Canonical singularity

In mathematics, canonical singularities are a class of singularities that appear on the canonical model of an algebraic variety, and terminal singularities are a narrower class that occur as singularities of minimal models. These classes of singularities were introduced by Miles. Terminal singularities are important in the minimal model program because smooth minimal models do not exist in the desired generality, and hence certain "mild" singularities must be allowed.

Definition

Let X be a normal variety over a field whose canonical class K_X is -Cartier, and let be a resolution of singularities of X. Using that Cartier divisors can be pulled back, one can write
where the sum is over the exceptional divisors of f. The a_i are rational numbers, called the discrepancies.
Then X is said to be

terminal if for all i,
canonical if for all i.

These properties are independent of the choice of resolution.
Suppose, more strongly, that is a log resolution, meaning that Y is nonsingular and the exceptional locus of f is a divisor with simple normal crossings in Y. Then X is said to be

Kawamata log terminal if for all i,
log canonical if for all i.

These two properties are independent of the choice of log resolution. They were introduced in the early 1980s by Yujiro Kawamata.
If some log resolution of X has an exceptional divisor with discrepancy less than, then X has other log resolutions with arbitrarily negative discrepancies. As a result, "log canonical" is the most general condition that can be defined along these lines, independent of the choice of log resolution.

Explanation

These properties mean that the differential forms on X behave like those on a smooth variety, to a greater or lesser extent. When X is a smooth variety of dimension n over a field k, its canonical line bundle is defined as the sheaf of n-forms on X,. For any morphism of smooth n-folds, there is a natural way to pull sections of back to sections of. In local coordinates, this pullback operation is given by the Jacobian determinant of f. When is a birational morphism of smooth varieties, the pullback of a section of vanishes along every exceptional divisor of f, because the Jacobian determinant vanishes there.
For a variety X over a perfect field which is normal but not smooth, the singular locus of X has codimension at least 2. The relevant sheaf of "volume forms", called the canonical sheaf, is defined by: a section of on an open subset U of X is simply an n-form on the smooth locus of U. This sheaf need not be a line bundle on X. But the condition that is -Cartier, in the definition of "terminal" and so on, means that for some positive integer m, the mth tensor power line bundle on extends to a line bundle on X. The main part of the definition of "terminal" says that for a resolution of singularities, each section of pulls back to a section of that vanishes on every exceptional divisor of f, and likewise for all positive multiples of. Thus volume forms on a terminal variety behave "exactly like" those on a smooth variety.
Likewise, "canonical" means that sections of pull back to sections of, and likewise for positive multiples. "Klt" means that sections of pull back to sections of, without the same requirement for positive multiples of. Finally, "log canonical" means that sections of pull back to rational sections of that have at most a pole of order 1 along each exceptional divisor.
A direct consequence of the definition of canonical singularities is that if two projective varieties with canonical singularities are birational, then they have the same plurigenera, the dimensions of the vector spaces for all. By Caucher Birkar, Paolo Cascini, Christopher Hacon, and James McKernan, every smooth projective variety X of general type over a field of characteristic zero is birational to a unique projective variety with canonical singularities and ample canonical class, called the canonical model of X. Moreover, X is also birational to a projective variety with terminal singularities and nef canonical class, called a minimal model of X. These are fundamental tools for the birational classification of algebraic varieties.
The motivation for defining terminal or canonical singularities was that they are the smallest classes of singularities for which minimal or canonical models can be expected to exist. But many techniques of the minimal model program turned out to work in the greater generality of klt or even log canonical singularities. As a result, one can say more by considering these broader classes of singularities. For example: the Kodaira vanishing theorem and the Cone theorem extend to projective log canonical varieties in characteristic zero. Or again: klt Fano varieties in characteristic zero are rationally connected.

