In mathematics, the canonical bundle of a non-singular algebraic variety
V
{\displaystyle V}
of dimension
n
{\displaystyle n}
over a field is the line bundle
Ω
n
=
ω
{\displaystyle \,\!\Omega ^{n}=\omega }
, which is the
n
{\displaystyle n}
th exterior power of the cotangent bundle
Ω
{\displaystyle \Omega }
on
V
{\displaystyle V}
.
Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle
T
∗
V
{\displaystyle T^{*}V}
. Equivalently, it is the line bundle of holomorphic
n
{\displaystyle n}
-forms on
V
{\displaystyle V}
.
This is the dualising object for Serre duality on
V
{\displaystyle V}
. It may equally well be considered as an invertible sheaf.
The canonical class is the divisor class of a Cartier divisor
K
{\displaystyle K}
on
V
{\displaystyle V}
giving rise to the canonical bundle — it is an equivalence class for linear equivalence on
V
{\displaystyle V}
, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −
K
{\displaystyle K}
with
K
{\displaystyle K}
canonical.
The anticanonical bundle is the corresponding inverse bundle
ω
−
1
{\displaystyle \omega ^{-1}}
. When the anticanonical bundle of
V
{\displaystyle V}
is ample,
V
{\displaystyle V}
is called a Fano variety.
Contents
The adjunction formula
Suppose that
X
{\displaystyle X}
is a smooth variety and that
D
{\displaystyle D}
is a smooth divisor on
X
{\displaystyle X}
. The adjunction formula relates the canonical bundles of
X
{\displaystyle X}
and
D
{\displaystyle D}
. It is a natural isomorphism
ω
D
=
i
∗
(
ω
X
⊗
O
(
D
)
)
.
{\displaystyle \omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D)).}
The canonical bundle formula
Let
X
{\displaystyle X}
be a normal surface. A genus
g
{\displaystyle g}
fibration
f
:
X
→
B
{\displaystyle f:X\to B}
of
X
{\displaystyle X}
is a proper flat morphism
f
{\displaystyle f}
to a smooth curve such that
f
∗
O
X
≅
O
B
{\displaystyle f_{*}{\mathcal {O}}_{X}\cong {\mathcal {O}}_{B}}
and all fibers of
f
{\displaystyle f}
Singular case
On a singular variety
X
{\displaystyle X}
, 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
X
{\displaystyle X}
. It is this class, denoted by
K
X
{\displaystyle K_{X}}
that is referred to as the canonical divisor on
X
.
{\displaystyle X.}
Alternately, again on a normal variety
X
{\displaystyle X}
, one can consider
h
−
d
(
ω
X
.
)
{\displaystyle h^{-d}(\omega _{X}^{.})}
, the
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 projective curves. Here, the canonical bundle is the same as the (holomorphic) cotangent bundle. 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.
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 origin on the Riemann sphere. 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.
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), x dx/√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.
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 curves. 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 divisors 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.
Canonical rings
The canonical ring of V is the graded ring
R
=
⨁
d
=
0
∞
H
0
(
V
,
K
V
d
)
.
{\displaystyle R=\bigoplus _{d=0}^{\infty }H^{0}(V,K_{V}^{d}).}
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 Proj R is not a variety, and so it cannot be birational to V; in particular, V admits no canonical model. One can show that if the canonical divisor K of V is a nef divisor and the self intersection of K is greater than zero, then V will admit a canonical model (more generally, this is true for normal complete Gorenstein algebraic spaces).