Invariants of Lagrangian surfaces

Original · EN

We define a nonnegative integer (L,L₀;ϕ) for a pair of diffeomorphic closed Lagrangian surfaces L₀,L embedded in a symplectic 4-manifold (M,) and a diffeomorphism ϕ∈+(M) satisfying ϕ(L₀)=L. We prove that if there exists ϕ∈+ₒ(M) with ϕ(L₀)=L and (L,L₀;ϕ)=0, then L₀,L are symplectomorphic. We also define a second invariant n(L₁,L₀;[Lₜ])=n(L₁,L₀,[ϕₜ]) for a smooth isotopy Lₜ=ϕₜ(L₀) between two Lagrangian surfaces L₀ and L₁ with (L₁,L₀;ϕ₁)=0, which serves as an obstruction of deforming Lₜ to a Lagrangian isotopy with L₀,L₁ preserved.