Examples

Terminal varieties of dimension at most 2 over a perfect field are smooth. This explains why minimal models of surfaces can be taken to be smooth. More generally, the singular locus of any terminal variety has codimension at least 3. Therefore, terminal singularities in dimension 3 are isolated; over the complex numbers, they were classified by Shigefumi and Reid. In particular, a terminal 3-fold singularity is the quotient of a hypersurface singularity with multiplicity 2 by a finite cyclic group. Some simple examples of terminal singularities in dimension 3 are the 3-fold node, in, and the quotient singularity.
Two-dimensional canonical singularities are also called du Val singularities. Over the complex numbers, they are locally analytically isomorphic to quotients of the affine plane by finite subgroups of the special linear group. In any dimension, quotient singularities are canonical when G is a finite subgroup of.
Two-dimensional klt singularities over are locally analytically isomorphic to quotients of by finite subgroups of the general linear group. In any dimension, all quotient singularities are klt. For another example, toric varieties X with -Cartier are klt.
Two-dimensional log canonical singularities were classified by : they are either simple elliptic, cusp, or smooth, divided by the action of a finite group. For example, the surface in is terminal for, canonical for, and log canonical for. For, this is the affine cone over a smooth conic curve in the projective plane , which can also be viewed as the quotient. For, this is the affine cone over an elliptic curve.
Generalizing the previous example, there is a clear description of these conditions for a cone singularity of any dimension. Let X be a smooth projective variety over a field, A an ample line bundle on X, and Y the affine cone over X with respect to A:
Then

Y is terminal if and only if A is -linearly equivalent to for some rational number ;
Y is canonical if and only if A is -linearly equivalent to for some rational number ;
Y is klt if and only if A is -linearly equivalent to for some rational number ;
and Y is log canonical if and only if is -linearly equivalent to for some rational number.

In particular, if Y is klt, then X must be a Fano variety, meaning that is ample. If Y is log canonical, then X is either Fano or a Calabi-Yau variety, meaning that is -linearly equivalent to zero. For example, a cone over an elliptic curve is log canonical but not klt, and a cone over a curve of genus at least 2 is not log canonical.
Another example: because the canonical bundle of the projective line has degree, these results give that the affine cone over the rational normal curve of degree d in is terminal for, canonical for, and klt for all positive integers d.
For positive integers with n at least 2, the hypersurface singularity in is canonical if and only if.

Relation to rational singularities

These classes of singularities are closely related to the older notion of rational singularities. Assume here that the base field is the complex numbers. Then every klt variety X has rational singularities. In particular, X is Cohen-Macaulay.
Conversely, if is Cartier, then "canonical", "klt", and "rational singularities" are all equivalent, and they imply that X is Gorenstein. More generally, every klt variety X is the quotient by a finite cyclic group G of a canonical singularity Y with Cartier, called the index-1 cover of X near p. Moreover, G acts freely in codimension 1. As a result, klt singularities over are exactly the quotients of rational Gorenstein singularities by a cyclic group acting freely in codimension 1.

Pairs

More generally, following Kawamata, these concepts can be defined for a pair . Namely, let X be a normal variety, and let be a -divisor on X such that is -Cartier. The basic idea is to think of a pair as a generalization of a variety, with the "canonical class" of the pair being.
Let be a log resolution of, meaning that Y is nonsingular and the union of the exceptional locus of f with the strict transform of is a divisor with simple normal crossings on Y. There is a well-defined -divisor on Y such that
The pair is called

Kawamata log terminal if all coefficients of are,
log canonical if all coefficients of are.

These properties can also be defined without specifying a particular log resolution of. Namely, let be any proper birational morphism from a normal variety Y to X. There is a well-defined -divisor on Y as above. For each codimension-1 subvariety E in Y, define the discrepancy of E with respect to as the negative of the coefficient of E in. Then is said to be klt if every irreducible divisor E over X has discrepancy. Likewise, is log canonical if every irreducible divisor E over X has discrepancy.
Using this terminology, several related classes of pairs can be defined as follows. Namely, a pair is

terminal if has coefficients and every exceptional divisor over X has discrepancy ,
canonical if has coefficients and every exceptional divisor over X has discrepancy,
purely log terminal if has coefficients and every exceptional divisor over X has discrepancy.

Here an exceptional divisor over means an irreducible divisor E in some normal variety as above such that the image of E in X has codimension at least 2. When, these definitions of "terminal" and "canonical" agree with those given above for varieties rather than pairs. When X has dimension at least 2, the assumption in these definitions about the coefficients of can be omitted.
One last related class of pairs is: is divisorial log terminal if has coefficients at most 1 and the discrepancy is for every exceptional divisor over X whose image in X is contained in the closed subset where the pair does not have simple normal crossings.

