Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217
2. Resolutions of Sheaves
omorphism (s˜provides a local inverse atsx forπ for a given representative s of sx∈ ˜F).
Thus we have associated to each presheaf F over X an étalé space.
Moreover, if the presheaf has algebraic properties preserved by direct limits, then the étalé spaceF˜ inherits these properties. For example, suppose that F is a presheaf of abelian groups. Then F˜ has the following properties:
(a) Each stalk is an abelian group.
(b) If F˜◦ ˜F:= {(s, t )∈ ˜F× ˜F:π(s)=π(t )}, then the map µ: ˜F◦ ˜F−→ ˜F
given by (sx, tx)−→sx−tx is continuous. This is true since if (s−t )∼(U ) is a basic open set of sx−tx for U open in X and s, t ∈F(U ), then the inverse image of (s−t )∼(U ) by the above map is just s(U )˜ ◦ ˜t (U ), which is a basic open set in F˜ ◦ ˜F.
(c) For U open in X, the set of sections of F˜ over U,Ŵ(U,F)˜ is an abelian group under pointwise addition, i.e., for s, t∈Ŵ(U,F)˜
(s−t )(x)=s(x)−t (x) for all x∈U.
We see thats−t is continuous since it is given by the following composition of continuous maps:
U−→ ˜(s,t ) F◦ ˜F−→ ˜µ F.
In associating an étalé spaceF˜ to a presheaf F, we have also associated a sheaf to F, namely the sheaf of sections ofF. We call this sheaf the˜ sheaf generated byF. We would now like to look more closely at the relationship between the presheaf,F, and the sheaf of sections of F˜ which we shall call
¯
Ffor the time being. We have already used the fact that there is a presheaf morphism, which we now denote by
τ :F−→ ¯F,
namely τU :F(U )−→ ¯F(U )[:=Ŵ(U,F)˜ ] is given by τU(s)= ˜s. Recall that
˜
s(x) = rxU(s) for all x ∈ U. In the case that F is a sheaf, we have the following basic result. Its proof will illustrate the use of the sheaf axioms in an abstract setting.
Theorem 2.2: IfF is a sheaf, then τ :F−→ ¯F is a sheaf isomorphism.
Proof: It suffices to show thatτU is bijective for eachU.
(a) τU is injective: Suppose that s′, s′′ ∈ F(U ) and τU(s′) = τU(s′′).
Then
[τU(s′)](x)= [τU(s′′)](x) for all x∈U;
i.e.,rxU(s′)=rxU(s′′)for allx∈U. But whenrxU(s′)=rxU(s′′)for somex ∈U, the definition of direct limit implies that there is a neighborhood V of x such that rVU(s′)=rVU(s′′). Since this is true for each x ∈U, we can cover U with open setsUi such that
rUU
i(s′)=rUU
i(s′′)
for all i. So since F is a sheaf, we have, by Axiom S1, s′=s′′.
(b) τU is surjective: Suppose σ ∈ Ŵ(U,F). Then for˜ x ∈ U there is a neighborhood V of x and s∈F(V ) such that
σ (x)=Sx= [τV(S)](x).
Since sections of an étalé space are local inverses for π, any two sections which agree at a point agree in some neighborhood of that point. Hence we have for some V∗ a neighborhood of x:
σ|V∗ =τV(s)|V∗ =τV∗(rVV∗(s)).
Since this is true for any x ∈ U, we can cover U with neighborhoods Ui such that there exists si∈F(Ui) and
τUi(si)=σ|Ui. Moreover, we have
τUi(si)=τUj(sj) on Ui∩Uj, so by part (a)
rUUi
i∩Uj(si)=rUUj
i∩Uj(sj).
Since F is a sheaf andU = ∪i Ui, there existss∈F(U ) such that rUU
i(s)=si. Thus
τU(s)|Ui =τUi(rUU
i(s))=τUi(si)=σ|Ui, and finally τU(s)=σ.
Q.E.D.
The content of this theorem is that to each sheaf F one can associate an étalé space F˜ whose sheaf of sections is the original F; i.e.,F˜ contains the same amount of information as F, and for this reason, a sheaf is very often defined to be an étalé space with algebraic structure along its fibres, as discussed above (see, e.g., Bredon [1] and Gunning and Rossi [1]). For doing analysis, however, the principal object is the presheaf, with its axioms (since most sheaves occur naturally in this form), and the associated étalé space is an auxiliary construction which is useful in constructing the homological machinery which makes sheaves useful objects. One way, in particular, that the étalé space is useful is to pass from a presheaf to a sheaf.
