20250131 Jesse Straat

We pick up from the previous talk to define Gromov–Witten invariants. First, we provide a short introduction of Chow groups, the algebraic cousin of (co-)homology, in the case of algebraic varieties. Using the definition of Deligne–Mumford stacks from the previous talk, we construct the Kontsevich (moduli) stack of stable maps on projective algebraic varieties over ℂ using nodal curves. We discuss the virtual fundamental class (without proof) in the Chow group and push it forward along an evaluation map to define Gromov–Witten invariants. These will be rational numbers that, in nice cases, correspond to the number of stable maps that satisfy some property. Finally, we will look at the situation of projective space and explicitly calculate the Gromov–Witten invariants.