20241219 Cisca Kalmijn

In part I we already set up some theory by looking at pseudoholomorphic curves and their moduli spaces; we also compactified the space of pseudoholomorphic maps using stable maps. Now we are interested in quotienting out the reparametrisation group, which gives us a new moduli space to study. We extend our notions of simple curves and regular almost complex structures to stable maps.

We use these notions to impose the structure of a finite-dimensional manifold on the new moduli space, for which we must require that our target manifold is semipositive as well. Then we look at pseudocycles and their product structure, which allow us to define Gromov–Witten invariants. We conclude by considering some easy cases where we can actually calculate the invariants.