Journal of Symplectic Geometry

Volume 17 (2019)

Number 4

Real Gromov–Witten theory in all genera and real enumerative geometry: Properties

Pages: 1083 – 1158



Penka Georgieva (Institut de Mathématiques de Jussieu, Sorbonne Université, Paris, France)

Aleksey Zinger (Department of Mathematics, State University of New York, Stony Brook, N.Y., U.S.A.)


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.

The first author was partially supported by ERC grant STEIN-259118.

The second author was partially supported by NSF grants DMS 0846978 and 1500875.

Received 27 April 2017

Accepted 18 August 2018

Published 24 October 2019