sum of the line bundles dual to the components of the boundary; 3) the all genus open Gromov-Witten theory of a toric Calabi-Yau threefold associated to the Looijenga pair; 4) the Donaldson-Thomas theory of a symmetric quiver specified by the Looijenga pair and 5) BPS invariants associated to the various curve counting theories. In this joint work with Pierrick Bousseau and Andrea Brini, we provide closed-form solutions to essentially all of the associated invariants and show that the theories are equivalent. I will start by describing the geometric transitions from one geometry to the other, then give an overview of the curve counting theories and their relations. I will end by describing how the scattering diagrams of Gross and Siebert are a natural place to count stable log maps.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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