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?


Any quiver variety where all the v_i are 1's is a hypertoric variety. 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.

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    $\begingroup$ Thanks! Though I was (maybe non-modestly) hoping for some answer outside HT varieties :) $\endgroup$ – Filip92 Oct 26 '18 at 20:14
  • $\begingroup$ Sure. I just had to spread the gospel :-P $\endgroup$ – Justin Hilburn Oct 26 '18 at 21:09

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