Let $\pi: \mathfrak{M}\_{(\theta,0)}(Q,\text{v},\text{w}) \rightarrow \mathfrak{M}\_{(0,0)}(Q,\text{v},\text{w})$ be the projective morphism between Nakajima quiver varieties when complex moment parameter is equal to zero $\zeta\_\mathbb{C}=0.$ We define the **Lagrangian core** to be $\Lambda\_{\theta}=\pi^{-1}(0).$


Now, when one varies the GIT-chamber of stability parameter $\theta$ (for rep theorists) $\leftrightarrow$ real moment map parameter (for hyperkähler geometers), it is expected that the corresponding core change in principle. Does anyone have examples of such different cores?



Lagrangian cores of quiver variety in different GIT chambers
Any quiver variety where all the v\_i are 1's is a [hypertoric variety](https://arxiv.org/abs/0705.4236). They are determined combinatorially by an arrangement of affine hyperplanes and one can compute the core by looking at the compact chambers of the arrangement. Changing the git parameters corresponds to translating the hyperplanes. By drawing a few pictures it is easy to see that the core changes.


