Basic Results on Sheaves and Analytic Sets

Basic results on Sheaves and Analytic Sets

Jean-Pierre Demailly Universitéde Grenoble I, Institut Fourier, BP 74, Laboratoire associéau C.N.R.S. n◦188, F-38402 Saint-Martin d’Hères

The goal of these notes is mainly to ﬁx the notation and prove a few fundamental results concerning sheaves, sheaf cohomology, in relation with coherent sheaves and complex analytic sets. Since we adopt later an analytic point of view, it is enough to develope sheaf theory in the case of paracompact spaces. The reader is referred to (Godement 1958) for a more complete treatment of sheaf theory. There are also many good references dealing with analytic sets and coherent sheaves, e.g. (Cartan 1950), (Gunning-Rossi 1965), (Narasimhan 1966) and (Grauert-Remmert 1984).

1. Presheaves and Sheaves

1.A. Deﬁnitions

Sheaves have become a very important tool in analytic or algebraic geometry as well as in algebraic topology. They are especially useful when one wants to relate global properties of an object to its local properties (the latter being usually easier to establish). We ﬁrst introduce presheaves and sheaves in full generality and give some basic examples.

(1.1) Definition. Let X be a topological space. A presheaf on X is a collection of non empty sets (U) associated to every open set U X A(the sets (U) are called sets of sections ofA over U), together with maps ⊂ A A ρU,V : (V ) (U) A −→ A defined whenever U V and satisfying the transitivity property ⊂ ρU,V ρV,W = ρU,W for U V W, ρU,U = IdU for every U. ◦ ⊂ ⊂ Moreover, the presheaf is said to be a sheaf if it satisfies the following additional A gluing axioms, where (Uα) and U = Uα are arbitrary open subsets of X :

(i) If Fα (Uα) are such thatSρU ∩U ,U (Fα) = ρU ∩U ,U (Fβ) ∈A α β α α β β for all α, β, there exists F (U) such that ρU ,U (F ) = Fα ; ∈A α

(ii) If F, G (U) and ρU ,U (F ) = ρU ,U (G) for all α, then F = G ; ∈A α α in other words, local sections over the sets Uα can be glued together if they coincide in the intersections and the resulting section on U is uniquely deﬁned. 2 J.P. Demailly: Basic results on Sheaves and Analytic Sets

(1.2) Examples. Not all presheaves are sheaves. Let E be an arbitrary set with a distinguished element 0. The constant presheaf EX on X is defined to be EX (U) = E for all = U X and EX ( ) = 0 , with restriction maps ρU,V = IdE for = U V∅. 6 Then⊂ clearly axiom (∅i) is not{ } satisfied if U is the union of two disjoint ∅ 6 ⊂ open sets U1, U2 (and if E contains at least two elements). On the other hand, if we assign to each open set U X the set (U) of all real ⊂ C valued continuous functions on U and let ρU,V be the obvious restriction morphism (V ) (U), then is a sheaf on X, because continuity is a purely local notion. SimilarlyC → C if X is a differentiableC (resp. complex analytic) manifold, there are well defined sheaves of rings k (resp. ) of functions of class Ck (resp. of holomorphic C O functions) on X. Because of these examples, the maps ρU,V in Def. 1.1 are often viewed intuitively as “restriction homomorphisms”, although the sets (U) are not necessarily sets of functions defined over U. A

Most often, the presheaf is supposed to carry an additional algebraic structure. A

(1.3) Definition. A presheaf is said to be a presheaf of abelian groups (resp. rings, R-modules, algebras) if allA sets (U) are abelian groups (resp. rings, R-modules, A algebras) and if the maps ρU,V are morphisms of these algebraic structures. In this case, we always assume that ( ) = 0 . A ∅ { } If , are presheaves of abelian groups (or of some other algebraic structure) on theA sameB space X, a presheaf morphism ϕ : is a collection of morphisms A → B B ϕU : (U) (U) commuting with the restriction morphisms, i.e. such that ρU,V A → BA ◦ ϕV = ϕU ρU,V . We say that is a subpresheaf of in the case where ϕU : (U) (◦U) is the inclusion morphism;A the commutationB property then means A B⊂ B A B that ρU,V ( (V )) (U) for all U, V , and that ρU,V coincides with ρU,V on (V ). If is a subpresheafA ⊂A of a sheaf of abelian groups, there is a quotient presheafA =A / defined by (U) = (UB)/ (U). In a similar way, one can define direct CsumsB A of presheavesC of abelianB groups,A tensor products of presheaves of R-modules,A ⊕ B etc. A ⊗ B

(1.4) Definition. Let X and be topological spaces (not necessarily Hausdorff), and let π : X be a mappingS such that S −→ a) π maps onto X ; S b) π is a local homeomorphism, that is, every point in has an open neighborhood which is mapped homeomorphically by π onto an openS subset of X. Then is called a sheaf-space on X and π is called the projection of on X. If S −1 S x X, then x = π (x) is called the stalk of at x. ∈ S S If Y is a subset of X, we denote by Γ (Y, ) the set of sections of on Y , i.e. S S the set of continuous functions F : Y such that π F = IdY . It is clear that the presheaf defined by the collection of sets→ S ′(U) := Γ (U,◦ ) for all open sets U X S S ′⊂ together with the restriction maps ρU,V satisfies axioms 1.1(i) and (ii). hence is a S 1. Presheaves and Sheaves 3

sheaf. If Y is a subset of X, we denote by Γ (Y, ) the set of sections of on Y , i.e. S S the set of continuous functions F : Y such that π F = IdY . It is clear that the presheaf defined by the collection of sets→ S ′(U) := Γ (U,◦ ) for all open sets U X S S ⊂′ together with the restriction maps ρU,V satisfies axioms 1.1(i) and (ii). hence is a sheaf. S Conversely, it is possible to assign in a natural way a sheaf-space to any presheaf. If is a presheaf, we define the set x of germs of at a point x X to be the abstractA inductive limit A A ∈ e (1.5) x = lim (U),ρU,V . A A −→U∋x e For every F (U), there is a natural map which assigns to F its germ Fx x. ∈A ∈ A Now, the disjoint sum = x∈X x can be equipped with a natural topology as follows: for all F (UA), we set A e ∈A ` e e ΩF,U = Fx ; x U ∈ and choose the ΩF,U to be a basis of the topology of ; note that this family is A stable by intersection: ΩF,U ΩG,V = ΩH,W where W is the (open) set of points ∩ x U V at which Fx = Gx and H = ρW,U (F ). Thee obvious projection map ∈ ∩ π : X which sends x to x is then a local homeomorphism (it is actually a A→ A { } homeomorphism from ΩF,U onto U). Hence is a sheaf-space. A eIf is a sheaf-space, thee sheaf-space associated to the presheaf ′(U) := Γ (U, ) is canonicallyS isomorphic to , since e S S S lim Γ (U, ) = x S S −→U∋x thanks to the local homeomorphism property. Hence, we will usually not make any difference between a sheaf-space and its associated sheaf ′, both will be denoted by the same symbol . S S S Now, given a presheaf , there is an obvious presheaf morphism ′ defined by A A→ A ′ e (1.6) (U) (U) := Γ (U, ), F F =(U x Fx). A −→ A A 7−→ ∋ 7→ This morphism is clearly injective if and only if satisfies axiom 1.1(i) and it is e e Ae not difficult to see that 1.1(i) and (ii) together imply surjectivity. Therefore ′ A→ A is an isomorphism if and only if is a sheaf. According to what we said above, and ′ will be denoted by the sameA symbol : is called the sheaf associated toAe A A A the presheaf . It will be identified with itself if is a sheaf, but the notationale A A A differencee will be kept if is not a sheaf. e e A (1.7) Example. The sheaf associated to the constant presheaf of stalk E over X is the sheaf of locally constant functions X E. This sheaf will be denoted merely → by EX or E if there is no risk of confusion with the corresponding presheaf.

