Computations of Gromov-Witten invariants of toric varieties

TL;DR

This paper introduces a Julia package that simplifies the computation of Gromov-Witten invariants for smooth projective toric varieties using the Atiyah-Bott formula, with extensive examples and applications.

Contribution

The paper presents a new Julia package, ToricAtiyahBott.jl, enabling efficient computations of Gromov-Witten invariants for toric varieties, expanding computational tools in algebraic geometry.

Findings

The package supports common cohomological cycles and is extensible.

It provides detailed algorithms and numerous examples.

Applications demonstrate practical computations of invariants.

Abstract

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah-Bott formula on the moduli space of genus $0$ stable maps $\overline{M}_{0,m}(X,\beta)$ where $X$ is any smooth projective toric variety, and $\beta$ is any effective $1$-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety $X$, as well as the cohomology class $\beta$, must be defined using the package Oscar.jl.