The survey is devoted to application of the technique of filling by holomorphic discs to different symplectic and complex analytic problems.

COMPLEX AND SYMPLECTIC RECOLLECTIONS

J-Convexity

Let X, J be an almost complex manifold of the real dimension 4 and Σ be an oriented hypersurface in X of the real codimension 1. Each tangent plane Tx(Σ), x ∈ Σ, contains a unique complex line ξx ⊂ Tx(Σ) which we will call a complex tangency to Σ at x. The complex tangency is canonically oriented and, therefore, cooriented. Hence the tangent plane distribution ξ on Σ can be defined by an equation α = 0 where the 1-form α is unique up to multiplication by a positive function. The 2-form dα ∣ξ is defined up to the multiplication by the same positive factor. We say that Σ is J-convex (or pseudo-convex) if dα(T, JT) > 0 for any non-zero vector T ∈ ξx, x ∈ Σ. We use the word “pseudo-convex” when the almost complex structure J is not specified.

An important property of a J-convex hypersurface Σ is that it cannot be touched inside (according to the canonical coorientation of Σ) by a J-holomorphic curve. In particular, if Ω is a domain in X bounded by a smooth J-convex boundary ∂Ω then all interior points of a J-holomorphic curve C ⊂ X with ∂C ⊂ ∂Ω belong to IntΩ. Moreover, C is transversal to ∂Ω in all regular points of its boundary ∂C.