Real Gromov-Witten Theory in All Genera and Real Enumerative Geometry: Properties

TL;DR

This paper investigates the properties of positive-genus real Gromov-Witten invariants in odd-dimensional symplectic manifolds, focusing on orientation compatibility and foundational aspects essential for their computation and comparison with enumerative geometry.

Contribution

It establishes the orientation properties of moduli spaces of real maps, ensuring their compatibility with Gromov-Witten theory's standard constructions, enabling further applications.

Findings

Orientation compatibility with node-identifying immersions confirmed

Comparison of different orientation methods in special cases

Foundational results for computing and relating real Gromov-Witten invariants

Abstract

The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working with these invariants. We determine the compatibility of the orientations on the moduli spaces of real maps constructed in the first part with the standard node-identifying immersion of Gromov-Witten theory. We also compare these orientations with alternative ways of orienting the moduli spaces of real maps that are available in special cases. In a sequel, we use the properties established in this paper to compare real Gromov-Witten and enumerative invariants, to describe equivariant localization data that computes the real Gromov-Witten invariants of odd-dimensional projective spaces, and to establish vanishing results for these invariants in the spirit of Walcher's predictions.