In the sequel, we usually work in the category of sheaves rather than in the category of presheaves themselves. For instance, the quotient / of a sheaf by B A B 4 J.P. Demailly: Basic results on Sheaves and Analytic Sets

a subsheaf generally refers to the sheaf associated with the quotient presheaf: its A stalks are equal to x/ x, but a section G of / over an open set U need not necessarily come fromB aA global section of (U);B whatA can be only said is that there B is a covering (Uα) of U and local sections Fα (Uα) representing G↾U such that ∈ B α (Fβ Fα)↾U ∩U belongs to (Uα Uβ). A sheaf morphism ϕ : is said to − α β A ∩ A → B be injective (resp. surjective) if the germ morphism ϕx : x x is injective (resp. surjective) for every x X. Let us note again that a surjectiveA → B sheaf morphism ϕ ∈ does not necessarily give rise to surjective morphisms ϕU : (U) (U). A → B

1.B. Direct and Inverse Images of Sheaves

Let X, Y be topological spaces and let f : X Y be a continuous map. If is a → A presheaf on X, the direct image f⋆ is the presheaf on Y deﬁned by A −1 (1.8) f⋆ (U) = f (U) A A

for all open sets U Y . When is a sheaf, it is clear that f⋆ also satisﬁes axioms ⊂ A A 1.1(i) and 1.1(ii), thus f⋆ is a sheaf. Its stalks are given by A −1 (1.9) (f⋆ )y = lim f (V ) A A −→V ∋y where V runs over all open neighborhoods of y Y . ∈ Now, let be a sheaf on Y , viewed as a sheaf-space with projection map π : Y . WeB deﬁne the inverse image f −1 by B → B −1 (1.10) f = Y X = (s,x) X ; π(s) = f(x) B B × ∈ B × with the topology induced by the product topology on X. It is then easy to ′ −1 B × see that the projection π = pr2 : f X is a local homeomorphism, therefore f −1 is a sheaf on X. By construction,B the → stalks of f −1 are B B −1 (1.11) (f )x = , B Bf(x) and the sections σ f −1 (U) can be considered as continuous mappings s : U such that π σ = f∈. In particular,B any section s (V ) on an open set V Y→has B a pull-back ◦ ∈ B ⊂

(1.12) f ⋆s = s f f −1 f −1(V ) . ◦ ∈ B There are always natural sheaf morphisms

−1 −1 (1.13) f f⋆ , f⋆f A −→A B −→ B −1 defined as follows. A germ in (f f⋆ )x = (f⋆ )f(x) is defined by a local section −1 A A s (f⋆ )(V ) = (f (V )) for some neighborhood V of f(x) ; this section can ∈ A A ⋆ be mapped to the germ sx x. In the opposite direction, the pull-back f s of a ∈ A −1 section s (V ) can be seen by (1.12) as a section of f⋆f (V ). It is not difficult to see that∈ theseB natural morphisms are not isomorphisms inB general. For instance, if f is a finite covering map with q sheets and if we take = EX , = EY to be A B 2. Coherent Sheaves 5

q −1 −1 q constant sheaves, then f⋆EX EY and f EY = EX , thus f f⋆EX EX and −1 q ≃ ≃ f⋆f EY E . ≃ Y

2. Coherent Sheaves

2.A. Locally Free Sheaves and Vector Bundles

Usually, the geometric structures introduced in analytic or algebraic geometry can be described in a convenient way by means of a structure sheaf which, most often, is a sheaf of commutative rings. For instance, the holomorphic propertiesA of a complex manifold X are well described by the sheaf X of holomorphic functions. Before introducing the more general notion of a coherentO sheaf, we discuss the notion of locally free sheaves over a sheaf a ring. All rings occurring in the sequel are supposed to be commutative with unit (the non commutative case is also of considerable interest, e.g. in view of the theory of -modules, but this subject is beyond the scope of these notes). D

(2.1) Definition. Let be a sheaf of rings on a topological space X and let a sheaf of modules over (orA briefly a -module). Then is said to be locally freeS of rank r over , if isA locally isomorphicA to ⊕r on a neighborhoodS of every point, i.e. for A S A every x0 X one can find a neighborhood Ω and sections F1,...,Fr (Ω) such that the sheaf∈ homomorphism ∈ S

⊕r ⊕r F : ↾Ω , (w ,...,wr) wj Fj,x x A↾Ω −→ S Ax ∋ 1 7−→ ∈ S ≤j≤r 1X is an isomorphism.

By deﬁnition, if is locally free, there is a covering (Uα)α∈I by open sets on S 1 r which admits free generators Fα,...,Fα (Uα). Because the generators can be uniquelyS expressed in terms of any other system∈ S of independent generators, there is for each pair (α, β) a r r matrix × jk jk Gαβ =(G ) ≤j,k≤r, G (Uα Uβ), αβ 1 αβ ∈A ∩ such that k j jk F = F G on Uα Uβ. β α αβ ∩ ≤j≤r 1X In other words, we have a commutative diagram

⊕r Fα ↾U ∩U A↾Uα∩Uβ −→ S α β Gαβ ⊕rx ↾Uα∩Uβ ↾ Uα ∩Uβ A  −→Fβ S 6 J.P. Demailly: Basic results on Sheaves and Analytic Sets

−1 It follows easily from the equality Gαβ = Fα Fβ that the transition matrices Gαβ are invertible matrices satisfying the transition◦ relation

(2.2) Gαγ = GαβGβγ on Uα Uβ Uγ ∩ ∩ −1 for all indices α,β,γ I. In particular Gαα = Id on Uα and G = Gβα on Uα Uβ. ∈ αβ ∩ Conversely, if we are given a system of invertible r r matrices Gαβ with × coefficients in (Uα Uβ) satisfying the transition relation (2.2), we can define A ∩ ⊕r a locally free sheaf of rank r over by taking over each Uα, the S A S ≃A identification over Uα Uβ being given by the isomorphism Gαβ. A section H of ∩ over an open set Ω X can just be seen as a collection of sections Hα = S 1 r ⊕r ⊂ (H ,...,H ) of (Ω Uα) satisfying the transition relations Hα = GαβHβ over α α A ∩ Ω Uα Uβ. ∩ ∩ The notion of locally free sheaf is closely related to another essential notion of differential geometry, namely the notion of vector bundle (resp. topological, differentiable, holomorphic ..., vector bundle). To describe the relation between these notions, we assume that the sheaf of rings is a subsheaf of the sheaf K of conti- nous functions on X with values in the fieldAK = R or K = C, containingC the sheaf of locally constant functions X K. Then, for each x X, there is an evaluation map → ∈ x K, w w(x) A → 7→ whose kernel is a maximal ideal mx of x, and x/mx = K. Let be a locally free A A S sheaf of rank r over . To each x X, we can associate a K-vector space Ex = m A ⊕r ∈ m ⊕r Kr x/ x x: since x x , we have Ex ( x/ x) = . The set E = x∈X Ex Sis equippedS withS a natural≃A projection ≃ A ` π : E X, ξ Ex π(ξ) := x, → ∈ 7→ −1 and the fibers Ex = π (x) have a structure of r-dimensional K-vector space: such a structure E is called a K-vector bundle of rank r over X. Every section s (U) ∈ S gives rise to a section of E over U by setting s(x) = sx mod mx. We obtain a function (still denoted by the same symbol) s : U E such that s(x) Ex for every x U, → ∈ ∈ i.e. π s = IdU . It is clear that (U) can be considered as a (U)-submodule of the K◦-vector space of functions US E mapping a point x AU to an element in → ∈ the fiber Ex. Thus we get a subsheaf of the sheaf of E-valued sections, which is in a natural way a -module isomorphic to . This subsheaf will be denoted by (E) and will be calledA the sheaf of -sectionsSof E. If we are given a K-vector bundleA E over X and a subsheaf = A(E) of the sheaf of all sections of E which is in a natural way a locally free S-moduleA of rank r, we say that E (or more precisely the pair (E, (E))) is a -vectorA bundle of rank r over X. A A

(2.3) Example. In case = X,K is the sheaf of all K-valued continuous functions on X, we say that E isA a topologicalC vector bundle over X. When X is a manifold and = p , we say that E is a Cp-diﬀerentiable vector bundle; ﬁnally, when X A CX,K is complex analytic and = X , we say that E is a holomorphic vector bundle. A O Let us introduce still a little more notation. Since (E) is a locally free sheaf A of rank r over any open set Uα in a suitable covering of X, a choice of generators 2. Coherent Sheaves 7

1 r 1 r (F ,...,F ) for (E)↾U yields corresponding generators (e (x),...,e (x)) of the α α A α α α ﬁbers Ex over K. Such a system of generators is called a -admissible frame of E A over Uα. There is a corresponding isomorphism

