The primary goal of this appendix is to explain a more or less self-contained proof of local positivity of intersections for holomorphic curves. The exposition begins with a survey of the main results needed from elliptic regularity theory; here most of the proofs are only sketched, but an effort is made to include all of the important ideas and avoid unnecessary technical overhead (e.g., we do not need to use the Calderon–Zygmund inequality). This leads to a proof of the similarity principle, and the latter is the main tool needed for proving a local representation formula that may be viewed as a “weak version” of the Micallef–White theorem. This representation formula is then used to prove positivity of intersections and give a precise definition of the local singularity index for a nonconstant holomorphic curve in dimension 4.

This lecture concludes our survey of closed holomorphic curves with a discussion, in the first section, of local intersection numbers, positivity of intersections and the adjunction formula for closed holomorphic curves, and then, in the second section, with an explanation of how these figure into the proof of McDuff’s theorem on symplectic ruled surfaces. The last two sections then begin a shift in focus toward punctured holomorphic curves: this discussion starts with a general introduction to contact manifolds and their symplectic fillings and then continues by defining the moduli space of punctured asymptotically cylindrical holomorphic curves in a completed symplectic cobordism between contact manifolds.