20241213 Jesse Straat

In this first part of a journey of constructing Gromov–Witten invariants, we set the stage by defining the necessary concepts used in the construction. We start by discussing the shortcomings of the classical concept of moduli spaces — in particular the absence of a fundamental class — and sketch how this provides an obstruction to defining Gromov–Witten invariants. It turns out that there is a concept of virtual fundamental classes on generalisations of spaces called Deligne–Mumford stacks. We define stacks in the context of complex geometry without relying too much on categorical concepts such as limits. Given enough time, we will discuss how stacks are applied in moduli problems and use vector bundles as an example. In the next part, the developed theory will be used to discuss the moduli problem of stable maps and define Gromov–Witten invariants.