Enumeration of real curves in CP2n−1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

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 $4$.