# How to define the canonical sheaf on singular varieties

Let $X$ be a variety which might be singular, how to defined the canonical sheaf $K_X$ on $X$?

When $X$ is a proper, irreducible variety over $\mathbb{C}$, Ueno defined $K_X$ as the pushforward of the canonical sheaf of its nonsigular model (see Chapter2 in his book " Classification Theory of Algebraic Varieties and Compact Complex Spaces"). However, I was wondering if there is a well-defined canonical sheaf on singular variety which is not proper (over $\mathbb{C}$ is OK for me). I will be very appreciated if someone can point out referenced of this kind.

• You can find it in page 8 of the book singularities of minimal model program of Kollar
– user21574
Sep 29, 2016 at 19:59

For $$X$$ normal, saying that canonical divisor is the pushforward of the canonical divisor of a resolution of singularities is totally fine (it even works in characteristic $$p > 0$$ if you happen to have a resolution). In particular, if $$\pi : Y \to X$$ is a resolution of singularities, then $$\pi_* K_Y$$ is $$K_X$$ (here we define $$\pi_* K_Y$$ by simply throwing away any components of $$K_Y$$ that get contracted to non-divisorial varieties).

However: The pushforward of the canonical sheaf $$\omega_Y$$ is not the canonical sheaf $$\omega_X$$ in general. Indeed, $$\pi_* \omega_Y = \omega_X$$ is very close to requiring that $$X$$ has rational singularities. (Actually one definition of rational singularities, typically attributed to Kempf, is that $$X$$ is Cohen-Macaulay and $$\pi_* \omega_Y = \omega_X$$.

A good exercise is to show that if $$X = \text{Spec} k[x,y,z]/(x^3+y^3+z^3)$$ and $$\pi : Y \to X$$ is the blowup of the cone point, then $$\pi_* \omega_Y = \mathfrak{m} \cdot \omega_X$$ where $$\mathfrak{m}$$ is the maximal ideal of the origin. (Hint: use the adjunction formula and the formula for the canonical divisor when blowing up a point on $$\mathbb{A}^3$$).

## The canonical sheaf

Ok, so what is the right definition of the sheaf $$\omega_X$$ in general?

Well, for a projective variety of dimension $$d$$, $$i : X \hookrightarrow \mathbb{P}^N$$, define $$i_* \omega_X := \text{Ext}^{N-d}\big(i_* O_X, O_{\mathbb{P}^N}(-N-1)\big).$$ Since $$i$$ is a closed embedding, this uniquely determines $$\omega_X$$. For $$X$$ quasi-projective, you can define this by localizing. There are generalizations which apply to other schemes of finite type over $$k$$ defined using the $$f^!$$ functor for $$f : X \to k$$ the structural map, but I won't get into that here.

## The canonical sheaf is S2

It turns out that for any variety, $$\omega_X$$ is an S2 sheaf, this means it satisfies Hartog's phenomena (search math overflow). In particular, the sheaf is determined by its codimension-1 behavior. There's an easy way to see this, it turns out that if $$h : X \to Z = \mathbb{P^d}$$ is a generic projection to a hyperplane of the same dimension, then $$h_* \omega_X = \text{Hom}(h_* O_X, O_Z(-d-1)).$$ Now, it easily follows that this sheaf is S2 since it is reflexive on $$Z$$ and reflexive sheaves on $$Z$$ are always S2 (see for example Hartshorne's Generalized divisors on Gorenstein schemes).

Why does this matter? Well it means that if $$U$$ is the regular locus of $$X$$, and furthermore $$X \setminus U$$ has codimension 2 on $$X$$ (which happens for example if $$X$$ is normal), then if $$j : U \hookrightarrow X$$ is the inclusion, then $$j_* \omega_U = \omega_X$$ since both sheaves are S2 and they agree outside a codimension-2 set.

## Back to divisors

This also explains our first statement about divisors. Indeed, any divisor, like $$\pi_* K_Y$$ is determined outside a codimension 2 set, it is determined on $$U$$ in fact. And so if $$X$$ is normal, $$\pi : Y \to X$$ is an isomorphism outside of a codimension-2 set of $$X$$, and so the canonical divisor on that set works fine as a canonical divisor everywhere. In particular, it can be computed on $$Y$$ as claimed.

For non-normal $$Y$$, something can be done, but the formula isn't quite so simple (you also have to describe by what exactly you mean by a divisor on a non-normal variety).

• Thank you for your answer, when $X$ is normal, one can defined the canonical divisor on its smooth locus, and then pullback to $X$. This is explained in the book Toric variety by Cox etc. But in my case, the singularity is worse than that, I hope something can be said when the scheme is Cohen-Macaulay. My original problem is to do adjunction formula on such scheme, and it is mention in the book Birational geometry of algebraic varieties (Page182, Prop5.73) one has adjuntion formula on CM-scheme, but I do not know how could one define $K$ in this setting. Beside, is $\omega_X$ dualizing sheaf? Jun 10, 2013 at 2:40
• Indeed, if $X$ is Cohen-Macaulay, then $\omega_X$ is the dualizing sheaf. But the canonical sheaf is still defined by the same Ext formula even for non-Cohen-Macaulay schemes. If your singularity is regular in codimension-1, then you can still use the pushforward from the regular locus. What exactly is your adjunction problem? Maybe it would be better to state that. Jun 10, 2013 at 3:05
• Oh, let me explain this perhaps in a better way. Every integral scheme is S1, and hence Cohen-Macaulay outside a set of codimension 2. Thus for some purposes you can just work on the Cohen-Macaulay locus. Jun 10, 2013 at 3:27
• (1)In your definition of $K_X$, do you require $X$ to be projective? Moreover, I do not understand the difference between $K_X$ and $\omega_X$. (2)My adjunction problem is: $X$ is a normal, Cohen-Macaulay variety with anticanonical sheaf $−K_X=D_1+\dots +D_n$, where $D_i$ are effective Cartier divisors. Suppose $Y$ is a complete intersection of $D_i′s$. I want to use adjunction formula to show $K_Y=0$. By doing adjunction formula as in Birational geometry of algebraic varieties (Page182, Prop5.73) for each complete intersection , Jun 10, 2013 at 3:36
• con't, and the complete intersection preserve the CM-condition, the result can be proved. But the first thing confused me is the meaning of $K_X,K_Y$. Jun 10, 2013 at 3:36