Exposition: Enumerative Geometry and Tree-Level Gromov-Witten Invariants

TL;DR

This paper reviews the mathematical background of enumerative geometry and localization techniques, then discusses Gromov-Witten invariants and their computation, including recursive formulas for counting rational curves in projective spaces.

Contribution

It provides a comprehensive overview of Gromov-Witten invariants and demonstrates how to recover recursive formulas for enumerating rational curves, connecting differential topology with algebraic geometry.

Findings

Derivation of genus 0 Gromov-Witten potentials for projective spaces

Recovery of Kontsevich-Manin recursive formula for rational curves

Illustration of localization techniques in enumerative geometry

Abstract

Here we review background in differential topology related to the calculation of an euler characteristic, and background on localization in equivariant cohomology. We then outline Gromov-Witten invariants in algebraic geometry and give examples of the genus 0 Gromov-Witten potential for $\PP^1, \PP^2$, and a genus $g>0$ Riemann surface. Kontsevich-Manin's recursive formula for $N_d$, the number of degree $d$ rational curves through $3d-1$ points in general position on $\PP^2$ is recovered.