Reference request for Gromov-Witten Invariants of non compact manifolds

Question

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?

Answer 1

In general like you said - you can't integrate a cohomology class of a manifold if it is not compact. The idea is to "localize" the class to some compact subspace of your moduli space $M$, and integrate there. In particular you would like to find some compact subspace $i:Z\hookrightarrow M$ and a class $[M]^{vir}_{loc} \in A_*(Z)$ such that $i_*[M]^{vir}_{loc} = [M]^{vir}$. With this, it follows that $deg([M]^{vir}_{loc}) = deg([M]^{vir})$. When the RHS is not defined, you can take the LHS as your definition.

It's similar to the notion of localized Euler class as discussed in Fulton's book on intersection theory. Given a section $\sigma$ of a vector bundle, he produces a localized Euler class supported on the zero locus $Z(\sigma))$, where if you take the pushforward to your ambient space you recover the usual Euler class.

Torus localization gives you one way of doing this where you localize your virtual class to your fixed point locus. Another way is to do cosection localization, where there is a cosection of your obstruction sheaf and you can localize your virtual class to the degeneracy locus of the cosection. In fact one quick application of this is that if the cosection is surjective, then you can immediately deduce that the virtual class is $0$.

I don't know a reference that addresses this specifically - but here are some examples where one starts with a non-compact moduli space and gets numbers out of them using localization techniques - Jiang and Thomas on defining virtual Euler characteristics -https://arxiv.org/pdf/1408.2541.pdf, Kiem Li's cosection localization - https://arxiv.org/pdf/1007.3085.pdf, Chang Li's moduli space of p-fields - https://arxiv.org/abs/1101.0914.