Chapter 10

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 11 June 2013

The Riemann-Roch Theorem

Throughout this chapter, X denotes a nonsingular, irreducible, projective algebraic variety defined over an algebraically closed field k (of any characteristic). A divisor D on X is an element of the free abelian group generated by the irreducible closed subvarieties of codimension 1 in X: D=niDi, where the ni are integers and the Di are irreducible subvarieties of codimension 1. D is positive (notation D0) if each ni0.

Since X is irreducible it has a field of rational functions, k(X). Any non-zero fk(X) defines a divisor (f)= (zeros of f)- (poles of f). Two divisors D1, D2 are linearly equivalent (notation D1D2) if D1-D2 is the divisor of some rational function. Clearly this is an equivalence relation. The set of all positive divisors linearly equivalent to a divisor D is denoted by |D|. A closely related object is the k-vector space L(D), which consists of 0 and all fk(X) such that D+(f)0. Thus the fL(D) give rise to the divisors in |D|, and |D| may be regarded as the projective space associated to the vector space L(D).

We shall see in a moment that L(D) is finite-dimensional. Its dimension is denoted by (D), and dim|D|=(D)-1. It is largely a matter of taste whether we work with |D| or L(D).

The Riemann-Roch theorem, in its original conception, is concerned with evaluating (D) (or dim|D|) in terms of other characters of D and X. One such character of X is the arithmetic genus pa(X). defined by

1+(-1)npa(X) =χ(X)=i=0d (-1)idimk Hi(X,𝒪X),

where d=dimX.

There is a distinguished equivalence class of divisors on X, called the canonical divisor class (definition later). A canonical divisor is denoted by K.

The Riemann-Roch theorem for a curve

If X is a curve, a divisor D on X is of the form niPi, where Pi are points of X. Hence we may define the degree of D: degD=ni. If D is the divisor of a rational function, then degD=0 (number of zeros = number of poles); hence degD depends only on the equivalence class of D. Riemann proved (for the case where k is the field of complex numbers) that


where g=pa(X)=dimkH1(X,𝒪X) is the genus of X; and Roch a few years later made this inequality more precise:

dim|D|=degD-g+ i(D) (1)

where i(D), the index of speciality of D, is defined to be (K-D), that is to say the number of linearly independent divisors DK, where K is a fixed canonical divisor. Thus (1) may be rewritten in the form

(D)-(K-D) =degD+χ(X) (1')

where χ(X)=1-g. In particular (D=0) (K)=g, hence (D=K) degK=2g-2.

The Riemann-Roch theorem for a surface

If X is a surface and C, D are divisors on X their intersection number C·D is defined; C·D is a symmetric bilinear function of C and D, and is zero if either C or D is linearly equivalent to O. The degree of a divisor D is degD=D·D; again this depends only on the equivalence class of D. A divisor D has another numerical invariant, its virtual genus π(D), which is defined as follows. Suppose first that C is an irreducible non-singular curve on X, and K any canonical divisor. Then K+C cuts out a canonical divisor on the, curve C, hence the genus g of C is given by 2g-2=C·(K+C). We use this formula to define the virtual genus of a divisor D, namely

2π(D)-2=D· (K+D).

Then the Riemann-Roch theorem for a surface (Castelnuovo, 1896) is

dim|D|degD+1 -π(D)+pa(X) -i(D) (2)

where as before i(D) is the 'index of speciality' of D, i.e. i(D)=(K-D). Thus (2) may be rewritten in the form

(D)+(K-D) >D·D-12D· (K+D)+χ(X)= 12D·(D-K)+ χ(X). (2')

In contrast to (1'), this is still an inequality. The difference between the two sides is called the superabundance s(D): thus

(D)-s(D)+ (K-D)=12D· (D-K)+χ(X) (2'')

where s(D) is some non-negative integer.

The next stage is to reinterpret (1') and (2'') in cohomological terms.

The line-bundle associated with a divisor

Let X be of arbitrary dimension, D=niDi a divisor on X, and let (Uα) be a covering of X by affine open sets. In the affine variety Uα each hypersurface Di is given by a single equation diα=0, where fiα belongs to the coordinate ring A(Uα) of Uα, hence we may associate with D the rational function gα=ifiαni; gα belongs to the field of fractions of A(Uα) [since X is irreducible, so is Uα, hence A(Uα) is an integral domain], and this field of fractions is just k(X). The divisor cut out by D on the open set U is the divisor of the rational function gα. Thus for each α we have gαk(X), such that hαβ=gαgβ-1 is finite and non-zero at every point of UαUβ: hence hαβ defines a regular map UαUβk (the multiplicative group of k), such that hαα=1, hαβhβγ=hαγ in UαUβUγ. Hence the functions hαβ define a line-bundle {D}, and it is not difficult to see that (i) {D} depends (up to isomorphism) only on D, and not on the covering (Uα); (ii) equivalent divisors give rise to isomorphic line-bundles. Conversely, a line bundle on X gives rise to a class of divisors, and L(D) is isomorphic to the vector space of global cross-sections of the bundle {D}.

