Presheaves and Sheaves

Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217

1. Presheaves and Sheaves

In this section we shall introduce the basic concepts of presheaves and sheaves, giving various examples to illustrate the main ideas. We shall start with some formal definitions.

Definition 1.1: A presheaf F over a topological space X is 36

(a) An assignment to each nonempty open set U⊂X of a setF(U ).

(b) A collection of mappings (called restriction homomorphisms) r_V^U :F(U )−→F(V )

for each pair of open sets U and V such that V ⊂U, satisfying (1) r_U^U =identity on U (=1U).

(2) For U⊃V ⊃W, r_W^U =r_W^V ◦r_V^U.

If F and G are presheaves over X, then a morphism(of presheaves) h:F−→G

is a collection of maps

h_U :F(U )−→G(U )

for each open set U in X such that the following diagram commutes:

F(U ) r_V^U

// G(U ) r_V^U

F(V ) // G(V ), V ⊂U ⊂X.

F is said to be a subpresheaf of G if the maps h_U above are inclusions.

Remark: We shall be dealing primarily with presheaves,F, whereF(U ) has some algebraic structure (e.g., abelian groups). In this case we also require that the subpresheaves have the induced substructure (e.g., sub- groups) and that restriction homomorphisms and morphisms preserve the algebraic structure (e.g.,r_V^U andhU are group homomorphisms). Moreover, we shall call the elements of F(U ) sections ofF over U for reasons which will become apparent later.

Definition 1.2: A presheafF is called asheaf if for every collectionU_i of open subsets of X with U = ∪U_i then F satisfies

Axiom S1: Ifs, t ∈F(U ) andr_U^U

i(s)=r_U^U

i(t ) for alli, then s=t. Axiom S2: Ifs_i ∈F(U_i)and if for U_i∩U_j = ⊘ we have

r_U^Ui

i∩Ui(s_i)=r_U^Uj

i∩Uj(s_j) for all i, then there exists an s∈F(U ) such that r_U^U

i(s)=si for all i.

Morphisms of sheaves (orsheaf mappings) are simply morphisms of the underlying presheaf. Moreover, when a subpresheaf of a sheaf F is also a sheaf, then it will be called asubsheaf ofF. Anisomorphismof sheaves (or presheaves) is defined in the obvious way, namely h_U is an isomorphism in the category under consideration for each open setU. Note that Axiom S1

for a sheaf says that data defined on large open sets U can be determined uniquely by looking at it locally, and Axiom S2 asserts that local data of a given kind (in a given presheaf) can be pieced together to give global data of the same kind (in the same presheaf).

We would now like to give some examples of presheaves and sheaves.

Example 1.3: Let X and Y be topological spaces and let C_X,Y be the presheaf over X defined by

(a) C_X,Y(U ):= {f :U→Y :f is continuous}.

(b) Forf ∈C_X,Y(U ), r_V^U(f ):=f|^V, the natural restriction as a function.

It is easy to see that this presheaf satisfies Axioms S1 and S2 and hence is a sheaf.

Example 1.4: Let X be a topological space and let K be R or C. Let C_X = C_X,K, as in the above example. This is a sheaf of K-algebras; i.e., C_X(U ) is a K-algebra under pointwise addition, multiplication, and scalar multiplication of functions.

Example 1.5: Let X be anS-manifold (as in Definition 1.1 in Chap. I).

Then we see that the assignment S_X given by

S_X(U ):=S(U )=the S-functions onU

defines a subsheaf ofC_X. This sheaf is called thestructure sheaf of the mani- foldX. In particular, we shall be dealing withE_X,A_X, andO_X, the sheaves of differentiable, real-analytic, andholomorphic functions on a manifold X.

Example 1.6: Let X be a topological space and let G be an abelian group. The assignmentU →G, forU connected, determines a sheaf, called theconstant sheaf (with coefficients inG). This sheaf will often be denoted simply by the same symbol G when there is no chance of confusion.