−1 r (2.4) θα : E↾U := π (Uα) Uα K α −→ × 1 r Kr which to each ξ Ex associates the pair (x, (ξα,...,ξα)) Uα composed of ∈ j 1 ∈ ×r x and of the components (ξα)1≤j≤r of ξ in the basis (eα(x),...,eα(x)) of Ex. The bundle E is said to be trivial if it is of the form X Kr, which is the same as saying ⊕r × that (E) = . For this reason, the isomorphisms θα are called trivializations of A A E over Uα. The corresponding transition automorphisms are

−1 r r θαβ := θα θ :(Uα Uβ) K (Uα Uβ) K , ◦ β ∩ × −→ ∩ × r θαβ(x,ξ)=(x,gαβ(x) ξ), (x,ξ) (Uα Uβ) K , · ∈ ∩ ×

where (gαβ) GLr( )(Uα Uβ) are the transition matrices already described (except that∈ they areA just seen∩ as matrices with coeﬃcients in K rather than with coeﬃcients in a sheaf). Conversely, if we are given a collection of matrices jk gαβ =(g ) GLr( )(Uα Uβ) satisfying the transition relation αβ ∈ A ∩

gαγ = gαβgβγ on Uα Uβ Uγ, ∩ ∩ we can define a -vector bundle A r E = Uα K / × ∼ α∈I a r by gluing the charts Uα K via the identification (xα,ξα) (xβ,ξβ) if and only if × ∼ xα = xβ = x Uα Uβ and ξα = gαβ(x) ξβ. ∈ ∩ · (2.5) Example. When X is a real differentiable manifold, an interesting example of n real vector bundle is the tangent bundle TX ; if τα : Uα R is a collection of coor- →Rm dinate charts on X, then θα = π dτα : TX↾Uα Uα define trivializations of × → × β −1 TX and the transition matrices are given by gαβ(x) = dταβ(x ) where ταβ = τα τβ β ⋆ ◦ and x = τβ(x). The dual TX of TX is called the cotangent bundle of X. If X is complex analytic, then TX has the structure of a holomorphic vector bundle.

We now briefly discuss the concept of sheaf and bundle morphisms. If and ′ are sheaves of -modules over a topological space X, then by a morphism ϕ S: S′ we just mean aA -linear sheaf morphism. If = (E) and ′ = (E′) are locallyS → free S sheaves, this isA the same as a -linear bundleS morphism,A S thatA is, a fiber preserving K A ′ -linear morphism ϕ(x): Ex Ex such that the matrix representing ϕ in any local -admissible frames of E and→E′ has coefficients in . A A (2.6) Proposition. Suppose that is a sheaf of local rings, i.e. that a section of is invertible in if and only ifA it never takes the zero value in K. Let ϕ : A′ be a -morphismA of locally free -modules of rank r, r′. If the rank of theSr →′ Sr A A × matrix ϕ(x) Mr′r(K) is constant for all x X, then Ker ϕ and Im ϕ are locally free subsheaves∈ of , ′ respectively, and Coker∈ ϕ = ′/ Im ϕ is locally free. S S S 8 J.P. Demailly: Basic results on Sheaves and Analytic Sets

Proof. This is just a consequence of elementary linear algebra, once we know that non zero determinants with coeﬃcients in can be inverted. A p Note that all three sheaves X,K, X,K, X are sheaves of local rings, so Prop. 2.6 applies to these cases. However,C evenC if weO work in the holomorphic category ( = A X ), a diﬃculty immediately appears that the kernel or cokernel of an arbitrary Omorphism of locally free sheaves is in general not locally free.

(2.7) Examples. a) Take X = C, let = ′ = be the trivial sheaf, and let ϕ : be the morphism u(z) Sz u(Sz). ItO is immediately seen that ϕ is injectiveO → as O a sheaf 7→ morphism ( being an entire ring), and that Coker ϕ is the skyscraper sheaf C0 of stalk C atOz = 0, having zero stalks at all other points z = 0. Thus Coker ϕ is not a locally free sheaf, although ϕ is everywhere injective6 (note however that the corresponding morphism ϕ : E E′,(z,ξ) (z,zξ) of trivial rank 1 vector bundles E = E′ = C C is not injective→ on the7→ zero ﬁber E ). × 0 b) Take X = C3, = ⊕3, ′ = and S O S O ⊕3 ϕ : , (u , u , u ) zj uj(z , z , z ). O → O 1 2 3 7→ 1 2 3 ≤j≤ 1X3 Since ϕ yields a surjective bundle morphism on C3 r 0 , one easily sees that Ker ϕ is locally free of rank 2 over C3 r 0 . However,{ by} looking at the Taylor { } expansion of the uj’s at 0, it is not diﬃcult to check that Ker ϕ is the - ⊕3 O submodule of generated by the three sections ( z2, z1, 0), ( z3, 0, z1) and (0, z , z ), andO that any two of these three sections cannot− generate− the 0-stalk 3 − 2 (Ker ϕ)0. Hence Ker ϕ is not locally free.

Since the category of locally free -modules is not stable by taking kernels or cokernels, one is led to introduce a moreO general category which will be stable under these operations. This leads to the notion of coherent sheaves.

2.B. Notion of Coherence

The notion of coherence again deals with sheaves of modules over a sheaf of rings. It is a semi-local property which says roughly that the sheaf of modules locally has a finite presentation in terms of generators and relations. We describe here some general properties of this notion, before concentrating ourselves on the case of coherent X -modules. O (2.8) Definition. Let be a sheaf of rings on a topological space X and a sheaf of modules over (orA briefly a -module). Then is said to be locallyS finitely generated if for everyA point x AX one can find aS neighborhood Ω and sections 0 ∈ F ,...,Fq (Ω) such that for every x Ω the stalk x is generated by the germs 1 ∈ S ∈ S F ,x,...,Fq,x as an x-module. 1 A 2. Coherent Sheaves 9

(2.9) Lemma. Let be a locally ﬁnitely generated sheaf of -modules on X and S A G ,...,GN sections in (U) such that G ,x ,...,GN,x generate x at x U. 1 S 1 0 0 S 0 0 ∈ Then G ,x,...,GN,x generate x for x near x . 1 S 0

Proof. Take F1,...,Fq as in Def. 2.8. As G1,...,GN generate x0 , one can ﬁnd a ′ ′ S ′ neighborhood Ω Ω of x0 and Hjk (Ω ) such that Fj = HjkGk on Ω . Thus ⊂ ∈A′ G1,x,...,GN,x generate x for all x Ω . S ∈ P

If U is an open subset of X, we denote by ↾U the restriction of to U, i.e. S S the union of all stalks x for x U. If F1,...,Fq (U), the kernel of the sheaf ⊕Sq ∈ ∈ S homomorphism F : ↾U deﬁned by A↾U −→ S

⊕q 1 q j (2.10) (g ,...,g ) g Fj,x x, x U Ax ∋ 7−→ ∈ S ∈ ≤j≤q 1X ⊕q is a subsheaf (F ,...,Fq) of , called the sheaf of relations between F ,...,Fq. R 1 A↾U 1 (2.11) Definition. A sheaf of -modules on X is said to be coherent if: S A a) is locally finitely generated ; S b) for any open subset U of X and any F ,...,Fq (U), the sheaf of relations 1 ∈ S (F ,...,Fq) is locally finitely generated. R 1

Assumption a) means that every point x X has a neighborhood Ω such that ⊕q ∈ there is a surjective sheaf morphism F : ↾Ω ↾Ω, and assumption b) implies that the kernel of F is locally finitely generated.A −→ Thus, S after shrinking Ω, we see that admits over Ω a finite presentation under the form of an exact sequence S ⊕p G ⊕q F (2.12) ↾Ω 0, A↾Ω −→ A↾Ω −→ S −→ where G is given by a q p matrix (Gjk) of sections of (Ω) whose columns × A (Gj ),..., (Gjp) are generators of (F ,...,Fq). 1 R 1 It is clear that every locally finitely generated subsheaf of a coherent sheaf is coherent. From this we easily infer:

(2.13) Theorem. Let ϕ : be a -morphism of coherent sheaves. Then Im ϕ and ker ϕ are coherent. F −→ G A