Equivalently, we may consider the sheaf (D) of germs of cross-sections of the bundle {D}. (D) is an 𝒪X-Module, locally isomorphic to 𝒪X and therefore coherent. If U is an open set in X, then Γ(U,(D)) is the set of all fk(X) such that (f)+D0 on U, so that in particular (U=X) L(D) is the space of global sect ions of (D):


Since (D) is coherent, L(D) is finite-dimensional by Serre's theorem quoted at the end of Chapter 9.

Next, let T be the (covariant) tangent bundle of X, whose fibre Tx at a point xX is the space of all tangent vectors to X at x (this may be defined algebraically as the dual of the k-vector space mx/mx2, where mx is the maximal ideal of the local ring of X at x). The fibre Tx is of dimension n, hence the nth exterior power ΛnT is a line-bundle. The corresponding divisor class is the canonical class on X.

Serre's duality theorem

Let D be a divisor on X, K a canonical divisor. Let

hi(D)=dimk Hi(X,(D))

(finite since (D) is coherent). The duality theorem states (or rather implies) that

hi(D)=hd-i (K-D),0id (d=dimX).

Since (D)=dimL(D) =dimH0(X,(D)) =h0(D), the Riemann-Roch theorem (1') for a curve now takes the form

h0(D)- h1(D)= degD+χ(X)


χ(D)=degD+χ(X) (1'')

where in general

χ(D)=i0 (-1)ihi(D) ;

and for a surface it turns out that the superabundance s(D) is just h1(D), so that the Riemann-Roch theorem (2'') for a surface takes the form

χ(D)=12D· (D-K)+χ(X). (2''')

The Chow ring

Let X be as before (nonsingular, irreducible, projective). A cycle on X is a formal linear combination of irreducible subvarieties of X. Thus a divisor is a cycle of codimension 1. Two cycles D0,D1 on X are rationally eguivalent if there exists a cycle C on the product variety X×k such that C intersects X×{0} and X×{1} properly (i. e. so that all components of the intersection have the right dimensions) in the cycles D0×{0} and D1×{1} respectively. For divisors, rational equivalence is the same as linear equivalence.

If C, D are cycles, their intersection C·D is defined only if C, D intersect properly. If C, D do not intersect properly, it can be shown that D can be replaced by an equivalent cycle D such that C·D is defined, and the rational equivalence class of C·D is independent of the choice of the cycle D. Hence we have a product defined on the group A(X) of classes of cycles with respect to rational equivalence. A(X) is a graded group: A(X)=i=0dAi(X), where d=dimX and Ai(X) consists of the classes of cycles of codimension i in X. The multiplication just defined on A(X) respects this grading, so that A(X) is a graded ring, called the Chow ring of X. It is commutative and associative and has an identity element. A(X) serves for some purposes as a replacement for the cohomology ring H(X,) which is defined when k is the field of complex numbers; but in general it is much bigger (consider e.g. a curve of genus >0).

A(X) has good functorial properties, corresponding to those of the cohomology ring of a manifold. First, if f:XY is a regular map (or morphism of algebraic varieties) then f-1 (cycle) is a cycle on X, and this operation is compatible with intersections and rational equivalence, hence defines a graded ring homomorphism


Next, if f:XY is proper, then the image of a Zariski-closed set in X is closed in Y, which enables us to define


f is an additive homomorphism, but not multiplicative, and does not respect the grading. However, there is the so-called projection formula

f(x·f(y)) =f(x)·y ( xA(X),yA(Y) ) .

Chern classes of a vector bundle

Let E be a vector bundle on X, say of rank q (this means dimEx=q for each xX). We shall associate with E elements ci(E)Ai(X) (0iq), where in particular c0(E)=1, called the Chern classes of X. There are various ways of defining these classes constructively, and they can also be characterized uniquely by the following axioms:

(i) Functoriality. Given f:YX, then ci(f(E)) =fci(E) (i0), where f(E) is the inverse image bundle on Y;
(ii) Normalization. If E is a line bundle, say E={D}, then c1(E) is the class of D in A1(X).
(iii) Additivity. If 0EE E0 is an exact sequence of vector bundles on X, then ci(D)= j+k=i cj(E)ck (E).

