Canonical bundle
Encyclopedia
In mathematics
, the canonical bundle of a non-singular algebraic variety
of dimension 
is the line bundle
which is the nth exterior power of the cotangent bundle
Ω on V. Over the complex number
s, it is the determinant bundle of holomorphic n-forms on V.
This is the dualising object
for Serre duality
on V. It may equally well be considered as an invertible sheaf
.
The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle
ω−1.
In terms of canonical classes, it is
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D.
, there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on 
. It is this class, denoted by 
that is referred to as the canonical divisor on 
Alternately, again on a normal variety
, one can consider 
, the 
'th cohomology of the normalized dualizing complex of 
. This sheaf corresponds to a Weil divisor class, which is equal to the divisor class 
defined above. In the absence of the normality hypothesis, the same result holds if 
is 
and Gorenstein
in dimension one.
. A global section of the canonical bundle is therefore the same as an everywhere-regular differential form. Classically, these were called differentials of the first kind
. The degree of the canonical class is 2g − 2 for a curve of genus g.
. In particular, KC and its multiples are not effective. If g is one, then C is an elliptic curve, and KC is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the n-canonical map for any n is the map to a point.
is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by
This means that the canonical map is given by homogeneous coordinates [1: X] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in X.
s. All non-singular plane quartics arise in this way. There is explicit information for the case g = 4, when a canonical curve is an intersection of a quadric
and a cubic surface
; and for g = 5 when it is an intersection of three quadrics. There is a converse, which is a corollary to the Riemann-Roch theorem: a non-singular curve C of genus g embedded in projective space of dimension g − 1 as a linearly normal curve of degree 2g − 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves C (in the non-hyperelliptic case of g at least 3), Riemann-Roch, and the theory of special divisor
s is rather close. Effective divisors D on C consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.
More refined information is available, for larger values of g, but in these cases canonical curves are not generally complete intersection
s, and the description requires more consideration of commutative algebra
. The field started with Max Noether's theorem: the dimension of the space of quadrics passing through C as embedded as canonical curve is (g − 2)(g − 3)/2. Petri's theorem, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for g at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) trigonal curves and (b) non-singular plane quintics when g = 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, Oscar Chisini
and Federigo Enriques
). The terminology is confused, since the result is also called the Noether–Enriques theorem. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is normally generated: the symmetric powers of the space of sections of the canonical bundle map onto the sections of its tensor powers. This implies for instance the generation of the quadratic differential
s on such curves by the differentials of the first kind; and this has consequences for the local Torelli theorem. Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a ruled surface
and a Veronese surface
.
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.
If the canonical class of V is an ample line bundle
, then the canonical ring is the homogeneous coordinate ring
of the image of the canonical map. This can be true even when the canonical class of V is not ample. For instance, if V is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a k-canonical map, where k is any sufficiently divisible positive integer.
The minimal model program proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated. In particular, this was known to imply the existence of a canonical model, a particular birational model of V with mild singularities that could be constructed by blowing down V. When the canonical ring is finitely generated, the canonical model is Proj
of the canonical ring. If the canonical ring is not finitely generated, then is not a variety, and so it cannot be birational to V; in particular, V admits no canonical model.
A fundamental theorem of Birkar-Cascini-Hacon-McKernan from 2006 is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
The Kodaira dimension
of V is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean Krull dimension
or transcendence degree
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the canonical bundle of a non-singular algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
of dimension is the line bundleLine bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
which is the nth exterior power of the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
Ω on V. Over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, it is the determinant bundle of holomorphic n-forms on V.
This is the dualising object
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
for Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
on V. It may equally well be considered as an invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
.
The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle
Inverse bundle
In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation.Let E \rightarrow M be a fibre bundle. A bundle E' \rightarrow M is called the inverse bundle of E if their Whitney sum is a trivial bundle, namely ifAny vector bundle over a compact...
ω−1.
The adjunction formula
Suppose that X is a smooth variety and that D is a smooth divisor on X. The adjunction formula relates the canonical bundles of X and D. It is a natural isomorphismIn terms of canonical classes, it is
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D.
Singular case
On a singular variety, there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on . It is this class, denoted by that is referred to as the canonical divisor on Alternately, again on a normal variety
, one can consider , the 'th cohomology of the normalized dualizing complex of . This sheaf corresponds to a Weil divisor class, which is equal to the divisor class defined above. In the absence of the normality hypothesis, the same result holds if is and GorensteinGorenstein
Gorenstein may refer to:* Daniel Gorenstein , mathematician*Hilda Goldblatt Gorenstein , artist and inspiration for the documentary I Remember Better When I Paint* Mark Gorenstein, Russian conductor...
in dimension one.
