Scattering diagrams, sheaves, and curves

Abstract

We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $({\mathbb{P}}^2,E)$, where $E$ is a smooth cubic curve in the complex projective plane ${\mathbb{P}}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $({\mathbb{P}}^2,E)$ and the world of moduli spaces of coherent sheaves on ${\mathbb{P}}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on ${\mathbb{P}}^2$. 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.

Expository paper. 18 pages, 1 figure