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?