If we define the total Chern class of E to be the sum c(E)= i0ci(E), then (iii) takes the form

c(E)=c(E) c(E).

The followtnq formalism, due to Hirzebruch, is very convenient. Let t be an indeterminate, and factorize 1+ c1(E)t+ c2(E)t2++ cq(E)tq formally: say

1+c1t++ cqtq= i=1q (1+γit),

and call the γi the 'Chern roots' of E. Then it can be shown that, if E is another vector bundle on X with Chern roots γj, then the Chern roots of EE are γi+γj; the Chern roots of the dual E of E are -γi; and the Chern roots of the exterior power ΛpE are γi1+ γi2++ γip (i1<<ip). The Chern character of E is defined to be

ch(E)=eγ1+ eγ2++ eγq (q=rankE)A(X)

where eγ means the exponential series 1+γ+12γ2+, which here is effectively a finite sum since A(X) is zero in dimensions >d=dimX. From axiom (iii) it follows that if 0EEE0 is an exact sequence of vector bundles on X, then

ch(E)-ch(E) +ch(E)= 0

i. e. the function ch is additive. It is also multiplicative: ch(EF)= ch(E)· ch(F).

We have another additive function at hand: if E is a vector bundle, let denote its sheaf of germs of local sections; then is a coherent sheaf and therefore the expression

χ(X,E)=i0 (-1)idimkHi (X,)

is a well-defined integer. If 0EEE0 is an exact sequence of bundles, then the sequence of sheaves 00 is exact, and from the cohomology sequence of this we deduce that

χ(X,E)- χ(X,E)+ χ(X,E) =0

by counting up the dimensions.

Hirzebruch's Riemann-Roch theorem

Let T* be the contravariant tangent X, i.e. the dual of T. Its Chern classes ci(T*) are called the Chern classes of X: notation ci(X). If γi are the Chern roots of T*, then -γi are the Chern roots of T, hence c1(ΛnT)= -γi= -c1(X). By the second axiom for Chern classes -c1(X) is the class of a canonical divisor K.

The Todd class of X, τ(X), is defined to be

τ(X)=i=1d γi/(1-e-γi) (d=dimX)

with the usual understanding that the product on the right is to be expanded out as a power series in the γi; since it is a symmetric function of the γi it can be written as a power series in the Chern classes ci(X), hence is an element of A(X) (= field of rational numbers). Then Hirzebruch's theorem is the formula

χ(D)=xd [ch({D})τ(X)] (3)

where D is any divisor on X, {D} the associated line bundle, χ(D) the alternating sum i0(-1)ihi(D)= i-(-1)idimkHi(X,(D)); and the symbol xd[] means that we take the homogeneous component of degree d of the expression inside the brackets, which is an element of Ad(X) . (Thus the right hand side of (3) is a 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 D=0 in (3), then χ(D)= χ(X) (=1+pa(X)), hence

χ(X) = x2 [ γ11-e-γ1 · γ21-e-γ2 ] = x2 [ ( 1-12γ1+ 16γ12 ) -1 · ( 1-12γ2+ 16γ22 ) -1 ] = x2 [ ( 1+12γ1+ 112γ12 ) ( 1+12γ2+ 112γ22 ) ] = 112(γ12+γ22) +14γ1γ2= 112(c12+c2) (ci=ci(X)).

Hence, if d=c1({D}) is the class of D in A1(X), we have

χ(D) = x2 [ (1+d+12d2) (1+12c1+112(c12+c2)) ] = 112(c12+c2) +12d2+12dc1 = 12d·(d+c1) +χ(X)=12D· (D-K)+χ(X)

since x1 is the class of -K.

Remark. The theorem actually proved by Hirzebruch was the formula (3) for a divisor D on a complex projective variety, the Chern classes being elements of the cohomology ring H*(X,).

The formula (3) generalizes to any vector bundle E on X (not necessarily a line bundle):

χ(X,E)=xd [ch(E)·τ(X)] . (3')

This is the most general form of Hirzebruch's Riemann-Roch theorem.

The Grothendieck group K(X)

Let X be as before and let D(X) be the free abelian group generated by the (isomorphism classes of) coherent 𝒪X-Modules: so that an element of F(X) is a formal linear combination x=nii of coherent 𝒪X-Modules. Corresponding to each short exact sequence (E):0 0, let Q(E) denote the element -+F(X), and let K*(X) denote the quotient of F(X) by the subgroup generated by all elements Q(E), as E runs through all exact sequences.

