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.

This lecture presents the technical asymptotic results underlying Siefring’s intersection theory for punctured holomorphic curves, including the necessary prerequisites on asymptotic operators and the relations proved by Hofer, Wysocki and Zehnder between winding numbers of asymptotic eigenfunctions and the Conley–Zehnder index. Siefring’s relative asymptotic formulas are stated largely without proof but are motivated in terms of an asymptotic analogue of the similarity principle. The last section then discusses the punctured analogue of the question about holomorphic foliations considered in Lecture 2, which motivates the definition of the normal Chern number for punctured holomorphic curves.

Using the relative asymptotic results of the previous lecture as a black box, this lecture explains the main definitions and results of Siefring’s intersection theory for punctured holomorphic curves. This includes the definition of a homotopy-invariant algebraic intersection number (the so-called “star-pairing”), the notion of hidden intersections at infinity and asymptotic positivity of intersections, and a punctured generalization of the adjunction formula.