Count of Genus Zero J-Holomorphic Curves in Dimensions Four and Six

TL;DR

This paper reviews genus zero Gromov-Witten invariants and explores their application in four and six-dimensional symplectic geometry, highlighting limitations in distinguishing certain symplectic structures.

Contribution

It demonstrates the restricted ability of genus zero Gromov-Witten invariants to differentiate symplectic structures on specific six-manifolds.

Findings

Gromov-Witten invariants reviewed and applied in 4D and 6D cases

Limitations shown in using these invariants to distinguish certain symplectic structures

Examples include products of minimal, simply connected 4-manifolds with S^2

Abstract

In this note, genus zero Gromov-Witten invariants are reviewed and then applied in some examples of dimension four and six. It is also proved that the use of genus zero Gromov-Witten invariants in the class of embedded $J$-holomorphic curves to distinguish the deformation types of symplectic structures on a smooth $6$-manifold is restricted in the sense that they can not distinguish the symplectic structures on $X_1\times S^2$ and $X_2\times S^2$ for two minimal, simply connected, symplectic $4$-manifolds $X_1$ and $X_2$ with $b_2^+(X_1)>1$ and $b_2^+(X_2)>1$.