# Image of a quiver variety under natural morphism

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:

1. $$\mathfrak{M}_1$$ is a stratified closed subspace of $$\mathfrak{M}_0$$ = closed union of strata
2. $$\mathfrak{M}_1$$ is an afine quiver variety
3. Moreover, it is an affine quiver variety for the same quiver, though possibly with different dimension vectors
4. 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.

1. Holds for general quiver varieties, as $$\pi$$ is a projective morphism so its image is a closed Poisson subvariety of $$\mathfrak{M}_0$$ so it is a closure of a stratum, which is then the closed union of that strata.
2. Holds for ADE quiver varieties as there $$\mathfrak{M}_0(\textbf{v},\textbf{w})=\sqcup_{\textbf{v}'\leq \textbf{v}} \mathfrak{M}_0^{reg}(\textbf{v}',\textbf{w}).$$ is a stratification, hence the image $$\mathfrak{M}_1$$ is a closure of $$\mathfrak{M}_0^{reg}(\textbf{v}',\textbf{w})$$ for some $$\textbf{v}',$$ hence $$\mathfrak{M}_0(\textbf{v}',\textbf{w}).$$
4. Holds for ADE quiver varieties as for them the action of the Weyl group $$w *_{\textbf{w}} \textbf{v}$$ (that is involved in Nakajima reflection functors) on the vector $$\textbf{v}$$ can get from an arbitrary $$\textbf{v}$$, via some $$w$$, to a dominant vector $$\textbf{v}'$$, that is a dimension vector for which $$\textbf{w}-C\textbf{v}\geq 0$$. This is not true for non-ADE quivers in general.