Last update: 11 June 2013
Throughout this chapter, denotes a nonsingular, irreducible, projective algebraic variety defined over an algebraically closed field (of any characteristic). A divisor on is an element of the free abelian group generated by the irreducible closed subvarieties of codimension 1 in where the are integers and the are irreducible subvarieties of codimension 1. is positive (notation if each
Since is irreducible it has a field of rational functions, Any non-zero defines a divisor (zeros of (poles of Two divisors are linearly equivalent (notation if is the divisor of some rational function. Clearly this is an equivalence relation. The set of all positive divisors linearly equivalent to a divisor is denoted by A closely related object is the space which consists of and all such that Thus the give rise to the divisors in and may be regarded as the projective space associated to the vector space
We shall see in a moment that is finite-dimensional. Its dimension is denoted by and It is largely a matter of taste whether we work with or
The Riemann-Roch theorem, in its original conception, is concerned with evaluating (or in terms of other characters of and One such character of is the arithmetic genus defined by
There is a distinguished equivalence class of divisors on called the canonical divisor class (definition later). A canonical divisor is denoted by
If is a curve, a divisor on is of the form where are points of Hence we may define the degree of If is the divisor of a rational function, then (number of zeros number of poles); hence depends only on the equivalence class of Riemann proved (for the case where is the field of complex numbers) that
where is the genus of and Roch a few years later made this inequality more precise:
where the index of speciality of is defined to be that is to say the number of linearly independent divisors where is a fixed canonical divisor. Thus (1) may be rewritten in the form
where In particular hence
If is a surface and are divisors on their intersection number is defined; is a symmetric bilinear function of and and is zero if either or is linearly equivalent to The degree of a divisor is again this depends only on the equivalence class of A divisor has another numerical invariant, its virtual genus which is defined as follows. Suppose first that is an irreducible non-singular curve on and any canonical divisor. Then cuts out a canonical divisor on the, curve hence the genus of is given by We use this formula to define the virtual genus of a divisor namely
Then the Riemann-Roch theorem for a surface (Castelnuovo, 1896) is
where as before is the 'index of speciality' of i.e. Thus (2) may be rewritten in the form
In contrast to (1'), this is still an inequality. The difference between the two sides is called the superabundance thus
where is some non-negative integer.
The next stage is to reinterpret (1') and (2'') in cohomological terms.
Let be of arbitrary dimension, a divisor on and let be a covering of by affine open sets. In the affine variety each hypersurface is given by a single equation where belongs to the coordinate ring of hence we may associate with the rational function belongs to the field of fractions of [since is irreducible, so is hence is an integral domain], and this field of fractions is just The divisor cut out by on the open set is the divisor of the rational function Thus for each we have such that is finite and non-zero at every point of hence defines a regular map (the multiplicative group of such that in Hence the functions define a line-bundle and it is not difficult to see that (i) depends (up to isomorphism) only on and not on the covering (ii) equivalent divisors give rise to isomorphic line-bundles. Conversely, a line bundle on gives rise to a class of divisors, and is isomorphic to the vector space of global cross-sections of the bundle
Equivalently, we may consider the sheaf of germs of cross-sections of the bundle is an locally isomorphic to and therefore coherent. If is an open set in then is the set of all such that on so that in particular is the space of global sect ions of
Since is coherent, is finite-dimensional by Serre's theorem quoted at the end of Chapter 9.
Next, let be the (covariant) tangent bundle of whose fibre at a point is the space of all tangent vectors to at (this may be defined algebraically as the dual of the space where is the maximal ideal of the local ring of at The fibre is of dimension hence the exterior power is a line-bundle. The corresponding divisor class is the canonical class on
Let be a divisor on a canonical divisor. Let
(finite since is coherent). The duality theorem states (or rather implies) that
Since the Riemann-Roch theorem (1') for a curve now takes the form
where in general
and for a surface it turns out that the superabundance is just so that the Riemann-Roch theorem (2'') for a surface takes the form
Let be as before (nonsingular, irreducible, projective). A cycle on is a formal linear combination of irreducible
subvarieties of Thus a divisor is a cycle of codimension
If are cycles, their intersection is defined only if intersect properly. If do not intersect properly, it can be shown that can be replaced by an equivalent cycle such that is defined, and the rational equivalence class of is independent of the choice of the cycle Hence we have a product defined on the group of classes of cycles with respect to rational equivalence. is a graded group: where and consists of the classes of cycles of codimension in The multiplication just defined on respects this grading, so that is a graded ring, called the Chow ring of It is commutative and associative and has an identity element. serves for some purposes as a replacement for the cohomology ring which is defined when is the field of complex numbers; but in general it is much bigger (consider e.g. a curve of genus
has good functorial properties, corresponding to those of the cohomology ring of a manifold. First, if is a regular map (or morphism of algebraic varieties) then (cycle) is a cycle on and this operation is compatible with intersections and rational equivalence, hence defines a graded ring homomorphism
Next, if is proper, then the image of a Zariski-closed set in is closed in which enables us to define
is an additive homomorphism, but not multiplicative, and does not respect the grading. However, there is the so-called projection formula
Let be a vector bundle on say of rank (this means for each We shall associate with elements where in particular called the Chern classes of There are various ways of defining these classes constructively, and they can also be characterized uniquely by the following axioms:
|(i)||Functoriality. Given then where is the inverse image bundle on|
|(ii)||Normalization. If is a line bundle, say then is the class of in|
|(iii)||Additivity. If is an exact sequence of vector bundles on then|
If we define the total Chern class of to be the sum then (iii) takes the form
The followtnq formalism, due to Hirzebruch, is very convenient. Let be an indeterminate, and factorize formally: say
and call the the 'Chern roots' of Then it can be shown that, if is another vector bundle on with Chern roots then the Chern roots of are the Chern roots of the dual of are and the Chern roots of the exterior power are The Chern character of is defined to be
where means the exponential series which here is effectively a finite sum since is zero in dimensions From axiom (iii) it follows that if is an exact sequence of vector bundles on then
i. e. the function ch is additive. It is also multiplicative:
We have another additive function at hand: if is a vector bundle, let denote its sheaf of germs of local sections; then is a coherent sheaf and therefore the expression
is a well-defined integer. If is an exact sequence of bundles, then the sequence of sheaves is exact, and from the cohomology sequence of this we deduce that
by counting up the dimensions.