Definition 2.3: Let F be a presheaf over a topological space X and let F¯ be the sheaf of sections of the étalé space F˜ associated with F. Then F¯ is the sheaf generated by F.
By Theorem 2.2 above, we see that a presheaf, which is a sheaf, generates itself; i.e., F= ¯F. Moreover, we shall use both notationsF(U )andŴ(U,F) to denote the set (or group or module) of sections of FoverU, depending on the context (the word section, of course, coming from the étalé space picture of a sheaf ).
We now want to study the elementary homological algebra of sheaves of abelian groups; all the concepts we shall encounter generalize in a natural manner to sheaves of modules.
Definition 2.4: Suppose that F and G are sheaves of abelian groups over a space X withG a subsheaf ofF, and letQbe the sheaf generated by the presheaf U →F(U )/G(U ). Then Q is called the quotient sheaf of F by G and is denoted by F/G.
The quotient mapping on presheaves above induces a natural sheaf surjec- tionF−→F/Gby going to the direct limit, inducing a continuous mapping of étalé spaces, and then considering the induced map on continuous sections.
This is then the desired sheaf mapping onto the quotient sheaf.
One of the fundamental concepts of homological algebra is that of exactness.
Definition 2.5: If A, B, andC are sheaves of abelian groups over X and A−→g B−→h C
is a sequence of sheaf morphisms, then this sequence is exact at B if the induced sequence on stalks
Ax−→gx Bx−→hx Cx
is exact for all x ∈X. A short exact sequence is a sequence 0−→A−→B−→C−→0,
which is exact at A, B, andC, where 0 denotes the (constant) zero sheaf.
Remark: Note that exactness is a local property. The sheaves are not defined to be exact at the presheaf level [i.e., exactness of
A(U )−→B(U )−→C(U )
for eachU open inX], which, of course, was possible since homomorphism of sheaves were so defined. The usefulness of sheaf theory is precisely in finding and categorizing obstructions to the “global exactness” of sheaves.
We shall now give some examples of short exact sequences of sheaves.
Example 2.6: LetXbe a connected complex manifold. LetObe the sheaf of holomorphic functions on X and let O∗ be the sheaf of nonvanishing holomorphic functions on X which is a sheaf of abelian groups under multiplication. Then we have the following sequence:
(2.1) 0−→Z−→i O−→exp O∗−→0
where Z is the constant sheaf of integers, i is the inclusion map, and exp:O−→O∗ is defined by
expU (f )(z)=exp(2πif(z)).
Moreover, for some (sufficiently small) simply-connected neighborhood U of x∈X and for some representativeg∈O∗(U ) of a germgx atx, we can choose fx =((1/2π i)logg)x for some branch of the logarithm function, and we have expx(fx)=gx. Also, expx(fx)=0 implies that†
exp 2πif(z)≡1, z∈U,
for any f ∈O(U ) which is a representative of the germfx on a connected neighborhood U of x. Therefore f is constant on U and is, in fact, an integer, so that
Ker(expx)=Z, and the sequence (2.1) is exact.
Example 2.7: Let Abe a subsheaf of B. Then 0−→A−→i B−→q B/A−→0
is an exact sequence of sheaves, where i is the natural inclusion and q is the natural quotient mapping.
Example 2.8: As a special case of Example 2.7, we letX=Cand letO be the holomorphic functions on C. Let I be the subsheaf ofOconsisting of those holomorphic functions which vanish atz=0∈C(Example 1.10).
Then we have the following exact sequence of sheaves:
0−→I−→O−→O/I−→0.
We note that
(O/I)x∼=
C, if x =0 0, if x =0.
Example 2.9: Let X be a connected Hausdorff space and let a, b be two distinct points inX. LetZdenote the constant sheaf of integers on X and I denote the subsheaf of Zwhich vanishes at a and b. Then
0−→I−→Z−→Z/I−→0 is exact and
(Z/I)x ∼=
Z, if x=a or x =b 0, if x=a and x=b.
†Note that “0” here is the identity element in an abelian group.
Remark: Example 2.9 shows the necessity of using the generated sheaf for the quotient sheaf in Definition 2.4, since the presheaf of quotients of sections of Z by sections of I violates Axiom S2.
Following the terminology of homological algebra for modules, we make the following definitions where sheaf means sheaf of abelian groups or sheaf of modules. Agraded sheaf is a family of sheaves indexed by integers, F∗= {Fα}α∈Z. A sequence of sheaves (or sheaf sequence) is a graded sheaf connected by sheaf mappings:
(2.2) · · · −→F0−→α0 F1−→α1 F2−→α2 F3−→ · · ·.