Proof. Clearly Im ϕ is a locally ﬁnitely generated subsheaf of , so it is coherent. G Let x0 X, let F1,...,Fq (Ω) be generators of on a neighborhood Ω of x0, ∈ ′ ⊕q ∈F F and G1,...,Gr (Ω ) be generators of ϕ(F1),...,ϕ(Fq) on a neighborhood Ω′ Ω of x . Then∈A ker ϕ is generated overRΩ′ by the sections ⊂ 0 q k ′ Hj = G Fk (Ω ), 1 j r. j ∈F ≤ ≤ k X=1 10 J.P. Demailly: Basic results on Sheaves and Analytic Sets

(2.14) Theorem. Let 0 0 be an exact sequence of -modules. If two of the sheaves −→, , Fare −→ coherent, S −→ G −→ then all three are coherent. A F S G Proof. If and are coherent, then = ker( ) is coherent by Th. 2.13. If and S areG coherent, then is locallyF finitelyS → generated; G to prove the co- S F G herence, let G ,...,Gq (U) and x U. Then there is a neighborhood 1 ∈ G 0 ∈ Ω of x and sections G˜ ,..., G˜q (Ω) which are mapped to G ,...,Gq on 0 1 ∈ S 1 Ω. After shrinking Ω, we may assume also that ↾Ω is generated by sections F F1,...,Fp (Ω). Then (G1,...,Gq) is the projection on the last q-components ∈F R p+q of (F ,...,Fp, G˜ ,..., G˜q) , which is finitely generated near x by the co- R 1 1 ⊂A 0 herence of . Hence (G ,...,Gq) is finitely generated near x and is coherent. S R 1 0 G Finally, assume that and are coherent. Let x X be any point, let F G 0 ∈ F1,...,Fp (Ω) and G1,...,Gq (Ω) be generators of , on a neighbor- ∈ F ∈ G ′ F G hood Ω of x0. There is a neighborhood Ω of x0 such that G1,...,Gq admit liftings ′ G˜ ,..., G˜q (Ω ). Then (F ,...,Fq, G˜ ,..., G˜q) generate ↾Ω′ , so is locally 1 ∈ S 1 1 S S finitely generated. Now, let S ,...,Sq be arbitrary sections in (U) and S ,..., Sq 1 S 1 their images in (U). For any x0 U, the sheaf of relations (S1,..., Sq) is gen- G ∈ ⊕q R erated by sections P1,...,Ps (Ω) on a small neighborhood Ω of x0. Set k ∈1 A q Pj = (P ) ≤k≤q. Then Hj = P S + ... + P Sq, 1 j s, are mapped to 0 in j 1 j 1 j ≤ ≤ G so they can be seen as sections of . The coherence of shows that (H1,...,Hs) ′Fs F ′ R has generators Q1,...,Qt (Ω ) on a small neighborhood Ω Ω of x0. Then ∈ A ′ k ′ ⊂ (S1,...,Sq) is generated over Ω by Rj = Qj Pk (Ω ), 1 j t, and is Rcoherent. ∈ A ≤ ≤ S P (2.15) Corollary. If and are coherent subsheaves of a coherent analytic sheaf , the intersection F is aG coherent sheaf. S F ∩ G Proof. Indeed, the intersection sheaf is the kernel of the composite morphism and / is coherent.F ∩ G , / ֒ F −→ S −→ S G S G

Case of a Coherent Sheaf of Rings.. A sheaf of rings is said to be coherent if it is coherent as a module over itself. By Def. 2.11, thisA means that for any open set U X and any sections Fj (U), the sheaf of relations (F1,...,Fq) is ﬁnitely generated.⊂ The above results∈A then imply that all free moduleRs ⊕p are coherent. As a consequence: A

(2.16) Theorem. If is a coherent sheaf of rings, any locally ﬁnitely generated subsheaf of ⊕p isA coherent. In particular, if is a coherent -module and A S ⊕q A F ,...,Fq (U), the sheaf of relations (F ,...,Fq) is also coherent. 1 ∈ S R 1 ⊂A Let be a coherent sheaf of modules over a coherent sheaf of ring . By an iterationS of construction (2.12), we see that for every integer m 0 and everyA point x X there is a neighborhood Ω of x on which there is an exact≥ sequence of sheaves ∈ ⊕pm Fm ⊕pm−1 ⊕p1 F1 ⊕p0 F0 (2.17) ↾Ω 0, A↾Ω −→ A↾Ω −→··· −→A↾Ω −→ A↾Ω −→ S −→ where Fj is given by a pj− pj matrix of sections in (Ω). 1 × A 3. Cechˇ Cohomology 11

2.C. Analytic Sheaves and the Oka Theorem

Many interesting properties of holomorphic functions can be expressed in terms of sheaves. Among them, analytic sheaves play a central role.

(2.18) Deﬁnition. Let M be a n-dimensional complex analytic manifold and let M be the sheaf of germs of analytic functions on M. An analytic sheaf over M isO by deﬁnition a sheaf of modules over M . S O

(2.19) Coherence theorem of Oka (1950). The sheaf of rings M is coherent for any complex manifold M. O

We refer e.g. to (Narasimhan 1965) for a proof of this theorem, which is certainly the most fundamental fact in the theory of analytic sheaves. This theorem can be seen as a semi-local version of the noetherianity property of rings Cn, of germs O 0 of holomorphic functions: more precisely, let F1,...,Fq (U). Since M,x is ⊕∈q O O Noetherian, we know that every stalk (F1,...,Fq)x M,x is ﬁnitely generated; the important new fact expressed byR the theorem is⊂ that O the sheaf of relations is locally ﬁnitely generated, namely that the “same” generators can be chosen to generate each stalk in a neighborhood of a given point. A by-product of the proof is the following:

(2.22) Strong Noetherian property. Let be a coherent analytic sheaf on a complex manifold M and let ... be anF increasing sequence of coherent subsheaves F1 ⊂F2 ⊂ of . Then the sequence ( k) is stationary on every compact subset of M. F F

3. Cechˇ Cohomology

3.A. Deﬁnitions

There are many ways of introducing sheaf cohomology. The simplest way is proba- bly the original deﬁnition of Cechˇ cohomology used by (Leray 1950, Cartan 1950). Although Cechˇ cohomology does not always work on arbitrary topological spaces, it is well behaved on the category of paracompact spaces and will be suﬃcient for our purposes. Let =(Uα)α∈I be an open covering of X. For the sake of simplicity, we denote U Uα α ...α = Uα Uα ... Uα . 0 1 q 0 ∩ 1 ∩ ∩ q The group Cq( , ) of Cechˇ q-cochains is the set of families U A

c =(cα0α1...αq ) Γ (Uα0α1...αq , ). ∈ q+1 A (α0,...,αYq)∈I The group structure on Cq( , ) is the obvious one deduced from the addition law on sections of . The Cechˇ U diﬀerentialA δq : Cq( , ) Cq+1( , ) is deﬁned by the formula A U A −→ U A 12 J.P. Demailly: Basic results on Sheaves and Analytic Sets

q j (3.1) (δ c)α0...αq+1 = ( 1) cα ...α ...α ↾U , − 0 j q+1 α0...αq+1 ≤j≤q 0 X+1 and we set Cq( , )=0, δq = 0 for q < 0. In degreesb 0 and 1, we get for example U A 0 (3.2) q =0, c =(cα), (δ c)αβ = cβ cα ↾U , − αβ ′ 1 (3.2 ) q =1, c =(cαβ), (δ c)αβγ = cβγ cαγ + cαβ ↾U . − αβγ Easy veriﬁcations left to the reader show that δq+1 δq = 0. We get therefore a cochain complex C•( , ), δ , called the complex of◦Cechˇ cochains relative to the covering . U A U (3.3) Deﬁnition. The Cechˇ cohomology group of relative to is A U Hq( , ) = Hq C•( , ) . U A U A

Formula (3.2) shows that the set of Cechˇ 0-cocycles is the set of families (cα) ∈ (Uα) such that cβ = cα on Uα Uβ. Such a family deﬁnes in a unique way a A ∩ global section f (X) with f↾U = cα. Hence Q ∈A α (3.4) H0( , ) = Γ (X, ). U A A

