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.

The first lecture introduces the study of closed symplectic manifolds via closed pseudoholomorphic curves, focusing in particular on McDuff’s famous theorem from 1990 that characterizes symplectic ruled surfaces via symplectically embedded spheres. The theorem is stated in a slightly more modern formulation using Lefschetz fibrations, and a proof modulo of some technical lemmas is sketched. The topic of intersection theory is then introduced by considering the natural question of what kinds of holomorphic curves can be proven to form leaves of smooth foliations by holomorphic curves.