Explanation

In dimension at least 2, if a pair is not log canonical, meaning that some irreducible divisor over X has discrepancy less than, then there are irreducible divisors over X with arbitrarily negative discrepancies. That explains why "log canonical" is the most general condition that can be formulated along these lines for normal varieties.
One practical reason for studying pairs is that many results of the minimal model program have been extended from varieties to pairs. For example, the Kodaira vanishing theorem and the Cone theorem hold for projective log canonical pairs with effective. Or again: for a dlt Fano pair , X is rationally connected. Thus, it may be possible to say more about a given variety X by finding a suitable.
Just as terminal and canonical singularities occur on minimal and canonical models of smooth projective varieties, dlt and log canonical pairs naturally arise as "minimal" or "log canonical" models of pairs with X smooth projective and D a simple normal crossing divisor with coefficients 1. One early motivation for studying such pairs was to classify smooth noncompact varieties, as far as possible. Indeed, every smooth quasi-projective variety over can be written as for some smooth projective variety X and some divisor D with simple normal crossings. A given pair can then be simplified via the minimal model program.
Philosophically, the definition of terminal singularities already indicates that many properties of a variety X can be encoded in a resolution of singularities together with the -divisor such that. This suggests that pairs can be viewed as geometric objects comparable to varieties: for some purposes, X can be replaced by the pair. One says that X is crepant birational to the pair.

Examples

For pairs, the most important conditions are "log canonical" and the various versions of "log terminal". These conditions measure the singularities of the variety X together with those of the -divisor. Letting the coefficients of vary gives a quantitative measure of how singular the components of are. For example: for C the cuspidal cubic curve in the affine plane, the pair is klt if and only if, and it is log canonical if and only if. That is, a cuspidal curve in a smooth surface has log canonical threshold.
For a simple case, let X be a smooth variety, and let be a -divisor on X whose support has simple normal crossings. Then the pair is klt if and only if the coefficients of are less than 1. Next, "dlt" and "log canonical" are both equivalent to the coefficients of being at most 1. Finally, "plt" means that the coefficients of are at most 1, and no two components with coefficient 1 meet. More generally, a pair is plt if and only if it is dlt and no two components of with coefficient 1 meet.
The "dlt" condition is local in the Zariski topology, but not in the classical topology over . For example, let C be the nodal cubic curve in the affine plane over. Then the pair is log canonical but not dlt, even though the singular point of C is locally analytically isomorphic to the two coordinate axes in. The point is that C has normal crossings but not simple normal crossings. The use of "simple normal crossings" in the definition makes dlt pairs easier to work with, while still being broad enough to include minimal models of simple normal crossing pairs.

Properties

For a pair, the following implications hold:
Also, "terminal" implies "klt" which implies "plt". A canonical pair need not be klt, as shown by the pair with X a smooth variety and D a smooth codimension-1 subvariety with coefficient 1.
The notion of "plt" is well-suited to arguments by induction on dimension. For example, let S be a subvariety locally defined by one equation in a normal variety X. Then the adjunction formula says that. As a result, one can hope to pass from information about the lower-dimensional variety S to information about the pair. Thus it becomes natural to study pairs with coefficient 1, even if the goal is to prove things about varieties. A useful tool in such arguments is the theorem on inversion of adjunction. A simple version says: if S is Cartier in X and is -Cartier, then the pair is plt on some open neighborhood of S if and only if S is klt.
An important consequence of the dlt condition is: for a dlt pair over a field of characteristic zero with effective, X has rational singularities. This fails in general for log canonical pairs, as mentioned earlier.
A variety X is said to be of klt type if every point has a Zariski open neighborhood U that contains an effective -divisor such that the pair is klt. For example, every toric variety is of klt type. For another example, the affine cone Y over the Segre embedding, also known as the determinantal variety of matrices with rank at most 1, is of klt type but not klt. By the previous paragraph, klt type implies rational singularities in characteristic zero.