Now, let = (Vβ)β∈J be another open covering of X that is finer than ; this V U means that there exists a map ρ : J I such that Vβ Uρ(β) for every β J. Then we can define a morphism ρ• : C→•( , ) C•( , ⊂) by ∈ U A −→ V A q (3.5) (ρ c)β ...β = c ↾ ; 0 q ρ(β0)...ρ(βq ) Vβ0...βq the commutation property δρ• = ρ•δ is immediate. If ρ′ : J I is another refine- • →′• ment map such that Vβ Uρ′(β) for all β, the morphisms ρ , ρ are homotopic. To see this, we define a map⊂hq : Cq( , ) Cq−1( , ) by U A −→ V A q j ′ ′ (h c)β ...β − = ( 1) cρ(β )...ρ(β )ρ (β )...ρ (β − ) ↾V . 0 q 1 − 0 j j q 1 β0...βq−1 ≤j≤q− 0 X 1 The homotopy identity δq−1 hq + hq+1 δq = ρ′q ρq is easy to verify. Hence ρ• and ρ′• induce a map depending◦ only on ◦ , : − U V (3.6) Hq(ρ•) = Hq(ρ′•) : Hq( , ) Hq( , ). U A −→ V A Now, we want to define a direct limit Hq(X, ) of the groups Hq( , ) by means of the refinement mappings (3.6). In order toA avoid set theoretic difficulties,U A the coverings used in this definition will be considered as subsets of the power set (X), so that the collection of all coverings becomes actually a set. P (3.7) Definition. The Cechˇ cohomology group Hq(X, ) is the direct limit A Hq(X, )= lim Hq( , ) A U A −→U 3. Cechˇ Cohomology 13 when runs over the collection of all open coverings of X. Explicitly, this means that theU elements of Hq(X, ) are the equivalence classes in the disjoint union of the groups Hq( , ), with an elementA in Hq( , ) and another in Hq( , ) identified if their imagesU A in Hq( , ) coincide for someU A refinement of the coveringsV A and . W A W U V

3.B. Long Exact Sequence of Cohomology Groups

If ϕ : is a sheaf morphism, we have an obvious induced morphism ϕ• : C•( , A) → BC•( , ), and therefore we ﬁnd a morphism U A −→ U B Hq(ϕ•): Hq( , ) Hq( , ). U A −→ U B Let 0 0 be an exact sequence of sheaves. We have an exact sequence→A→B→C→ of groups

(3.8) 0 Cq( , ) Cq( , ) Cq( , ), −→ U A −→ U B −→ U C

but in general the last map is not surjective, because every section in (Uα0,...,αq ) • • C need not have a lifting in (Uα0,...,αq ). The image of C ( , ) in C ( , ) will be de- • B U B U C noted CB( , ) and called the complex of liftable cochains of in . By construction, the sequenceU C C B

(3.9) 0 Cq( , ) Cq( , ) Cq ( , ) 0 −→ U A −→ U B −→ B U C −→ is exact, thus we get a corresponding long exact sequence of cohomology

(3.10) Hq( , ) Hq( , ) Hq ( , ) Hq+1( , ) . U A −→ U B −→ B U C −→ U A −→ ··· (3.11) Theorem. Assume that X is paracompact. If

0 0 −→ A −→ B −→ C −→ is an exact sequence of sheaves, there is an exact sequence

0 Γ (X, ) Γ (X, ) Γ (X, ) H1(X, ) −→ A −→ B −→ C −→ A −→ ··· Hq(X, ) Hq(X, ) Hq(X, ) Hq+1(X, ) A −→ B −→ C −→ A −→··· which is the direct limit of the exact sequences (3.10) over all coverings . U

Proof. We have to show that the natural map

lim Hq ( , ) lim Hq( , ) B U C −→ U C −→ −→ is an isomorphism. This follows easily from the following lemma, which says essentially that every cochain in becomes liftable in after a reﬁnement of the covering. C B 14 J.P. Demailly: Basic results on Sheaves and Analytic Sets

q (3.12) Lifting lemma. Let =(Uα)α∈I be an open covering of X and c C ( , ). U ∈ U C If X is paracompact, there exists a ﬁner covering = (Vβ)β∈J and a reﬁnement map ρ : J I such that ρqc Cq ( , ). V → ∈ B V C

Proof. Since admits a locally finite refinement, we may assume that itself is U U locally finite. There exists an open covering =(Wα)α∈I of X such that W α Uα. W ⊂ For every point x X, we can select an open neighborhood Vx of x with the following properties: ∈ a) if x Wα, then Vx Wα ; ∈ ⊂ b) if x Uα or if Vx Wα = , then Vx Uα ; ∈ ∩ 6 ∅ ⊂ q c) if x Uα ...α , then cα ...α C (Uα ...α , ) admits a lifting in (Vx). ∈ 0 q 0 q ∈ 0 q C B Indeed, a) (resp. c)) can be achieved because x belongs to only finitely many sets Wα (resp. Uα), and so only finitely many sections of have to be lifted in . b) can ′ C B be achieved because x has a neighborhood Vx that meets only finitely many sets Uα ; then we take ′ ′ Vx V Uα (V r W α). ⊂ x ∩ ∩ x U ∋x U 6∋x \α \α Choose ρ : X I such that x W for every x. Then a) implies Vx W , → ∈ ρ(x) ⊂ ρ(x) so =(Vx)x∈X is finer than , and ρ defines a refinement map. If Vx0...xq = , we haveV U 6 ∅ Vx W Vx Vx = for 0 j q, 0 ∩ ρ(xj ) ⊃ 0 ∩ j 6 ∅ ≤ ≤

thus Vx0 Uρ(x0)...ρ(xq ) by b). Now, c) implies that the section cρ(x0)...ρ(xq ) admits ⊂ q a lifting in (Vx ), and in particular in (Vx ...x ). Therefore ρ c is liftable in . B 0 B 0 q B (3.13) Corollary. Let 0 0 be an exact sequence of sheaves on X, and let U be a paracompact−→ A −→ Bopen −→ subset. C −→ If H1(U, )=0, there is an exact sequence A 0 Γ (U, ) Γ (U, ) Γ (U, ) 0. −→ A −→ B −→ C −→

(3.14) Proposition. Let be a sheaf on X. Assume that either A a) is a sheaf of modules over a soft sheaf of rings on X (by this we mean A R that for every covering =(Uα)α∈I of X, there exists a -valued partition of U R unity (ψα)α∈I subordinate to ). U b) is ﬂabby, that is, every section in (U) extends to X, for any open set U. A A q Then H ( , )=0 for every q 1 and every open covering =(Uα)α∈I of X. In particularUHqA(X, )=0 for q ≥1. Such a sheaf is said to beU acyclic. U ≥

Proof. a) A typical example is the case where = ∞ is the sheaf of C∞ functions on a diﬀerentiable manifold. Under assumptionR C a), we can deﬁne a map hq : Cq( , ) Cq−1( , ) by U A −→ U A 3. Cechˇ Cohomology 15

q (3.15) (h c)α0...αq−1 = ψν cνα0...αq−1 ν∈I X ∁ where ψν cνα ...α − is extended by 0 on Uα ...α − Uν. It is clear that 0 q 1 0 q 1 ∩ q−1 q q (δ h c)α ...α = ψν cα ...α (δ c)να ...α , 0 q 0 q − 0 q ν∈I X i.e. δq−1hq + hq+1δq = Id. Hence δqc = 0 implies δq−1hqc = c if q 1. ≥ b) Let be the sheaf of sections of the sheaf-space which are not necessarily continuous,B that is, A (U) = x. e B A x∈U Y Then is a sheaf of modules over the sheaf ofe rings of arbitrary functions U Z, whichB clearly has partitions of unity with respect to any covering of X. Hence→ by a) we have Hq( , ) = 0 for q 1. Moreover, there is a canonical injection which allows usU toB consider the≥ exact sequence A → B

0 0 −→ A −→ B −→ C −→ where = / . It is not difficult to check that is also flabby and that every cochainC withB valuesA in is liftable to . Hence (3.10)C yields an exact sequence C B Γ (X, ) Γ (X, ) H1( , ) H1( , )=0, B −→ C −→ U A −→ U B 0 = Hq( , ) Hq( , ) Hq+1( , ) Hq+1( , )=0. U B −→ U C −→ U A −→ U B Clearly Γ (X, ) Γ (X, ) is surjective, thus H1( , ) = 0. Since is flabby, we can argue by inductionB → onCq to get Hq+1( , ) HUq(A, ) = 0 for qC 1. U A ≃ U C ≥

3.C. Leray’s Theorem for Acyclic Coverings

By deﬁnition of Cechˇ cohomology, for every exact sequence of sheaves 0 0 there is a commutative diagram →A→ B→C→ q q q q+1 q+1 H ( , ) H ( , ) HB( , ) H ( , ) H ( , ) (3.16) U A −→ U B −→ U C −→ U A −→ U B Hq(X, ) Hq(X, ) Hq(X, ) Hq+1(X, ) Hq+1(X, ).  A −→  B −→  C −→  A −→  B y y y y y in which the vertical maps are the canonical arrows to the inductive limit.

