Counting genus zero real curves in symplectic manifolds

TL;DR

This paper develops a new framework for counting genus zero real curves in symplectic manifolds, combining two types of $J$-holomorphic spheres and using localization techniques to analyze their invariants.

Contribution

It introduces a unified approach to count real genus zero curves by studying moduli spaces of both fixed point and non-fixed point spheres, extending existing theories.

Findings

Invariants are consistent across different involutions on projective space.

Moduli spaces for non-fixed point spheres are constructed and analyzed.

Localization shows invariants are essentially the same for standard involutions.

Abstract

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on $\mathbb{P}^{4n-1}$.