Canonical maps
If the canonical class is effective, then it determines a rational map from V into projective space. This map is called the canonical map. The rational map determined by the nth multiple of the canonical class is the n-canonical map. The n-canonical map sends V into a projective space of dimension one less than the dimension of the global sections of the nth multiple of the canonical class. n-canonical maps may have base points, meaning that they are not defined everywhere (i.e., they may not be a morphism of varieties). They may have positive dimensional fibers, and even if they have zero-dimensional fibers, they need not be local analytic isomorphisms.Canonical curves
The best studied case is that of curves. Here, the canonical bundle is the same as the (holomorphic) cotangent bundleCotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
. A global section of the canonical bundle is therefore the same as an everywhere-regular differential form. Classically, these were called differentials of the first kind
Differential of the first kind
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces and algebraic curves , for everywhere-regular differential 1-forms...
. The degree of the canonical class is 2g − 2 for a curve of genus g.
Low genus
Suppose that C is a smooth algebraic curve of genus g. If g is zero, then C is P1, and the canonical class is the class of −2P, where P is any point of C. This follows from the calculus formula d(1/t) = −dt/t2, for example, a meromorphic differential with double pole at the point at infinity on the Riemann sphereRiemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. In particular, KC and its multiples are not effective. If g is one, then C is an elliptic curve, and KC is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the n-canonical map for any n is the map to a point.
Hyperelliptic case
If C has genus two or more, then the canonical class is big, so the image of any n-canonical map is a curve. The image of the 1-canonical map is called a canonical curve. A canonical curve of genus g always sits in a projective space of dimension g − 1. When C is a hyperelliptic curve, the canonical curve is a rational normal curve, and C a double cover of its canonical curve. For example if P is a polynomial of degree 6 (without repeated roots) then- Y2 = P(X)
is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by
- dX/√P(X), XdX/√P(X).
This means that the canonical map is given by homogeneous coordinates [1: X] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in X.
General case
Otherwise, for non-hyperelliptic C which means g is at least 3, the morphism is an isomorphism of C with its image, which has degree 2g − 2. Thus for g = 3 the canonical curves (non-hyperelliptic case) are quartic plane curveQuartic plane curve
A quartic plane curve is a plane curve of the fourth degree. It can be defined by a quartic equation:Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0....
s. All non-singular plane quartics arise in this way. There is explicit information for the case g = 4, when a canonical curve is an intersection of a quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
and a cubic surface
Cubic surface
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
; and for g = 5 when it is an intersection of three quadrics. There is a converse, which is a corollary to the Riemann-Roch theorem: a non-singular curve C of genus g embedded in projective space of dimension g − 1 as a linearly normal curve of degree 2g − 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves C (in the non-hyperelliptic case of g at least 3), Riemann-Roch, and the theory of special divisor
Special divisor
In mathematics, in the theory of algebraic curves, certain divisors on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors...
s is rather close. Effective divisors D on C consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.
More refined information is available, for larger values of g, but in these cases canonical curves are not generally complete intersection
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
s, and the description requires more consideration of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
. The field started with Max Noether's theorem: the dimension of the space of quadrics passing through C as embedded as canonical curve is (g − 2)(g − 3)/2. Petri's theorem, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for g at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) trigonal curves and (b) non-singular plane quintics when g = 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, Oscar Chisini
Oscar Chisini
Oscar Chisini was an Italian mathematician. He introduced the Chisini mean in 1929.-Biography:Chisini was born in Bergamo....
and Federigo Enriques
Federigo Enriques
Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
). The terminology is confused, since the result is also called the Noether–Enriques theorem. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is normally generated: the symmetric powers of the space of sections of the canonical bundle map onto the sections of its tensor powers. This implies for instance the generation of the quadratic differential
Quadratic differential
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle.If the section is holomorphic, then the quadratic differentialis said to be holomorphic...
s on such curves by the differentials of the first kind; and this has consequences for the local Torelli theorem. Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
and a Veronese surface
Veronese surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
.
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.
Canonical rings
The canonical ring of V is the graded ringIf the canonical class of V is an ample line bundle
Ample line bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...
, then the canonical ring is the homogeneous coordinate ring
Homogeneous coordinate ring
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring...
of the image of the canonical map. This can be true even when the canonical class of V is not ample. For instance, if V is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a k-canonical map, where k is any sufficiently divisible positive integer.
The minimal model program proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated. In particular, this was known to imply the existence of a canonical model, a particular birational model of V with mild singularities that could be constructed by blowing down V. When the canonical ring is finitely generated, the canonical model is Proj
Proj construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties...
of the canonical ring. If the canonical ring is not finitely generated, then is not a variety, and so it cannot be birational to V; in particular, V admits no canonical model.
A fundamental theorem of Birkar-Cascini-Hacon-McKernan from 2006 is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
The Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
of V is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
or transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
.