Gromov-Witten Invariant

A Gromov-Witten invariant is an intersection number on a moduli space of stable maps, evaluated against its virtual fundamental class. For a smooth projective complex target , genus , curve class , and marked cohomology insertions , the descendant invariant is

Here evaluates the map at the th marking, and is the first Chern class of its cotangent line. A stable map has a nodal marked source and only finitely many automorphisms preserving the map. The virtual class accounts for deformation obstructions and is used even when the moduli space is singular or has excess dimension.

The ordinary theory uses the usual virtual class without a reduced-obstruction correction. Its total descendant potential packages every genus and integral effective curve class. The Virasoro conjecture imposes differential equations on this generating function.