A differential sheaf is a sequence of sheaves where the composite of any pair of mappings is zero; i.e., αj◦αj−1=0 in (2.2). Aresolutionof a sheaf F is an exact sequence of sheaves of the form
0−→F−→F0−→F1−→ · · · −→Fm−→ · · ·, which we also denote symbolically by
0−→F−→F∗, the maps being understood.
We shall see later that various types of information for a given sheaf F can be obtained from knowledge of a given resolution. We shall close this section with various examples of resolutions of sheaves. Their utility in computing cohomology will be demonstrated in the next section.
Example 2.10: Let X be a differentiable manifold of real dimension m and let EpX be the sheaf of real-valued differential forms of degree p. Then there is a resolution of the constant sheaf Rgiven by
(2.3) 0−→R−→i E0X−→d E1X−→ · · · −→d EmX −→0,
whereiis the natural inclusion anddis the exterior differentiation operator.
Since d2=0, it is clear that the above is a differential sheaf. However, the classical Poincaré lemma (see, e.g., Spivak [1], p. 94) asserts that on a star-shaped domain U in Rn, if f ∈ Ep(U ) is given such that df = 0, then there exists a u ∈ Ep−1(U ) (p > 0) so that du = f. Therefore the induced mapping dx on the stalks at x ∈ X is exact, since we can find representatives in local coordinates in star-shaped domains. At the term E0X, exactness is an elementary result from calculus [i.e.,df ≡0 implies that f is a constant (locally)]. We shall denote this resolution by 0−→R−→E∗X (or 0−→C−→E∗X if we are using complex coefficients).
Example 2.11: LetXbe a topological manifold. We want to derive a re- solution for the constant sheafGoverX, whereGis an abelian group (which will hold also for more general spaces). LetSp(U, G)be the group of singular cochains in U with coefficients in G; i.e.,Sp(U, G)=HomZ(Sp(U,Z), G), where Sp(U,Z) is the abelian group of integral singular chains of degree
pinUwith the usual boundary map (see, e.g., MacLane [1] or any standard algebraic topology text). Letδdenote the coboundary operator,δ:Sp(U, G)
−→Sp+1(U, G), and letSp(G)be the sheaf overXgenerated by the presheaf U −→Sp(U, G), with the induced differential mappingSp(G)−→δ Sp+1(G).
Consider the unit ball U in Euclidean space. Then the sequence (2.4) · · · −→Sp−1(U, G)−→δ Sp(U, G)−→δ Sp+1(U, G)−→ · · · is exact, since Ker δ/Im δ is the classical singular cohomology for the unit ball, which is well known to be zero for p > 0 (see MacLane [1], pp. 54–61, for an elementary proof of this fact, using barycentric subdivi- sion). Therefore the sequence
0−→G−→S0(G)−→δ S1(G)−→δ S2(G)−→ · · · −→Sm(G)−→ · · · is a resolution of the constant sheaf G, noting that
Ker(δ:S0(U, G)−→S1(U, G))∼=G.
We remark that we could also have consideredC∞chains ifXis a differ- entiable manifold, i.e. (linear combinations of) mapsf :p−→U, wheref is aC∞mapping defined in a neighborhood of the standardp-simplex p. The corresponding results above still hold [in particular, the elementary proof of the exactness of (2.4) still works in the C∞ case], and we have a resolution by differentiable cochains with coefficients in G:
0−→G−→S0∞(G)−→S1∞(G)−→ · · · −→Sm∞(G)−→ · · ·, which we abbreviate by
(2.5) 0−→G−→S∗∞(G).
