Why does a Gorenstein isolated three-fold singularity have a canonical (3, 0) form on the singularity?

Question

I'm trying to read a physics paper, and when talking about rational, graded, Gorenstein, isolated three-fold singularities they say:

"Here graded means that the singularity should have a $\mathbb{C}^*$ action, and Gorenstein means that there is a canonical well defined $(3, 0)$ form on the singularity, finally rational means that the weights of the $(3, 0)$ form under the $\mathbb{C}^∗$ action is positive."

I'm struggling to see how these conditions follow from the definitions of rational and Gorenstein that I know and love. I know that the canonical sheaf is invertible on a Gorenstein variety; but I don't see how that gives a $(3, 0)$ form.

Answer 1

Let $X$ be the three-fold, let $x$ be the singular point, and let $U = X \setminus \{x\}.$ Let $\omega_X$ be the canonical bundle on $X$. Then $(\omega_X)_{| U} = \Omega^3_U,$ since $U$ is smooth (by assumption $x$ is an isolated singularity).

If $j: U \to X$ is the inclusion, then adjunction between $j^*$ (i.e. restriction to $U$) and $j_*$ gives a canonical morphism $\omega_X \to j_*j^*\omega_X = j_*\Omega^3_U$. Now $\omega_X$ is locally free (of rank one), and $X$ is Gorenstein, hence Cohen--Macaulay, hence $S_2$, and so this morphism is an isomorphism. (Since $X$ is $S_2$, any locally free sheaf over $X$ is $S_2$, and $S_2$ coherent sheaves are characterized by $j_*j^*\mathcal F \to F$ is an isomorphism whenever $j$ is an open immersion whose complement is in codimension $ > 1$.)

Thus $\omega_X = j_*\Omega^3_U$. If we shrink $X$ around $x$ sufficiently to ensure that $\omega_X$ is free (and not just locally free) of rank one, then we may choose a generating global section $\omega$, which by definition of $j_*$ is a global section of $\Omega^3_U$ over $U$. That is your $3$-form.

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I have to confess that I can't yet derive the precise relationship between the condition on a presumed $\mathbb C^{\times}$-action and having rational singularities. One way to get an example of an isolated singularity on a $3$-fold with a $\mathbb C^{\times}$-action is to take the affine cone on a smooth surface. I started to think through the properties of these examples to see what was going on, but didn't sort it out yet. Maybe you can?

Answer 2

The Gorenstein condition can be phrased in different languages depending on whether you are looking at it from the commutative algebra point of view, or the algebraic geometry point of view.

From the algebraic geometry point of view, the Gorenstein condition means that the canonical sheaf is invertible (is a line bundle). If we look at the smooth part of our variety, the canonical sheaf is represented by a top $(3, 0)$ form, which is non-vanishing everywhere, we can denote it as $\Omega$.

The rational condition also has many characterizations, it can be defined using the resolution, etc. Since our variety is Gorenstein, it can also be phrased using the condition that the integral$$\int \Omega \wedge \overline{\Omega} < \infty.$$This condition is satisfied if $\Omega$ has positive scaling under the $\mathbb{C}^\times$-action.

I have to confess that I can't yet derive the precise relationship between the condition on a presumed $\mathbb C^{\times}$-action and having rational singularities. One way to get an example of an isolated singularity on a $3$-fold with a $\mathbb C^{\times}$-action is to take the affine cone on a smooth surface. I started to think through the properties of these examples to see what was going on, but didn't sort it out yet. Maybe you can?

The rational condition can be defined for any isolated singularity using the resolution. If the singularity has a $\mathbb{C}^\times$-action and the singularity is Gorenstein, the rational condition can be checked using the sign of the weights of the generator of the canonical sheaf, i.e. if the weight of $\Omega$ is positive, the singularity is rational.