## Duke Mathematical Journal

- Duke Math. J.
- Volume 166, Number 17 (2017), 3291-3347.

### Enumeration of real curves in $\mathbb{C}{\mathbb{P}}^{2n-1}$ and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

Penka Georgieva and Aleksey Zinger

#### Abstract

We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus $0$ real invariants with conjugate pairs of constraints to genus $0$ invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the $3$-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo $4$.

#### Article information

**Source**

Duke Math. J., Volume 166, Number 17 (2017), 3291-3347.

**Dates**

Received: 26 January 2016

Revised: 20 February 2017

First available in Project Euclid: 11 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1507687598

**Digital Object Identifier**

doi:10.1215/00127094-2017-0023

**Mathematical Reviews number (MathSciNet)**

MR3724219

**Zentralblatt MATH identifier**

1384.53070

**Subjects**

Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

**Keywords**

real Gromov–Witten invariants recursion enumeration of real curves

#### Citation

Georgieva, Penka; Zinger, Aleksey. Enumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants. Duke Math. J. 166 (2017), no. 17, 3291--3347. doi:10.1215/00127094-2017-0023. https://projecteuclid.org/euclid.dmj/1507687598