(3.17) Theorem (Leray). Assume that

s H (Uα ...α , )=0 0 t A q q for all indices α ,...,αt and s 1. Then H ( , ) H (X, ). 0 ≥ U A ≃ A We say that the covering is acyclic (with respect to ) if the hypothesis of Th. 3.17 is satisﬁed. Leray’s theoremU asserts that the cohomAology groups of on A 16 J.P. Demailly: Basic results on Sheaves and Analytic Sets

X can be computed by means of an arbitrary acyclic covering (if such a covering exists), without using the direct limit procedure.

Proof. By induction on q, the result being obvious for q = 0. Consider the exact sequence 0 0 where is the sheaf of non necessarily continuous sections of →A→B→C→and = / . As isB acyclic, the hypothesis on and the long A C B A B s A exact sequence of cohomology imply H (Uα0...αt , ) = 0 for s 1, t 0. Moreover • • C ≥ ≥ CB( , ) = eC ( , ) thanks to Cor. 3.13 applied on each open set Uα0...αq . The inductionU C hypothesisU C in degree q and diagram (3.16) give Hq( , ) Hq( , ) Hq+1( , ) 0 U B −→ U C −→ U A −→ Hq(X,≃ ) Hq(X,≃ ) Hq+1(X, ) 0,  B −→  C −→  A −→ y y y hence Hq+1( , ) Hq+1(X, ) is also an isomorphism. U A −→ A (3.18) Remark. The morphism H1( , ) H1(X, ) is always injective. Indeed, (3.10) yields U A −→ A ≃ H0 ( , )/ Im H0( , ) H1( , ) B U C U B −→ U A

≃ H0(X, )/ Im H0(X, ) H1(X, ) C y B −→ y A 0 0 0 0 and H ( , ) = H (X, ) = Γ (X, ), while HB( , ) H (X, ) is an injection. As a consequence,U B the reﬁnementB mappingsB H1( U, C) −→H1( , )C are also injective. U A → V A

3.D. Alternate Cechˇ Cochains and Topological Dimension

For explicit calculations, it is sometimes useful to consider a slightly modiﬁed Cechˇ complex which has the advantage of producing much smaller cochain groups. If is q q A a sheaf and = (Uα)α∈I an open covering of X, we let AC ( , ) C ( , ) be U ˇ ˇ U A ⊂ U A the subgroup of alternate Cech cochains, consisting of Cech cochains c = (cα0...αq ) such that

cα ...α =0 if αi = αj, i = j, (3.19) 0 q 6 ( cασ(0)...ασ(q) = ε(σ) cα0...αq for any permutation σ of 1,...,q of signature ε(σ). Then the Cechˇ diﬀerential (3.1) of an alternate cochain{ is still} alternate, so AC•( , ) is a subcomplex of C•( , ). U A U A Select a total ordering on the index set I. For each such ordering, we can deﬁne a projection πq : Cq( , ) ACq( , ) Cq( , ) by U A −→ U A ⊂ U A c alternate c such that cα ...α = cα ...α whenever α <...<αq. 7−→ 0 q 0 q 0 One can check that π• is a morphism of complexes and that π• is homotopic to the e e identity on C•( , ). Hence there is an isomorphism U A 3.E. The De Rham-Weil Isomorphism Theorem 17

(3.20) Hq AC•( , ) Hq C•( , ) = Hq( , ). U A ≃ U A U A (3.21) Corollary. Assume that X is a paracompact space of topological dimension n, i.e. that X has arbitrarily ﬁne open coverings = (Uα) such that more than ≤ U q n +1 distinct sets Uα0 ,...,Uαn have empty intersection. Then H (X, )=0 for q>n and any sheaf on X. A A

3.E. The De Rham-Weil Isomorphism Theorem

Let ( •,d) be a resolution of a sheaf , that is, an exact sequence of sheaves L A j d0 dq 0 0 1 q q+1 . −→ A −→ L −→ L −→··· −→L −→ L −→ ··· We assume in addition that all q are acyclic on X, i.e. Hs(X, q) =0 for all q 0 and s 1. Set q = ker dq. ThenL 0 = and for every q 1 weL get a short exact≥ sequence≥ Z Z A ≥ − dq 1 0 q−1 q−1 q 0. −→ Z −→ L −→ Z −→ Theorem 3.11 yields an exact sequence

− dq 1 ∂s,q (3.22) Hs(X, q−1) Hs(X, q) Hs+1(X, q−1) Hs+1(X, q−1)=0. L −→ Z −→ Z → L If s 1, the ﬁrst group is also zero and we get an isomorphism ≥ ≃ ∂s,q : Hs(X, q) Hs+1(X, q−1). Z −→ Z For s = 0 we have H0(X, q−1) = Γ (X, q−1) and H0(X, q) = Γ (X, q) is the q-cocycle group of •(X), soL the connectingL map ∂0,q givesZ an isomorphismZ L ∂0,q Hq •(X) = Γ (X, q)/dq−1Γ (X, q−1) H1(X, q−1). L Z L −→ Z The composite map ∂q−1,1 ∂1,q−1 ∂0,q thereforee deﬁnes an isomorphism ◦···◦ ◦ − − ∂0,q ∂1,q 1 ∂q 1,1 (3.23) Hq Γ (X, •) H1(X, q−1)e Hq(X, 0)=Hq(X, ). L −→ Z −→ ··· −→ Z A This isomorphism behaves e functorially with respect to morphisms of resolutions.

(3.24) De Rham-Weil isomorphism theorem. If ( •,d) is a resolution of by sheaves q which are acyclic on X, there is a functorialL isomorphism A L Hq Γ (X, •) Hq(X, ). L −→ A (3.25) Example: De Rham cohomology. Let X be a n-dimensional paracompact diﬀerential manifold. Consider the resolution d d 0 R 0 1 q q+1 n 0 → →E →E →···→E →E →···→E → 18 J.P. Demailly: Basic results on Sheaves and Analytic Sets

given by the exterior derivative d acting on the sheaves of germs of C∞ diﬀerential q-forms. The De Rham cohomology groups of X are by deﬁnition

(3.26) Hq (X, R) = Hq Γ (X, •) . DR E All sheaves q are ∞-modules, so q is acyclic by Prop. (3.14 a). Therefore, we get an isomorphismE C E

≃ (3.27) Hq (X, R) Hq(X, R) DR −→ from the De Rham cohomology onto the cohomology with values in the constant sheaf R. Instead of using C∞ differential forms, one can consider the resolution of R given by the exterior derivative d acting on currents. Let ′q be the sheaf of germs of currents of degree q (that is, differential forms of degreeDq with distribution coefficients). The Poincarélemma still holds for currents, so we have a resolution

d d 0 R ′0 ′1 ′q ′q+1 ′n 0. → →D →D →···→D →D →···→D → Again, the sheaves ′q are ∞-modules, hence they are acyclic. Now, the inclusion q ′q induces anD isomorphismC E ⊂D (3.28) Hq Γ (X, •) Hq Γ (X, ′•) , E ≃ D both groups being isomorphic to Hq(X, R). The isomorphism between cohomology of diﬀerential forms and singular cohomology (another topological invariant) was ﬁrst established by (De Rham 1931). The above proof follows essentially the method given by (Weil 1952), in a more abstract setting.