We want to give at least one example of a presheaf which is not a sheaf, although our primary interest later on will be sheaves of the type mentioned above.

Example 1.7: Let X be the complex plane, and define the presheaf B by letting B(U ) be the algebra of bounded holomorphic functions in the open set U. LetU_i = {z: |z|< i}, and then C= ∪ U_i. Let f_i ∈B(U_i) be defined by settingf_i(z)=z. Then it is quite clear that there is nof ∈B(C) with the property thatf|U_i =f_i. In fact, by Liouville’s theorem,B(C)=C.

Consequently, B is not a sheaf, since it violates Axiom S₂.

We see in the above example that the basic reason B was not a sheaf was that it was not defined by a local property (such as holomorphicity, differentiability, or continuity).

Remark: A presheaf that violates Axiom S1 can be obtained by taking the sections ofC_X,K withXa two point discrete space but letting all proper restrictions be zero.

A natural structure on presheaves which occurs quite often is that of a module.

Definition 1.8: Let R be a presheaf of commutative rings and let M be a presheaf of abelian groups, both over a topological space X. Suppose that for any open set U ⊂ X,M(U ) can be given the structure of an R(U )-module such that if α∈R(U ) and f ∈M(U ), then

r_V^U(αf )=ρ_V^U(α)r_V^U(f )

for V ⊂U, where r_V^U is the M-restriction homomorphism and ρ_V^U is the R-restriction homomorphism. Then M is called a presheaf of R-modules.

Moreover, if M is a sheaf, then Mwill be a sheaf of R-modules.

Example 1.9: Let E → X be an S-bundle. Then define a presheaf S(E) (=S_X(E))† by settingS(E)(U )=S(U, E), forU open inX, together with the natural restrictions. Then S(E) is, in fact, a subsheaf ofC_X,E and is called thesheaf of S-sections of the vector bundleE. As special cases, we have the sheaves of differential formsE^∗_Xon a differentiable manifold, or the sheaf of differential forms of type (p, q),E^p,q_X , on a complex manifold X.

These sheaves are examples of sheaves of E_X-modules, and, more generally, S(E)is a sheaf of S_X-modules for an S-bundle E→X.

Example 1.10: LetO_Cdenote the sheaf of holomorphic functions in the complex plane Cand let Jdenote the sheaf defined by the presheaf

U−→O(U ), if 0∈/U U−→ {f ∈O(U ):f (0)=0}, if 0∈U.

Then, clearly, this presheaf is a sheaf, and it is also a sheaf of modules over the sheaf of commutative rings O_C (in fact, it is a sheaf of ideals in the sheaf of rings, going one step further).

The most commonly occurring sheaves of modules in complex analysis have names.

Definition 1.11: Let X be a complex manifold. Then a sheaf of modules over the structure sheaf O_X of X is called an analytic sheaf.

As one knows from algebra, the simplest type of modules are the free modules. We have a corresponding definition for sheaves. First, we note that there is a natural (and obvious) notion of restriction of a sheaf (or presheaf) F on X to a sheaf (or presheaf) on an open subset U of X, to be denoted by F|^U.

Definition 1.12: Let Rbe a sheaf of commutative rings over a topological space X.

†S_X(E)is not to be confused with S_E(E), which are the global S-functions defined on the manifold E. In context it will be clear which is meant.

(a) Define R^p, for p≥0, by the presheaf

U −→R^p(U ):=R(U )⊕ · · · ⊕R(U )

!

p terms

.

R^p, so defined, is clearly a sheaf of R-modules and is called the direct sum of R (p times;p=0 corresponds to the 0-module).

(b) If Mis a sheaf of R-modules such that M∼=R^p for some p≥0, then M is said to be a free sheaf of modules.

(c) If M is a sheaf of R-modules such that each x ∈ X has a neighborhood U such that M|^U is free, then Mis said to be locally free.

The following theorem demonstrates the relationship between vector bundles and locally free sheaves.

