2008/08/18 by Her, Hai-Long
#53D05 #53D12 #53D45 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.0808.2228
Let (X,ω) be a compact symplectic manifold, L be a Lagrangian submanifold and V be a codimension 2 symplectic submanifold of X, we consider the pseudoholomorphic maps from a Riemann surface with boundary (Σ,∂Σ) to the pair (X,L) satisfying Lagrangian boundary conditions and intersecting V. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If L∩ V=∅, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution ϕ on X such that L is the fixed point set of ϕ and V is ϕ-anti-invariant, then we define the so-called "relatively open" invariants for the tuple (X,ω,V,ϕ) if L is orientable and dimX≤ 6. If L is nonorientable, we define such invariants under the condition that dimX≤4 and some additional restrictions on the number of marked points on each boundary component of the domain.