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

Reginald Anderson

TL;DR

The paper surveys the construction and computation of tree-level Gromov–Witten invariants via stable maps, localization in equivariant cohomology, and Schubert-calculus techniques, beginning with classical enumerative geometry (e.g., $27$ lines on a cubic surface and $2875$ lines on a quintic threefold) to motivate the GW framework. It develops the genus-0 theory for $\mathbb{P}^1$ and $\mathbb{P}^2$, derives the small and big quantum cohomology rings, and derives a recursive Kontsevich–Manin formula for the degree $d$ invariants $N_d$, all through localization and the WDVV equations. The work presents explicit generating functions for the GW potentials, e.g., for $\mathbb{P}^1$ the big potential is $\Phi(\gamma)=\tfrac{1}{2}t_0^2t_1 + e^{t_1} q^{\ell}$, and for $\mathbb{P}^2$ the potential encodes the $N_d$-dependent contributions to $QH^*(\mathbb{P}^2)$. Beyond toric examples, the manuscript outlines the genus-$g$-surface potential and discusses the interplay between GW invariants and mirror symmetry, paving the way for generalizations to other spaces via localization and quantum cohomology structures.

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.

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

TL;DR

The paper surveys the construction and computation of tree-level Gromov–Witten invariants via stable maps, localization in equivariant cohomology, and Schubert-calculus techniques, beginning with classical enumerative geometry (e.g., lines on a cubic surface and lines on a quintic threefold) to motivate the GW framework. It develops the genus-0 theory for and , derives the small and big quantum cohomology rings, and derives a recursive Kontsevich–Manin formula for the degree invariants , all through localization and the WDVV equations. The work presents explicit generating functions for the GW potentials, e.g., for the big potential is , and for the potential encodes the -dependent contributions to . Beyond toric examples, the manuscript outlines the genus--surface potential and discusses the interplay between GW invariants and mirror symmetry, paving the way for generalizations to other spaces via localization and quantum cohomology structures.

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 , and a genus Riemann surface. Kontsevich-Manin's recursive formula for , the number of degree rational curves through points in general position on is recovered.
Paper Structure (24 sections, 2 theorems, 163 equations, 9 figures)

This paper contains 24 sections, 2 theorems, 163 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Warm-up and Review
  3. Chern class argument for 27 lines on non-singular cubic surface
  4. Differential Geometry Overview and Equivariant cohomology
  5. Differential Geometry Overview
  6. Equivariant Cohomology
  7. Schubert Cell Decomposition
  8. $\mathbb{P}^n$ (Euler characteristic and Schubert cell decomposition)
  9. $Gr_\mathbb{C}(k,n)$
  10. $Gr(1,3)$
  11. $Gr(2,5)$
  12. $Gr_\mathbb{C}(2,6)$
  13. Complete flag variety
  14. Localization argument for 27 lines on a nonsingular cubic surface $X$
  15. Localization argument for 2875 lines on a generic nonsingular quintic threefold $Q \subset \mathbb{P}^4$
  16. ...and 9 more sections

Key Result

Lemma 2.2

For $V$ a $4$-dimensional vector space, with $\mathcal{S} \subset V \otimes \mathcal{O}_\mathbb{G}$ the tautological rank $2$ subbundle on $\mathbb{G}=\mathbb{G}r(1,\mathbb{P}(V))$ of lines in $\mathbb{P} V \cong \mathbb{P}^3$. A form $f$ of degree $d=3$ on $\mathbb{P} V$ gives rise to a global sect in $A(\mathbb{G}r(1,\mathbb{P}^1))$.

Figures (9)

  • Figure 1: Macaulay2 output for computing 2875 lines on the quintic threefold
  • Figure 2: A localization in equivariant cohomology argument implies that $\chi(S^2)=2$.
  • Figure 3: Building the Morse complex on $S^2$
  • Figure 4: Computing the Morse complex for a homotopy deformation of $S^2$
  • Figure 5: Schubert cell decomposition for $\mathbb{P}^1$
  • ...and 4 more figures

Theorems & Definitions (25)

  • Remark 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.4
  • proof
  • proof
  • proof
  • Definition 3.1
  • Definition 3.3
  • Definition 3.4
  • ...and 15 more