Example 2.12: Let X be a complex manifold of complex dimension n, let Ep,q be the sheaf of (p, q) forms on X, and consider the sequence of sheaves, for p≥0, fixed,
0−→p−→i Ep,0−→∂¯ Ep,1−→ · · · −→∂¯ Ep,n−→0,
where p is defined as the kernel sheaf of the mappingEp,0−→∂¯ Ep,1, which is the sheaf ofholomorphic differential forms of type (p,0)(and we usually say holomorphic forms of degree p); i.e., in local coordinates,ϕ∈p(U ) if and only if
ϕ =
|I|=p
′ϕIdzI, ϕI ∈O(U ),
and we note that 0 =O(=OX). Then for each p we have a differential sheaf
(2.6) 0−→p−→Ep,∗,
since∂¯2=0, which is, in fact, a resolution of the sheafp, by virtue of the Grothendieck version of the Poincaré lemma for the ∂-operator. Namely,¯ if ω is a (p, q)-form defined in a polydisc in Cn,= {z: |zi|< r, i=1,
. . . , n}, and ∂ω¯ =0 in , then there exists a (p, q−1)-form u defined in a slightly smaller polydisc ′ ⊂⊂, so that ∂u¯ =ω in ′. See Gunning and Rossi [1], p. 27, for an elementary proof of this result using induction (as in one of the classical proofs of the Poincaré lemma) and the general Cauchy integral formula in the complex plane.†
Example 2.13: LetXbe a complex manifold and consider the differential sheaf over X,
0−→C−→0−→∂ 1−→ · · ·−→∂ n−→0,
where the p are defined in Example 2.12. Then we claim that this is a resolution of the constant sheaf C. First we note that ∂=d, when acting on holomorphic forms of degree p, since d =∂+ ¯∂, and ∂(¯ p) =0 for p=0, . . . , u; then exactness at 0 is immediate. Moreover, one can locally solve the equation ∂u=ω for u if ∂ω =0 by the same type of proof as for the operator ∂¯ indicated in Example 2.12.
Suppose that L∗ andM∗ are differential sheaves. Then a homomorphism f : L∗ → M∗ is a sequence of homomorphism fj : Lj → Mj which commutes with the differentials of L∗ andM∗. Similarly, a homomorphism of resolutions of sheaves
0 // A //
A∗
0 // B // B∗
is a homomorphism of the underlying differential sheaves.
Example 2.14: Let X be a differentiable manifold and let 0−→R−→E∗
0−→R−→S∗∞(R)
be the resolutions of Rgiven by Examples 2.10 and 2.11, respectively. Then there is a natural homomorphism of differential sheaves
I :E∗−→S∗∞(R)
which induces a homomorphism of resolutions in the following manner:
E∗
0 R
S∗∞(R).
i
I
i
The homomorphism I is given by integration over chains; i.e., IU :E∗(U )−→S∗∞(U,R)
†The same result holds for ∂:Ep,q−→Ep+1,q, as one can easily see by conjugation.
is given by
IU(ϕ)(c)=
$
c
ϕ,
wherecis aC∞chain (with real coefficients, in this case), and thenIU(ϕ)∈ S∗∞(U,R). Moreover, by Stokes’ theorem it follows that the mapping I commutes with the differentials.
We shall see in the next section how resolutions can be used to represent the cohomology groups of a space. In particular, we shall see that every sheaf admits a canonical abstract resolution with certain nice (cohomological) properties, and we shall then compare this abstract resolution with our more concrete examples of this section.
At this point we mention an analogue of the classical Poincaré lemma mentioned above, for which we shall have an application later on.
Lemma 2.15: Let ϕ∈Ep,q(U ) forU open inCn and suppose thatdϕ=0.
Then for any pointp∈U there is a neighborhoodN ofpand a differential form ψ∈Ep−1,q−1(N )such that
∂∂ψ¯ =ϕ in N.
Proof: The proof consists of an application of the Poincaré lemmas for the operators d,∂, and ∂¯ (see Examples 2.10 and 2.12). Namely, since dϕ=0, we have that there is au∈Er−1x (using germs at x), so thatdu=ϕ, where r = p+q is the total degree of ϕ. Thus we see that if we write u=ur−1,0+ · · · +u0,r−1, we have
du= ¯∂up,q−1+∂up−1,q
¯
∂up−1,q=∂up,q−1=0,
and then there exists (by the∂¯and∂ Poincaré lemmas, Example 2.12) forms ψ1∈Ep−1,q−1x and ψ2∈Ep−1,q−1x so that
∂ψ1=up,q−1
¯
∂ψ2=up−1,q which implies that
ϕ=du= ¯∂∂ψ1+∂∂ψ¯ 2
=∂∂(ψ¯ 2−ψ1).
Q.E.D.
Remark: Let H = Ker ∂∂¯ : E0,0 −→ E1,1 on a complex manifold X.
Then there is a fine resolution (see Definition 3.3)
0−→H−→E0,0−→∂∂¯ E1,1−→d E2,1⊕E1,2−→ · · ·,
where H is the sheaf of pluriharmonic functions, Lemma 2.15 showing exactness at theE1,1term (see Bigolin [1]). This is analogous to the resolution of O by E0,∗ and has a similar usefulness.