The group K*(X) has an obvious universal property. A function φ, defined on the class of coherent 𝒪X-Modules, with values in an abelian group G is said to be additive if φ()- φ()+ φ() =0 whenever 0 0 is exact. Then every additive function φ factors through K*(X), i.e. induces a homomorphism K*(X)G.

We may perform the same construction with vector bundles on X in place of coherent sheaves. This gives us another group K*(X). Each vector bundle E has a sheaf of local sections, which is locally free (i.e., locally isomorphic to 𝒪Xn for some n) and therefore coherent. Equivalently, we can define K*(X) in terms of locally free sheaves.

If E is a vector bundle on X, tensoring with E is an exact operation and therefore gives rise to a product in K*(X). 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 K*(X).

If is a locally free sheaf on X, tensoring with is an exact operation and therefore gives rise to a product K*(X)×K*(X) K*(X), which makes K*(X) into a K*(X)-module.

Let f:XY be a regular map. If E is a vector bundle on Y, then its inverse image f*(E) is a bundle on X. The functor f* is exact and therefore defines f!:K*(Y)K*(X), which is a ring homomorphism since f* is compatible with tensor product of bundles.

Next, let f:XU 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 𝒪X-Module, then by the finiteness theorem quoted at the end Chapter 9 the higher direct images Rqf*() (q0) are coherent 𝒪Y-Modules which vanish for q>dimX. Define

f!()=q0 (-1)qRqf* ().

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

f!:K*(X) K*(Y).

As in the case of the Chow ring, there is a "projection formula"

f!(f!(y)x) =f!(x) (yK*(Y),xK*(X))

which says that, if we regard K*(X) as a K*(Y)-module via f!, then f! is a K*(Y)-module homomorphism.

Since K*(X) can be defined in terms of locally free coherent sheaves, it follows that we have an (additive) homomorphism :K*(X)K*(X). It can be shown that, if X is irreducible, nonsingular and quasi-projective (which means isomorphic to an open subset of a projective variety) then is an isomorphism.

Remark. K*(X) has most of the formal properties of a cohomology ring, except for the dimension axiom (it is not a graded ring). Similarly K*(X) has the formal properties of homology, apart from dimension. The theorem K*K* when X is nonsingular and quasi-projective should be regarded as a statement of Poincaré duality. From now on we shall identify K* and K* by means of , and denote them both by K.

We remarked earlier than the Chern character ch is additive: if 0EEE0 is an exact sequence of vector bundles on X, then ch(E)- ch(E)+ ch(E) =0: hence we have

ch:K(X)A(X) Q

which is a ring homomorphism. How does this behave with respect to the homomorphisms f! and f!? Take f! first: let f:XY be a regular map. From the functoriality of Chern classes we have ch(f*(E))=f*(ch(E)) and therefore the diagram

K(X) ch A(X) f! K(Y) ch A(Y)

Grothendieck's Riemann-Roch theorem

The answer to the same question for f! (where the map f:XY is now proper) is the Riemann-Roch theorem of Grothendieck: the diagram

K(X) τ(X)ch A(X) f! K(Y) τ(Y)ch A(Y)

is commutative, i.e.

f*(τ(X)ch(x)) =τ(Y)ch (f!(x)) for anyxA(X). (4)

This includes Hirzebruch's Riemann-Roch theorem (3') as the special case in which Y is taken to be a single point. A coherent sheaf on Y is then a finite-dimensional vector space, hence the dimension function gives an isomorphism K(Y). If is a coherent sheaf on X, then f!()= (-1)qR1f*()= (-1)qHq(X,) (since f* is now the section functor Γ). We have A0(Y)=, Ai(Y)=0 for i>0, hence f*(τ(X)ch())= xd[ch()τ(X)]; finally τ(Y)=1 and hence (4) reduces to

χ(X,)=xd [ch()τ(X)] (3'')

which is Hirzebruch's Riemann-Roch theorem stated for a coherent sheaf rather than a vector bundle E. 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 f into an injection g:XP×Y (where P is a projective space containing X and g(x)=(x,f(x))) followed by a projection h:P×YY. It is enough to prove (4) for each of g and h separately; the proof for h can be reduced to the case where Y is a point , i.e. to the Hirzebruch theorem (3') for a projective space P; the proof for g is more difficult and is achieved by first taking the case where the subvariety g(X) of P×Y is of codimension 1, and then reducing the general case to this by blowing up the subvariety g(X).

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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