Filling by holomorphic discs and its applications

Summary

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 T_x(Σ), x ∈ Σ, contains a unique complex line ξ_x ⊂ T_x(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 ∈ ξ_x, 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.