Enumeration of real curves in C P^2n-1 and a Witten-Dijkgraaf-Verlinde-Verlinde relation for real Gromov-Witten invariants

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.