Let be the contravariant tangent i.e. the dual of Its Chern classes are called the Chern classes of notation If are the Chern roots of then are the Chern roots of hence By the second axiom for Chern classes is the class of a canonical divisor
The Todd class of is defined to be
with the usual understanding that the product on the right is to be expanded out as a power series in the since it is a symmetric function of the it can be written as a power series in the Chern classes hence is an element of field of rational numbers). Then Hirzebruch's theorem is the formula
where is any divisor on the associated line bundle, the alternating sum and the symbol means that we take the homogeneous component of degree of the expression inside the brackets, which is an element of (Thus the right hand side of (3) is priori only a rational number.)
Let us show for example how to recover from (3) the Riemann-Roch theorem for an algebraic surface, in the form (2'''). First take in (3), then hence
Hence, if is the class of in we have
since is the class of
Remark. The theorem actually proved by Hirzebruch was the formula (3) for a divisor on a complex projective variety, the Chern classes being elements of the cohomology ring
The formula (3) generalizes to any vector bundle on (not necessarily a line bundle):
This is the most general form of Hirzebruch's Riemann-Roch theorem.
Let be as before and let be the free abelian group generated by the (isomorphism classes of) coherent so that an element of is a formal linear combination of coherent Corresponding to each short exact sequence let denote the element and let denote the quotient of by the subgroup generated by all elements as runs through all exact sequences.
The group has an obvious universal property. A function defined on the class of coherent with values in an abelian group is said to be additive if whenever is exact. Then every additive function factors through i.e. induces a homomorphism
We may perform the same construction with vector bundles on in place of coherent sheaves. This gives us another group Each vector bundle has a sheaf of local sections, which is locally free (i.e., locally isomorphic to for some and therefore coherent. Equivalently, we can define in terms of locally free sheaves.
If is a vector bundle on tensoring with is an exact operation and therefore gives rise to a product in This product is clearly associative and commutative, and the class of the trivial line bundle is the identity element. Hence we have a commutative ring structure on
If is a locally free sheaf on tensoring with is an exact operation and therefore gives rise to a product which makes into a
Let be a regular map. If is a vector bundle on then its inverse image is a bundle on The functor is exact and therefore defines which is a ring homomorphism since is compatible with tensor product of bundles.
Next, let be a proper map. We cannot define the direct image of a bundle but we can define the direct image of a sheaf. If is a coherent then by the finiteness theorem quoted at the end Chapter 9 the higher direct images are coherent which vanish for Define
The right-hand side of this formula is additive in (from the exact sequence of derived functors, (8.1)) and hence induces a homomorphism of abelian groups
As in the case of the Chow ring, there is a "projection formula"
which says that, if we regard as a via then is a homomorphism.
Since can be defined in terms of locally free coherent sheaves, it follows that we have an (additive) homomorphism It can be shown that, if is irreducible, nonsingular and quasi-projective (which means isomorphic to an open subset of a projective variety) then is an isomorphism.
Remark. has most of the formal properties of a cohomology ring, except for the dimension axiom (it is not a graded ring). Similarly has the formal properties of homology, apart from dimension. The theorem when is nonsingular and quasi-projective should be regarded as a statement of Poincaré duality. From now on we shall identify and by means of and denote them both by
We remarked earlier than the Chern character ch is additive: if is an exact sequence of vector bundles on then hence we have
which is a ring homomorphism. How does this behave with respect to the homomorphisms and Take first: let be a regular map. From the functoriality of Chern classes we have and therefore the diagram
The answer to the same question for (where the map is now proper) is the Riemann-Roch theorem of Grothendieck: the diagram
is commutative, i.e.
This includes Hirzebruch's Riemann-Roch theorem (3') as the special case in which is taken to be a single point. A coherent sheaf on is then a finite-dimensional vector space, hence the dimension function gives an isomorphism If is a coherent sheaf on then (since is now the section functor We have for hence finally and hence (4) reduces to
which is Hirzebruch's Riemann-Roch theorem stated for a coherent sheaf rather than a vector bundle However this generality over (3') is illusory, since both sides of (3'') are additive in the argument
Grothendieck's proof consists in factorizing the morphism into an injection (where is a projective space containing and followed by a projection It is enough to prove (4) for each of and separately; the proof for can be reduced to the case where is a point , i.e. to the Hirzebruch theorem (3') for a projective space the proof for is more difficult and is achieved by first taking the case where the subvariety of is of codimension 1, and then reducing the general case to this by blowing up the subvariety
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".