The title of my question essentially explains what I am looking for, but let elaborate a bit, to put it in a more specific context.

There are quite a few papers, where the authors compute Gromov-Witten Invariants of a projective manifold $X$, where $X$ is non-compact. A very common example is where X is the total space of a vector bundle over some compact projective manifold. For example, $X$ could be the total space of $\mathcal{O}_{\mathbb{P}^2}(-1)\oplus \mathcal{O}_{\mathbb{P}^2}(-1)$. The Gromov-Witten Invariants of this space are computed in this paper by Klemm and Pandharipande (page 21)

https://arxiv.org/pdf/math/0702189.pdf

The basic idea of computing the invariants are to use virtual localization (as developed by Graber and Pandharipande in this paper https://arxiv.org/abs/alg-geom/9708001). I am not asking for details about how to use localization etc.

$\textbf{Question}$: I am looking for a reference (expository notes, book or paper) where the concept of GW invariants for non compact targets is explained (at least for genus zero). My main source of confusion with this concept is as follows: to define GW invariants, one considers the moduli space of stable maps (allowing nodal curves as well). One defines a topology on this space, called Gromov-Convergence (which is defined in chapter 5 of McDuff-Salamon's book on J-holomorphic curves). By Gromov-Compactness Theorem, the moduli space is compact, provided the target is a compact symplectic manifold. From that one can show it is possible to integrate on the compactified moduli space and get numbers (when the genus is greater than zero, this is more complicated, but never the less, we need the moduli space to be compact).

Of course, the entire thing I have said is a gross oversimplification (to do things precisely, one needs to understand the construction of Virtual Fundamental Class). However, as far as I am aware, we need the moduli space to be compact (and hence, the target manifold to be compact). How does one define the moduli space (and virtual fundamental class) when the target manifold is non compact?