Enumeration of real curves in CP^{2n-1} and a WDVV 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 WDVV-type relation for 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 non-trivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the three-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo 4.

45 pages, 7 figures, 3 tables; exposition revised in Sections 1 and 2; omitted proofs of corollaries added

References in corpus (2)

• Counting genus zero real curves in symplectic manifolds
• The orientability problem in open Gromov-Witten theory

Cited by in corpus (8)

• Counting genus zero real curves in symplectic manifolds
• Relative quantum cohomology
• Point-like bounding chains in open Gromov-Witten theory
• WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6
• Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants
• The cyclic open-closed map, u-connections and R-matrices
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
• A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants