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 is -Cartier (as discussed below), 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 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 if for all i, canonical if for all i. (One can also say that X has "terminal singularities" or "canonical singularities".) These properties are independent of the choice of resolution.Kollár (2013), Corollary 2.12.
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 (klt) if for all i, log canonical (lc) if for all i. These two properties are independent of the choice of log resolution.Kollár (2013), Corollary 2.13. They were introduced in the early 1980s (with slightly different terminology) 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 .Kollár and Mori (1998), Corollary 2.31. 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.Reid (1987), section I.1. 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, . 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.Reid (1987), section I.5. 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-form on .) 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 (on an open subset of X) 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 (but without being required to vanish on the exceptional divisors). "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. (The word "log" comes primarily from the notion of logarithmic forms, which amounts to this "pole of order 1" condition. For example, 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, 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 (not in general unique) with terminal singularities and nef canonical class, called a minimal model of X.Birkar, Cascini, Hacon, and McKernan (2010), Corollary 1.1.1. 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.Fujino (2014), Theorems 1.8 and 1.19. Or again: klt Fano varieties in characteristic zero are rationally connected.Hacon and McKernan (2007), Corollaries 1.3 and 1.5.
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.Reid (1987), Theorems 3.2 and 6.1; Kollár and Mori (1998), section 5.3. 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.Kollár and Mori (1998), Theorem 4.20. 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 .Kollár and Mori (1998), Proposition 4.18; Kollár (2013), section 3.40. In any dimension, all quotient singularities (with G a finite subgroup of ) 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.Kollár (2013), section 3.5. For example, the surface in is terminal (in fact, smooth) for , canonical (or also klt) for , and log canonical for . For , this is the affine cone over a smooth conic curve in the projective plane (called a surface node), 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:
ThenKollár (2013), Lemma 3.1. 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 (in fact, smooth) for , canonical for , and klt for all positive integers d. (Over the complex numbers, these surface singularities can also be viewed as quotient singularities , with G the cyclic group of order d acting by scalars.)
For positive integers with n at least 2, the hypersurface singularity in is canonical if and only if .Reid (1980), Proposition 4.3.
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.Kollár and Mori (1998), Corollary 5.22. 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.Kollár (2013), Example 3.6.)
Conversely, if is Cartier, then "canonical", "klt", and "rational singularities" are all equivalent, and they imply that X is Gorenstein.Kollár and Mori (1998), Corollary 5.24. 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 with 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). As a result, klt singularities over are exactly the quotients of rational Gorenstein singularities by a cyclic group acting freely in codimension 1.Kollár and Mori (1998), Corollary 5.21.
Pairs More generally, following Kawamata, these concepts can be defined for a pair (or log pair) . Namely, let X be a normal variety, and let be a -divisor on X (a finite linear combination of codimension-1 subvarieties with rational coefficients) such that is -Cartier. (The following properties depend on the specific -divisor , not just on its linear equivalence class. More generally, the definitions work the same way for -divisors.) 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. (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; this extends in the obvious way to a finite linear combination of subvarieties.) There is a well-defined -divisor on Y such that
(Here the strict transform of each component of has the same coefficient in as in .) The pair is called Kawamata log terminal (klt) if all coefficients of are , log canonical (lc) if all coefficients of are .