Theorem 1.13: Let X=(X,S)be a connected S-manifold. Then there is a one-to-one correspondence between (isomorphism classes of)S-bundles over X and (isomorphism classes of) locally free sheaves of S-modules overX.

Proof: The correspondence is provided by E−→S(E)

and it is easy to see that S(E)is a locally free sheaf ofS-modules. Namely, by local triviality, for some neighborhood U of a point x ∈ X, we have E|U ∼=U×K^r, wherer is the rank of the vector bundleE. It follows that S(E)|U ∼=S(U×K^r). We claim that

S(U×K^r)∼=S|U⊕ · · · ⊕S|U.

From the definition of a section, it follows that f ∈S(U×K^r)(V ) (for V open in U) if and only if f (x) =(x, g(x)), where g : V → K^r and g is an S-morphism (cf. Example I.2.12). Therefore g=(g1, . . . , g_r), g_j ∈S(V ), and the correspondence above is given by

f −→(g1, . . . , g_r)∈S_U(V )⊕ · · · ⊕S_U(V ),

which is clearly an isomorphism of sheaves. Therefore S(E)is a locally free S_X-module.

We shall now show how to construct a vector bundle from a locally free sheaf, which inverts the above construction. Suppose thatLis a locally free sheaf of S-modules. Then we can find an open covering {U_α} of X such that

gα:L|^Uα

−→∼ S^r|^Uα

for some r >0 (excluding the trivial case); note thatr does not depend on α, sinceX is connected. Then define

gαβ:S^r|^Uα∩U_β

−→∼ S^r|^Uα∩U_β

by settinggαβ=gα◦g⁻¹_β . Nowgαβis a sheaf mapping, so in particular (when

acting on the open set Uα∩Uβ) it determines an invertible mapping of vector-valued functions (g_αβ)_Uα∩Uβ, which we write as

g_αβ:S(U_α∩U_β)^r−→S(U_α∩U_β)^r,

which is then a nonsingular r×r matrix of functions in S(U_α∩U_β), i.e., g_αβ :U_α∩U_β → GL(r, K), and hence determines transition functions for a vector bundle E, since the compatibility conditions g_αβ ·g_βγ =g_αγ are trivially satisfied. Thus a vector bundle E can be defined by letting

˜

E= ∪_α Uα×K^r (disjoint union) and making the identification

(x, ξ )∼(x, gαβ(x)ξ ), if x∈Uα∩ ¯Uβ =∅. (Cf. the remark after Definition I.2.2.)

We leave it to the reader to verify that isomorphism classes are preserved under this correspondence.

Q.E.D.

Remark: Most of the sheaves we shall be dealing with will be locally free sheaves arising from vector bundles; however, there is a generalization which is of great importance for the study of function theory on complex manifolds and, more generally, complex manifolds with singularities—complex spaces.

An analytic sheaf Fon a complex manifoldX is said to be coherent if for each x ∈ X there is a neighborhood U of x such that there is an exact sequence of sheaves over U,

O^p|U−→O^q|U −→F|U −→0,

for some p andq. For a complete discussion of coherent analytic sheaves on complex spaces, see Gunning and Rossi [1]. For instance, let V be a subvariety of Cⁿ; i.e.,V is defined as a closed subset inCⁿ, which is locally given as the set of zeros of a finite number of holomorphic functions. LetI_V be the subsheaf ofOdefined by sections that vanish onV. ThereforeI_V is an ideal sheaf in the sheaf of rings O. ThenI_V is a coherent analytic sheaf (by results of Oka and Cartan; see Gunning and Rossi [1]) but not necessarily locally free. A simple example of this situation is the case whereV is simply the origin in C²; then we see that I_V =I_{0} is similar to Example 1.10.

Moreover, I_{0} is coherent because of the following exact sequence,

0−→O−→^ν O²−→^µ I_{0}−→0 (Koszul complex), where

µ(f₁, f₂)=z₁f₁−z₂f₂ ν(f )=(z₂f, z₁f ).

One can easily check that this is exact (by expanding the functions in power series at the origin and determining the relations between the coefficients).