3.F. Mayer-Vietoris Exact Sequence

Let U1, U2 be open subsets of X and U = U1 U2, V = U1 U2. For any sheaf on X, there exists a resolution ∪ ∩ A

j d0 dq 0 0 1 q q+1 −→ A −→ L −→ L −→··· −→L −→ L −→··· by flabby sheaves q. In fact, it is enough to take for j the injection of into the sheaf 0 of non necessarilyL continuous sections of , and then inductivelyA to embed 0/j(L), 1/d0( 0), ..., q/dq−1( q−1) into flabbyA sheaves 1, 2, ..., q+1. For L A L L L L L L L every q we get an exact sequence e 0 q(U) q(U ) q(U ) q(V ) 0 −→ L −→ L 1 ⊕L 2 −→ L −→ where the injection is given by f (f↾U ,f↾U ) and the surjection by (g ,g ) 7−→ 1 2 1 2 7−→ g2↾V g1↾V ; the surjectivity of this map follows immediately from the fact that q is− flabby. The snake lemma and the De Rham-Weil isomorphism Hq(X, ) HL q(Γ (X, •)) yield: A ≃ L

(3.29) Theorem. For any sheaf on X and any open sets U1, U2 X, set U = U U , V = U U . Then thereA is an exact sequence ⊂ 1 ∪ 2 1 ∩ 2 4. Complex Analytic Sets and Complex Spaces 19

Hq(U, ) Hq(U , ) Hq(U , ) Hq(V, ) Hq+1(U, ) A −→ 1 A ⊕ 2 A −→ A −→ A ···

4. Complex Analytic Sets and Complex Spaces

4.A. Deﬁnition. Irreducible Components

A complex analytic set is a set which can be defined locally by finitely many holomorphic equations; such a set has in general singular points, because no assumption is made on the differentials of the equations. For a detailed study of analytic set theory, we refer to H. Cartan’s seminar (Cartan 1950), to the books of (Gunning-Rossi 1965), (Narasimhan 1966) or the recent book by (Grauert-Remmert 1984).

(4.1) Deﬁnition. Let M be a complex analytic manifold. A subset A M is said to be an analytic subset of M if A is closed and if for every point x ⊂A there exist a 0 ∈ neighborhood U of x and holomorphic functions g ,...,gn in (U) such that 0 1 O A U = z U ; g (z) = ... = gN (z)=0 . ∩ { ∈ 1 }

Then g1,...,gN are said to be (local) equations of A in U.

It is easy to see that a finite union or intersection of analytic sets is analytic: if ′ ′′ ′ ′′ (gj), (gk ) are equations of A , A in the open set U, then the family of all products (g′ g′′) and the family (g′ ) (g′′) define equations of A′ A′′ and A′ A′′ respectively. j k j ∪ k ∪ ∩ (4.2) Remark. Assume that M is connected. The analytic continuation theorem shows that either A = M or A has no interior point. In the latter case, the Riemann extension theorem show that every function f (M r A) that is locally bounded near A can be extended to a function f˜ (M∈). O Moreover M r A is connected. ∈ O We now focus our attention on local properties of analytic sets. If A M is an analytic set, we denote by (A,x) the germ of A at any given point x⊂ M, ∈ and by A,x the ideal of germs f M,x which vanish on (A,x). Conversely, if I ∈ O =(g ,...,gN ) is an ideal of M,x, we denote by V ( ),x the germ at x of the J 1 O J zero variety V ( ) = z U ; g (z) = ... = gN (z)=0 , where U is a neighborhood J { ∈ 1 } of x such that gj (U). It is clear that ∈ O ′ (4.3 ) for every ideal in the ring M,x, , J O IV (J ),x ⊃J ′′ (4.3 ) for every germ of analytic set (A,x), V ( A,x),x =(A,x). I (4.4) Definition. A germ (A,x) is said to be irreducible if it has no decomposition (A,x)=(A ,x) (A ,x) with analytic sets (Aj,x) =(A,x), j =1, 2. 1 ∪ 2 6

(4.5) Proposition. A germ (A,x) is irreducible if and only if A,x is a prime ideal I of the ring M,x. O 20 J.P. Demailly: Basic results on Sheaves and Analytic Sets

Proof. Let us recall that an ideal is said to be prime if fg implies f J ∈ J ∈ J or g . Assume that (A,x) is irreducible and that fg A,x. As we can write (A,x)=(∈ J A ,x) (A ,x) with A = A f −1(0) and A ∈= IA g−1(0), we must 1 ∪ 2 1 ∩ 2 ∩ have for example (A ,x)=(A,x) ; thus f A,x and A,x is prime. Conversely, 1 ∈ I I if (A,x)=(A ,x) (A ,x) with (Aj,x) = (A,x), there exist f A ,x, g A ,x 1 ∪ 2 6 ∈ I 1 ∈ I 2 such that f,g / A,x. However fg A,x, thus A,x is not prime. ∈ I ∈ I I

(4.6) Theorem. Every decreasing sequence of germs of analytic sets (Ak,x) is stationary.

Proof. In fact, the corresponding sequence of ideals k = A ,x is increasing, thus J I k k = k for k k large enough by the Noetherian property of M,x. Hence J J 0 ≥ 0 O (Ak,x) = V ( k),x is constant for k k0. This result has the following straight- forward consequence:J ≥ (4.7) Theorem. Every analytic germ (A,x) has a ﬁnite decomposition

(A,x) = (Ak,x) ≤k≤N 1 [ where the germs (Aj,x) are irreducible and (Aj,x) (Ak,x) for j = k. The decomposition is unique apart from the ordering. 6⊂ 6

4.B. Local Structure of a Germ of Analytic Set

We are going to describe the local structure of a germ of analytic set, both from the holomorphic and topological points of view. By the above decomposition theorem, we may restrict ourselves to the case of irreducible germs. Let be a prime ideal J in n = Cn, and let A = V ( ) be its zero variety. O O 0 J

(4.8) Local parametrization theorem. Let be a prime ideal of n and let A = V ( ). There is an integer d, called the dimensionJ of A, such that afterO rotating the coordinatesJ by a generic linear transformation of Cn, the coordinates

′ ′′ (z ; z )=(z1,...,zd ; zd+1,...,zn) satisfy the following properties. a) The ring morphism

d = C z ,...,zd n/ = C z ,...,zn / O { 1 } −→ O J { 1 } J defines a finite integral extension of d. O ′ Let q be the degree of the extension and let δ(z ) d be the discriminant of the ∈′′ O ′ ′′ irreducible polynomial of a primitive element u(z ) = k>d ckzk. If ∆ , ∆ are polydisks of sufficiently small radii r′,r′′ and if r′ r′′/C with C large, the projection map π : A (∆′ ∆′′) ∆′ is a ramified covering≤ with q Psheets, whose ramification locus is contained∩ × in S−→= z′ ∆′; δ(z′)=0 . This means that: { ∈ } 4. Complex Analytic Sets and Complex Spaces 21

′ ′′ b) the open subset AS = A (∆ r S) ∆ is a smooth d-dimensional manifold, dense in A (∆′ ∆′′)∩ ; × ∩ × ′ c) π : AS ∆ r S is a covering ; −→ d) the ﬁbers π−1(z′) have exactly q elements if z′ / S and at most q if z′ S. ∈ ∈ ′ ′ ′′ Moreover, AS is a connected covering of ∆ r S, and A (∆ ∆ ) is contained in a cone z′′ C z′ (see Fig. 1). ∩ × | | ≤ | |

z′′ Cn−d ∈

0 ∆′′ π A

′ S S ∆ z′ Cd ∈

Ramiﬁed covering from A to ∆′ Cd. ⊂

We refer to the references already mentioned for a proof of these properties. Another fundamental result is:

(4.9) Hilbert’s Nullstellensatz. For every ideal n J ⊂ O = √ , IV (J ),0 J

where √ is the radical of , i.e. the set of germs f n such that some power f k lies inJ . J ∈ O J

In other words, if a germ (B, 0) is deﬁned by an arbitrary ideal n and if k J ⊂ O f n vanishes on (B, 0), then some power f lies in . ∈ O J 22 J.P. Demailly: Basic results on Sheaves and Analytic Sets

4.C. Regular and Singular Points. Dimension

The above powerful results enable us to investigate the structure of singularities of an analytic set. We ﬁrst give a few deﬁnitions.

(4.10) Deﬁnition. Let A M be an analytic set and x A. We say that x A is a regular point of A if A ⊂Ω is a C-analytic submanifold∈ of Ω for some neighborhood∈ Ω of x. Otherwise x is∩ said to be singular. The corresponding subsets of A will be denoted respectively Areg and Asing.

It is clear from the deﬁnition that Areg is an open subset of A (thus Asing is closed), and that the connected components of Areg are C-analytic submanifolds of M (non necessarily closed).

