Question

Let a finite group $G\subset SL(n,\mathbb{C})$ act on $\mathbb{C}^{n}$ in a natural way. Assume there is a crepant resolution of $f:X\rightarrow \mathbb{C}^{n}/G$. When is it possible to write $X$ as $Y/G$ for some $Y$ and $G$-action on $Y$?

This is true for example $\pm id_{\mathbb{C}^{2}} \subset SL(2,\mathbb{C})$ acting on $\mathbb{C}^{2}$. Is it still true for example $$ \langle diag(e^{\frac{2\pi i}{3}},e^{\frac{2\pi i}{3}},e^{\frac{2\pi i}{3}})\rangle \cong \mathbb{Z}/3\mathbb{Z} \subset SL(n,\mathbb{C}) $$ acting on $\mathbb{C}^{3}$? FYI, this has toric crepant resolution.

Answer 1

By definition $C^n/G = Spec(C[x_1,\dots,x_n]^G)$ and $X$ being a blow up of an ideal $I$ on $C^n/G$ can be written as $$ X = Proj_{Spec(C[x_1,\dots,x_n]^G)}(C[x_1,\dots,x_n]^G \oplus I \oplus I^2 \oplus \dots). $$ Now assume the ideal $I \subset C[x_1,\dots,x_n]^G$ can be written as $$ I = J \cap C[x_1,\dots,x_n]^G = J^G, $$ where $J \subset C[x_1,\dots,x_n]$ is a $G$-invariant ideal (the simplest thing to do is to take $J = C[x_1,\dots,x_n]\cdot I$). Take $$ Y = Proj_{C[x_1,\dots,x_n]}(C[x_1,\dots,x_n] \oplus J \oplus J^2 \oplus\dots). $$ Then $Y/G = X$.