Canonical Bundle

In mathematics, the canonical bundle of a non-singular algebraic variety $V$ of dimension $n$ over a field is the line bundle $\,\!\Omega^n = \omega$, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.

Over the complex numbers, 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. When the anticanonical bundle of V is ample, V is called a Fano variety.

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 isomorphism
:$\omega_D = i^*(\omega_X \otimes \mathcal(D)).$
In terms of canonical classes, it is
:$K_D = (K_X + D), _D.$
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 $X$, 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 $X$. It is this class, denoted by $K_X$ that is referred to as the canonical divisor on $X.$

Alternately, again on a normal variety $X$, one can consider $h^(\omega^._X)$, the $-d$'th cohomology of the normalized dualizing complex of $X$. This sheaf corresponds to a Weil divisor class, which is equal to the divisor class $K_X$ defined above. In the absence of the normality hypothesis, the same result holds if $X$ is S2 and Gorenstein 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 ''n''th 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 ''n''th 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 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 2''g'' − 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 −2''P'', where ''P'' is any point of ''C''. This follows from the calculus formula ''d''(1/''t'') = −''dt''/''t''², for example, a meromorphic differential with double pole at the point at infinity on the Riemann sphere. In particular, ''K''_''C'' and its multiples are not effective. If ''g'' is one, then ''C'' is an elliptic curve, and ''K''_''C'' 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

:''y''² = ''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''/,   ''x dx''/.

This means that the canonical map is given by homogeneous coordinates : ''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 2''g'' − 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 2''g'' − 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.

More refined information is available, for larger values of ''g'', but in these cases canonical curves are not generally complete intersections, 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

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases