Question

I read that in dimension $\geq 4$ there are Gorenstein abelian quotient singularities that have no crepant resolutions. What is the simplest example? I immagine that there should be toric examples. Is it the case that the cone does not admit a suitable subdivision in simplicial cones, corresponding to a crepant resolution? In your answers, please consider that I am only an amateur algebraic geometer, but I know some toric geometry.

Answer 1

[EDIT: added proof that $\mathbb Q$-factoriality implies that the exceptional set of any resolution is a divisor.]

Definition A variety is called $\mathbb Q$-factorial if every Weil divisor on it is $\mathbb Q$-Cartier, i.e., some multiple of it is a Cartier divisor.

The general statement you might want is that

Claim A $\mathbb Q$-factorial, terminal singularity does not admit a non-trivial crepant resolution.

Proof $\mathbb Q$-factoriality implies that the exceptional set of any resolution is a divisor, and being terminal implies that all the discrepancies are positive. $\square$

Remark One might think that a Gorenstein terminal singularity does not admit a non-trivial crepant resolution. Here is an example that this is not true. Consider a cone over a smooth quadric surface in $\mathbb P^3$. This is a hypersurface in $\mathbb A^4$, so it is clearly Gorenstein. Blowing up the vertex and a simple calculation using adjunction shows that this is a terminal singularity. However, blowing up a divisor that is a cone over a line on the quadric surface gives a small resolution which will be crepant by being an isomorphism in codimension $1$. This shows that it is necessary to add the $\mathbb Q$-factoriality condition for the above Claim.

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Example For the example in Jim Bryan's answer, $\mathbb C^4/\pm$, or more generally, $\mathbb C^{m}/\pm$, the point to notice is that this is just the cone over the Veronese embedding of $\mathbb P^{m-1}$. Isolated quotient singularities are $\mathbb Q$-factorial and an easy computation shows that the discrepancy of the single exceptional divisor of the blow up of the vertex is $\dfrac m2-1$. This implies that it is terminal as soon as $m>2$, but for $m$ odd it will not be Gorenstein, so the first example of the desired kind is for $m=4$.

Addendum Here is a proof that $\mathbb Q$-factoriality implies that the exceptional set of any resolution is a divisor:

Claim Let $X$ be a $\mathbb Q$-factorial variety and $f:Y\to X$ a proper birational morphism. Let $E=\mathrm{Exc}(f)$ denote the exceptional set of $f$, i.e., the largest (closed) subset of $Y$ such that $f|_{Y\setminus E}:Y\setminus E\to X\setminus f(E)$ is an isomorphism. Then $E$ is of pure codimension $1$ in $Y$.

Proof Let $y\in E$ and suppose that $\mathrm{codim}_YE\geq 2$ in a neighborhood of $y$. Let $C\subseteq E$ be an arbitrary proper curve such that $f(C)$ is a point and $y\in C$ and let $H\subseteq Y$ be an effective divisor such that $y\in H$, but $C\not\subseteq H$. This implies that $H\cdot C>0$. Consider the Weil(!) divisor $f_*H$ on $X$ (the push-forward is meant as a cycle). As $X$ is $\mathbb Q$-factorial, some multiple of $f_*H$ will be Cartier, so replacing $H$ with that multiple we may assume that actually $f_*H$ is Cartier. Then it makes sense to pull it back (as a Cartier divisor). So we get a (Cartier) divisor $f^*f_*H$ which agrees with $H$ on $Y\setminus E$. In particular, if $\mathrm{codim}_YE\geq 2$ in a neighborhood $U$ of $y$, then $H|_U=(f^*f_*H)|_U$. Now by construction $y\in C\cap U\neq\emptyset$, so along $C$, $f^*f_*H=H+F$ where $F$ is an effective (exceptional) divisor that does not contain $C$. Finally, this leads to a contradiction, because we get that $$ 0=f^*f_*H\cdot C \geq H\cdot C>0$$ since $f(C)$ is a point.

Answer 2

There is a nice explicit description, in this paper of Morrison and Stevens, of four dimensional cyclic quotient singularities that are Gorenstein and terminal (cf. this MO question---terminal implies the non-existence of crepant resolutions).

Answer 3

The simplest example is $\mathbb{C}^4/\pm1$ where $-1$ acts diagonally. I'm sure there is an elementary proof that this does not have a crepant resolution, maybe via toric geometry. Someone else on MO might know a reference. The proof I know uses a result of Yasuda which says that if $X$ is a Gorenstein orbifold and $Y\to X$ is a crepant resolution, then $H^*_{orb}(X) = H^*(Y)$ as graded vector spaces. In the case at hand, this would imply that the exceptional fiber of a crepant resolution $Y\to \mathbb{C}^4/\pm1$ would have cohomology in degree 0 and in degree 4 only, which is impossible for a projective variety.