Quiver variety, generically symplectic

Question

Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $M(Q,\mathbf{d}) = \mathrm{Rep}(\Pi,\mathbf{d})//G_\mathbf{d}$ is generically symplectic. Here, $\Pi = k\bar{Q}/\langle\mathbf{w}\rangle$: the quotient of the path algebra of the double of $Q$, and $\mathbf{w} = \sum_{a\in Q}[a,a^*]$.

My questions are:

(1) What is the precise statement of being generically symplectic? I'm also not 100% sure about the definition of being symplectic in this setting. (It would be equipped with a closed 2-form on a smooth open subscheme, but what does it mean to be non-degenerate?)

(2) Where can we read off the proof of generic symplecticity from the original paper by Crawley-Boevey? Theorem 11.3.1 is attributed to this original paper and therefore the proof is omitted in C-E-G paper. I could find the proof of (i)-(iii) of Theorem; am I overlooking something?

Thank you.

Answer 1

The definition of non-degeneracy for a closed 2-form is given at the beginning of 4.2 of CB-E-G. It says that the map $\theta\mapsto i_\theta(\omega)$ is a bijection.

For $d\in\Sigma_0$ we know from CB that there is a smooth open dense subset of $M(Q,d)$ corresponding to simple $\Pi$-modules. This is Theorem 1.2, but see also Theorem 6.7.

Also, as mentioned at the beginning of 11.2 of CB-E-G, we have a natural bi-simplectic structure on Rep$(\bar Q,d)$, since we can identify this with the cotangent bundle of Rep$(Q,d)$. See Section 5.1.

I therefore understand this part (iv) of the theorem to be saying that the natural simplectic structure on Rep$(\bar Q,d)$ descends to a closed 2-form on $M(Q,d)$, which is non-degenerate on the smooth open dense subset corresponding to the simple $\Pi$-modules. I haven’t however checked the details.

I hope this helps.