Real Gromov-Witten Theory in All Genera

The study of curves in projective varieties has been central to algebraic geometry since the nineteenth century. It was reinvigorated through its introduction into symplectic topology in Gromov’s seminal work [4] and now plays prominent roles in symplectic topology and string theory as well. The foundations of (complex) Gromov-Witten invariants, i.e. counts of J-holomorphic curves in symplectic manifolds, were established in the 1990s and have been spectacularly applied ever since. However, there has been much less progress in establishing the foundations of and applying real Gromov-Witten invariants, i.e. counts of J-holomorphic curves in symplectic manifolds preserved by anti-symplectic involutions. A real symplectic manifold is a triple (X,ω, φ) consisting of a symplectic manifold (X,ω) and an anti-symplectic involution φ. For such a triple, we denote by J φ ω the space of ω-compatible almost complex structures J on X such that φ∗J=−J . The fixed locus X of φ is then a Lagrangian submanifold of (X,ω) which is totally real with respect to any J ∈J φ ω . The basic example of a real Kahler manifold (X,ω, φ, J) is the complex projective space Pn−1 with the Fubini-Study symplectic form, the coordinate conjugation τn : Pn−1 −→ Pn−1, τn ( [z1, . . . , zn] ) = [ z1, . . . , zn ] ,