Scattering Diagrams, Sheaves, and Curves

Abstract

We review the recent proof of the N. Takahashi’s conjecture on genus 0 Gromov-Witten invariants of (ℙ2, E), where E is a smooth cubic curve in the complex projective plane ℙ2. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of (ℙ2, E) and the world of moduli spaces of coherent sheaves on ℙ2. Using this bridge, the N. Takahashi’s conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on ℙ2. This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9–12 September 2019.