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 KX-Cartier 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 codimension-1 subvarieties of Y, these being irreducible by definition, whose image in X has codimension at least 2). The ai are rational numbers, called the discrepancies.

Then X is said to be

• terminal for all i,

• canonical for all i.

(One can also say that X has "terminal singularities" or "canonical singularities".) These properties are independent of the choice of resolution.

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 for all i,

• log canonical for all i.

These two properties are independent of the choice of log resolution. They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata.

If some log resolution of X, then X. 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 (or "volume forms") on X of smooth n. In local coordinates, this pullback operation is given by the Jacobian determinant of f is a birational morphism 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 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 (a locally free sheaf of rank 1) on X. (That is, at a singular point p in X, there may not be any open neighborhood U of p for which there is a nowhere-vanishing n-Cartier, in the definition of "terminal" and so on, means that for some positive integer m, the mth tensor power extends to a line bundle on X (on an open subset of X that vanishes on every exceptional divisor of f. Thus volume forms on a terminal variety behave "exactly like" those on a smooth variety.

that have at most a pole of order 1 along each exceptional divisor. (The word "log" comes primarily from the notion of logarithmic forms is a 1-form on the affine line with "log poles" at the origin.)

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. 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 (not in general unique) 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 (and pairs, as discussed below) 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. (Over an arbitrary field, the correct statement is rather that terminal varieties of dimension at most 2 are regular schemes.) 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..

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 are canonical when G.

by finite subgroups of the general linear group. (with G) are klt. For another example, toric varieties X-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., this is the affine cone over a smooth conic curve in the projective plane, 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-linearly equivalent for some rational number;

• Y is canonical if and only if A;

• Y is klt if and only if A;

• and Y.

In particular, if Y is klt, then X must be a Fano variety is ample. If Y is log canonical, then X is either Fano or a Calabi-Yau variety-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, these results give that the affine cone over the rational normal curve of degree d, and klt for all positive integers d, with G the cyclic group of order d acting by scalars.)

with n.

Relation to rational singularities

These classes of singularities are closely related to the older notion of rational singularities. Then every klt variety X has rational singularities. In particular, X is Cohen-Macaulay. (These statements can fail for log canonical varieties. For example, an affine cone over an abelian surface is log canonical but not Cohen-Macaulay, and so it does not have rational singularities.)

is Cartier, then "canonical", "klt", and "rational singularities" are all equivalent, and they imply that X is Gorenstein. More generally, every klt variety X (in some neighborhood of a given point p) is the quotient by a finite cyclic group G of a canonical singularity Y Cartier, called the index-1 cover of X near p. Moreover, G acts freely in codimension 1 (meaning that it acts freely outside a closed subset of codimension at least 2 in Y 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 (or log pair. Namely, let X-divisor on X, not just on its linear equivalence class.

, meaning that Y is nonsingular and the union of the exceptional locus of f is a divisor with simple normal crossings on Y. (The strict transform of a codimension-1 subvariety S of X (understood to be irreducible) is the unique codimension-1 subvariety of Y that maps onto S on Y such that

is called

• Kawamata log terminal,

• log canonical.

be any proper birational morphism from a normal variety Y to X on Y as above. For each codimension-1 subvariety E in Y, define the discrepancy of E as the negative of the coefficient of E is said to be klt if every irreducible divisor E over X (that is, in every such variety Y is log canonical if every irreducible divisor E over X. (This agrees with the previous definitions if a log resolution exists, for example when the base field has characteristic zero. The definitions in this paragraph are the ones used when a log resolution is not known to exist, for example when the base field has positive characteristic and X has dimension greater than 3.)

is

• terminal and every exceptional divisor over X),

• canonical and every exceptional divisor over X,

• purely log terminal and every exceptional divisor over X.

Here an means an irreducible divisor E as above such that the image of E in X, these definitions of "terminal" and "canonical" agree with those given above for varieties rather than pairs. When X can be omitted (it follows from the condition on exceptional divisors).

is divisorial log terminal for every exceptional divisor over X whose image in X does not have simple normal crossings. (Roughly speaking, this means that "dlt" is the smallest class of pairs that includes klt pairs as well as simple normal crossing pairs with coefficients at most 1.)

Explanation

is not log canonical, meaning that some irreducible divisor over X, 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. (There is a useful generalization of "log canonical" to non-normal schemes, "semi-log canonical" or "slc". For example, stable curves are slc.)

). (meaning that X is effective), X is rationally connected. Thus, it may be possible to say more about a given variety X.

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 for some smooth projective variety X and some divisor D can then be simplified via the minimal model program.

Philosophically, the definition of terminal singularities (for example) already indicates that many properties of a variety X together with. This suggests that pairs can be viewed as geometric objects comparable to varieties: for some purposes, X. One says that X is crepant birational.

Examples

For pairs (as opposed to varieties), the most important conditions are "log canonical" and the various versions of "log terminal". These conditions measure the singularities of the variety X are. For example: for C. That is, a cuspidal curve in a smooth surface has log canonical threshold-Cartier" remains true.)

For a simple case, let X-divisor on X whose support has simple normal crossings. (For example, X with coefficient 1 meet.

The "dlt" condition is local in the Zariski topology (or in the étale topology, over a general field). For example, let C is log canonical but not dlt, even though the singular point of C. 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

, the following implications hold:

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 (so S is an irreducible Cartier divisor in X). Then the adjunction formula. As a result, one can hope to pass from information about the lower-dimensional variety S. 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 is plt on some open neighborhood of S if and only if S is klt.

effective, X has rational singularities. This fails in general for log canonical pairs (and even log canonical varieties), as mentioned earlier.

A variety X is said to be of klt type if every point has a Zariski open neighborhood U 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 matrices with rank at most 1, is of klt type but not klt. in the Picard group.) By the previous paragraph, klt type implies rational singularities in characteristic zero.

See also

• Multiplier ideal

Notes

References