We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily surjective, so denote by $\mathfrak{M}_1$ its image.

E.g. picking the $A_1$ Dynkin quiver, standard stability condition $\theta>0,$ and dimension vectors $\mathbf{v}=4,\mathbf{w}=6,$ one gets the smooth quiver variety $\mathfrak{M}=T^*Gr(4,6)$ being cotangent bundle of Grassmanian, whereas $$\mathfrak{M}_1(A_1,4,6)=\overline{\mathcal{O}_{2211}}\subsetneq \mathfrak{M}_0(A_1,4,6)=\overline{\mathcal{O}_{222}},$$ are closures of nilpotent orbits in $\mathfrak{sl}_6.$ Also, using Nakajima reflection functors one can pass to the surjective setup. Namely, we have $\mathfrak{M}_{\theta>0}(A_1,4,6)\cong \mathfrak{M}_{\theta<0}(A_1,2,6),$ whereas both $\mathfrak{M}_0(A_1,2,6)$ and $\mathfrak{M}_1(A_1,2,6)$ are equal to $\overline{\mathcal{O}_{2211}}.$ The summary of these two examples is that:

- $\mathfrak{M}_1$ is a stratified closed subspace of $\mathfrak{M}_0$ = closed union of strata
- $\mathfrak{M}_1$ is an afine quiver variety
- Moreover, it is an affine quiver variety for the same quiver, though possibly with different dimension vectors
- Using Nakajima reflection functors, we can get to the setup where $\mathfrak{M}_1$ and $\mathfrak{M}_0$ coincide.

The question is, whether any of 1-4 holds in general, or at least for ADE Dynkin quiver varieties?

NB We assume that the complex moment parameter is equal to zero.