(4.11) Deﬁnition. The dimension of an irreducible germ of analytic set (A,x) is deﬁned by dim(A,x) = dim(Areg,x). If (A,x) has several irreducible components (Al,x), we set

dim(A,x) = max dim(Al,x) , codim(A,x) = n dim(A,x). { } −

4.D. Coherence of Ideal Sheaves

Let A be an analytic set in a complex manifold M. The sheaf of ideals A is the I subsheaf of M consisting of germs of holomorphic functions on M which vanish O on A. Its stalks are the ideals A,x already considered; note that A,x = M,x if I I O x / A. If x A, we let A,x be the ring of germs of functions on (A,x) which can be∈ extended∈ as germs ofO holomorphic functions on (M,x). By deﬁnition, there is a surjective morphism M,x A,x whose kernel is A,x, thus O −→ O I A,x = M,x/ A,x, x A, O O I ∀ ∈ i.e. A =( M / A)↾A. Since A,x = M,x for x / A, the quotient sheaf M / A is zeroO on M rO A. I I O ∈ O I

(4.12) Theorem (Cartan 1950). For any analytic set A M, the sheaf of ideals A is a coherent analytic sheaf. ⊂ I

(4.13) Corollary. Asing is an analytic subset of A.

Proof. The statement is local. Assume ﬁrst that (A, 0) is an irreducible germ in n C . Let g1,...,gN be generators of the sheaf A on a neighborhood Ω of 0. Set d = dim A. In a neighborhood of every point xI A Ω, A can be deﬁned by ∈ reg ∩ holomorphic equations u1(z) = ... = un−d(z) = 0 such that du1,...,dun−d are linearly independant. As u1,...,un−d are generated by g1,...,gN , one can extract

a subfamily gj1 ,...,gjn−d that has at least one non zero Jacobian determinant of rank n d at x. Therefore A Ω is deﬁned by the equations − sing ∩ 4. Complex Analytic Sets and Complex Spaces 23

∂gj det j∈J =0, J 1,...,N ,K 1,...,n , J = K = n d. ∂zk k∈K ⊂{ } ⊂{ } | | | | − Assume now that (A, 0) = (Al, 0) with (Al, 0) irreducible. The germ of an analytic set at a regular point is irreducible, thus every point which belongs simultaneously to at least two componentsS is singular. Hence

(A , 0) = (Al, , 0) (Ak Al, 0), sing sing ∪ ∩ k6 l [ [=

and Asing is analytic.

4.E. Complex Spaces

A complex space is a space locally isomorphic to an analytic set A in an open subset n Ω C , together with a sheaf of rings A = Ω/ associated to a coherent sheaf of⊂ ideals on Ω such that V ( ) = A.O This ideaO J is made precise by the following deﬁnitions.J J

(4.14) Deﬁnition. A ringed space is a pair (X, X ) consisting of a topological space R X and of a sheaf of rings X on X, called the structure sheaf. A morphism R

F :(X, X ) (Y, Y ) R −→ R of ringed spaces is a pair (f, F ⋆) where f : X Y is a continuous map and −→ ⋆ −1 ⋆ F : f Y X , F : ( Y ) ( X )x R −→ R x R f(x) −→ R a homomorphism of sheaves of rings on X, called the comorphism of F .

If F : (X, X ) (Y, Y ) and G : (Y, Y ) (Z, Z) are morphisms of ringed spaces, theR composite−→ RG F is the pair consistingR −→ of theR map g f : X Z and of the comorphism (G F )◦⋆ = F ⋆ f −1G⋆ : ◦ −→ ◦ ◦ −1 ⋆ ⋆ ⋆ −1 ⋆ −1 −1 f G −1 F F f G : f g Z f Y X , (4.15) ⋆ ◦ ⋆ R −−−→ R −−→ R F G : ( Z ) ( Y ) ( X )x. x ◦ f(x) R g◦f(x) −−−→ R f(x) −−→ R We begin by a description of what will be the local model of an analytic complex n space. Let Ω C be an open subset, Ω a coherent sheaf of ideals and A = V ( ) the⊂ analytic set in Ω deﬁnedJ locally ⊂ O as the zero set of a system of J generators of . By Hilbert’s Nullstellensatz 4.9 we have A = √ , but A diﬀers J I J I in general from . The sheaf of rings Ω/ is supported on A, i.e. ( Ω/ )x = 0 J O J O J if x / A. Ringed spaces of the type (A, Ω/ ) will be used as the local models of analytic∈ complex spaces. O J

(4.16) Deﬁnition. A morphism

⋆ ′ ′ F =(f, F ):(A, Ω/ ↾A) (A , Ω′ / ′ ) O J −→ O J↾A 24 J.P. Demailly: Basic results on Sheaves and Analytic Sets

is said to be analytic if for every point x A there exists a neighborhood Wx of x ∈ ′ in Ω and a holomorphic function Φ : Wx Ω such that f↾A∩Wx = Φ↾A∩Wx and such that the comorphism −→

⋆ ′ F :( Ω′ / ) ( Ω/ )x x O J f(x) −→ O J ⋆ ⋆ ′ is induced by Φ : ′ u u Φ Ω,x with Φ . OΩ ,f(x) ∋ 7−→ ◦ ∈ O J ⊂J Cn 2 Cn−1 (4.17) Example. Take Ω = and =(zn). Then A is the hyperplane 0 , J ×{ }′ and the sheaf Cn / can be identiﬁed with the sheaf of rings of functions u + znu , ′ O J 2 u, u Cn−1 , with the relation z = 0. In particular, zn is a nilpotent element of ∈ O n Cn / . A morphism F of (A, Cn / ) into itself is induced (at least locally) by a O J O J n holomorphic map Φ =(Φ, Φn) deﬁned on a neighborhood of A in C with values in n C , such that Φ(A) A, i.e. Φn↾A = 0. We see that F is completely determined by ⊂ the data e n−1 n−1 f(z1,...,zn−1)= Φ(z1,...,zn−1, 0), f : C C , ′ ∂Φ ′ n−1 −→ n f (z1,...,zn−1)= (z1,...,zn−1, 0), f : C C , ∂zn −→ e which can be chosen arbitrarily.

(4.18) Deﬁnition. A complex space is a ringed space (X, X ) over a separable Haus- dorﬀ topological space X, satisfying the following property:O there exist an open covering (Uλ) of X and isomorphisms of ringed spaces

Gλ :(Uλ, X↾U ) (Aλ, Ω / λ ↾A ) O λ −→ O λ J λ N where Aλ is the zero set of a coherent sheaf of ideals λ on an open subset Ωλ C λ , −1 J ⊂ such that every transition morphism Gλ G is a holomorphic isomorphism from ◦ µ gµ(Uλ Uµ) Aµ onto gλ(Uλ Uµ) Aλ, equipped with the respective structure ∩ ⊂ ∩ ⊂ sheaves Ω / µ ↾A , Ω / λ ↾A . O µ J µ O λ J λ

We shall often consider the maps Gλ as identiﬁcations and write simply Uλ = Aλ. A morphism F : (X, X ) (Y, Y ) of analytic complex spaces obtained by O −→ O′ ′ ′ ′ gluing patches (Aλ, Ωλ / λ ↾Aλ ) and (Aµ, Ω / µA ), respectively, is a morphism O J O µ J µ F of ringed spaces such that for each pair (λ, µ), the restriction of F from Aλ f −1(A′ ) X to A′ Y is holomorphic in the sense of Def. 4.16. ∩ µ ⊂ µ ⊂ Nilpotent Elements and Reduced Complex Spaces.. Let (X, X ) be an analytic O complex space. The set of nilpotent elements is the sheaf of ideals of X deﬁned by O k (4.19) X = u X ; u = 0 for some k N . N { ∈ O ∈ }

Locally, we have X↾A =( Ω / λ)↾A , thus O λ O λ J λ

(4.20) X↾A =( λ/ λ)↾A , N λ J J λ (4.21) ( X / X )↾pA ( Ω / λ)↾A =( Ω / A )↾A = A . O N λ ≃ O λ J λ O λ I λ λ O λ

The complex space (X, X ) is saidp to be reduced if X = 0. The associated ringed O N space (X, X/ X ) is reduced by construction; it is called the reduced complex space O N 4. Complex Analytic Sets and Complex Spaces 25

of (X, X). We shall often denote the original complex space by the letter X merely, O the associated reduced complex space by Xred, and let X,red = X / X . There is a canonical morphism X X given by the comorphismO O N red → (4.22) X (U) X, (U)=( X / X )(U) O −→ O red O N for every open set U in X ; the kernel of (4.22) is the set of locally nilpotent sections of X on U. It is easy to see that a morphism F of reduced complex spaces X,Y is completelyO determined by the underlying set-theoretic map f : X Y . →