Sayan Chattopadhyay

Well so do I. From what I understand it is some funny business about how moduli spaces of stable maps might in general be too big/singular and have arbitrary components. So in order to get enumerative invariants out you want to integrate classes against fundamental classes on the moduli space of stable maps. This would require you to define a "correct" notion of fundamental class on this usually horrible space.

This is usually done by hoping for a fantasy. That is "locally" embedding your moduli space into some smooth projective variety, looking at some vector bundle such that some section…

Okay so here is what is happening: Let $X \subset \Bbb{P}^n$ be a projective submanifold and let $G$ be a reductive algebraic group action on $X$ and consider a representation of $G$ on $GL(n+1,\Bbb{C})$. Let $K \subset G$ be its maximal compact subgroup.
A point $p \in X$ is said to be polystable under $G$ (Mumford) if $G \cdot \tilde{p}$ is closed where $G$ acts on a lift of $p$ in $\Bbb{C}^{n+1}$ (this is called a linearization of the action to a line bundle (in this case O(1)). Now the Kempf-Ness theorem states the following: