Ever since Gromov published his '85 paper, pseudo-holomorphic curves have been one of the main tools in symplectic topology to prove “rigidity” results.

Discussions related to holomorphic curves could be inspired by the following theorem in dimension 3.

Theorem (Novikov Reebless). Let M be a 3-manifold and let F be a Reebless foliation. Then

1. every leaf λ is π_1-injective; and
2. every transverse loop γ is essential in π_1.

Could some similar result be proved with holomorphic curves? Even if it is not possible to state such a result directly using only topological notions like fundamental groups or homology, maybe one could still define some type of Floer type invariants for symplectic foliations to get a generalization of the Novikov statement.

Should one consider holomorphic curves in the leaves of the foliation? Should one do the symplectization of the foliated manifold to study holomorphic curves there like in an SFT-style?