Classical comparison theorems

4.3. Theorem. The canonical morphisms κ^•_DR,c and κ^DR_• are isomorphisms compatible with the Poincar´e duality.

Proof. We suppose that P_i are proper (and smooth). The compatibility with the Poincar´e duality is obvious by construction, and therefore it is sufficient to show that the K¨unneth morphisms for De Rham cohomology with compact supports are isomorphisms. By the previous Lemma we obtain the following isomorphism

The first term is isomorphic to lim←− In fact, since the Pi are proper, the K-vector spaces

H^•(Pi,IC^N_i^−•Ω_P^•_i are finite dimensional. The second term is isomorphic to

lim←−

In this section we keep the notation of our setting 1.1, and assume K = C. We compare the algebraic De Rham cohomology with compact supports of the algebraic variety X, with the singular cohomology with

compact supports of the corresponding complex analytic space X^an. We refer to chapter IV of [H.75] for the analogous statements for De Rham cohomology without supports (or with supports in a closed subvariety Z of X) and for De Rham homology.

We recall that in the construction of the algebraic pairing of (3.1) we assumed for simplicity P to be smooth. The algebraic pairing was then shown to be independent of the choice of P . When comparing the algebraic and analytic Poincar´e duality pairings we will make the same assumption. We then construct canonical morphisms relating the spectral sequence of algebraic De Rham cohomology with compact supports (resp. algebraic De Rham homology) and the analytic one. These morphisms will be compatible with the natural pairings of spectral sequences of section 3 and of [HL].

5.0. Results from [H.75, IV]. Let T be a complex analytic space and S be a closed analytic subspace defined by the coherentOT-IdealIS. Then T/S will denote the formal completion of T along S, namely the ringed space with underlying topological space S and structural sheaf OT_/S := lim

←−^N(OT/IS^N)_|S. Similarly, for a coherent OT-ModuleF , F/S will denote the OT_/S-Module lim

←−^N(F /JS^NF )_|S, which coincides with the inverse image of the OT-Module F on T/S. If κ : U ,→ T is an open immersion of analytic spaces, or formal completions of such, the functor

κ!:A b(U) −→ A b(T ) ,

left adjoint to κ⁻¹:A b(T ) −→ A b(U), is the usual topological extension by zero.

5.0.1. The formal analytic Poincar´e lemma [H.75, IV.2.1]. The complex (Ω^•_Wan)_/Xan is a resolution of the constant sheaf CX^an in the category of abelian sheaves on X^an.

5.0.2. [H.75, IV.1.1]. The canonical morphism

βⁱ: H_Xⁱan(W^an, C)−−→ HⁱX^an(W^an, Ω^•_Wan)

is an isomorphism for any i. We recall that, if dim W = n, H_X²ⁿan⁻ⁱ(W^an, C) ∼= H_i^BM(X^an, C), the Borel-Moore homology of X^an. We set

H_i^DR(X^an) := H²ⁿ_Xan⁻ⁱ(W^an, Ω_W^• an) . 5.0.3. [H.75, loc. cit.]. The canonical morphism

αⁱ: H_DRⁱ (X)−−→ HDRⁱ (X^an)

is an isomorphism for any i (apply αⁱ of Hartshorne with X, X, W as Z, X, Y ).

5.0.4. [H.75, loc. cit.]. The canonical morphism

α_i: H_i^DR(X)−−→ Hi^DR(X^an)

is an isomorphism for any i (apply αⁱ of Hartshorne with X, W , W as Z, X, Y ).

5.1. Analytic De Rham cohomology with compact supports. We define H_DR,c^• (X^an) := H^•_c(X^an, (Ω_W^• an)/X^an) ∼= H^•(X^an, j_!^an(Ω_W^• an)/X^an) and we recall that

H_cⁱ(X^an, C) ∼= Hⁱ(X^an, j_!^anC_Xan) . By 5.0.1 and the exactness of j^an_! , the canonical morphism

β_cⁱ: H_cⁱ(X^an, C)−−→ HDR,cⁱ (X^an) is an isomorphism for any i.

5.1.1. Lemma. Let κ : U ,→ T be an open immersion of complex analytic spaces and J be a coherent sheaf of OT-Ideals such that the support of OT/J is T r U. Let F be a coherent OU-Module and let F be any coherent extension ofF to T . Then the canonical morphism

κ!F −→ lim←−

N

J^NF is an isomorphism.

Proof. The assertion is easily checked on the fibers. In fact, if x is a point of T r U ,Jx is a proper ideal of the noetherian ringOT ,x. So, lim

←−^NJx^NFx=T

NJx^NFx= 0. On the other hand, for any M , the canonical morphism lim

←−^NJ^NF −→ J^MF is a monomorphism, so that (lim←−

N

J^NF )x⊆ lim←−

N

Jx^NFx= 0

is also zero. On the other hand, both sheaves restrict toF on U.  28

5.1.2. Lemma. LetF be a coherent OP^an-Module. The canonical morphism j_!^an((F_|W^an)_/Xan)−→ lim←−

N

IC^NanF_/X^an is an isomorphism.

Proof. We have to show that for any x∈ P^an, the fiber ( lim

M >N←−

IC^NanF /I_X^ManF )x=







0 if x∈ P^anrX^an

(lim←−^MF_|W^an/I_X^ManF_|W^an)_x if x∈ X^an

0 if x∈ C^an .

If x∈ P^anrX^anthe assertion is clear: on open neighborhoods U of x, x∈ U ⊆ P^anrX^an ( lim

M >N←−

IC^NanF /I_X^ManF )(U) = lim←−

M >N

((IC^NanF /I_X^ManF )(U)) = 0 ,

since (IC^an)_|PanrX^an = (I_X^an)_|PanrX^an= (OP^an)_|PanrX^an. If x∈ X^an, the assertion is also clear, since ( lim

M >N←−

IC^NanF /I_X^ManF )x= lim

x∈U⊆W−→ ^an ( lim

M >N←−

(IC^NanF /I_X^ManF )(U)) ,

where U varies among open neighborhoods of x contained in W^an. But on W^an, (IC^an)_|Wan =OW^an, so that

lim−→

x∈U⊆W^an

( lim

M >N←−

(IC^NanF /I_X^ManF )(U)) = lim

x∈U⊆W−→ ^an (lim←−

M

(F_|W^an/IX^ManF_|W^an)(U )) = (lim

←−M

F_|W^an/IX^ManF_|W^an)_x.

We are left to show that, for x∈ C^an, (lim

←−M >NIC^NanF /I_X^NanF )x= 0. To check this, we write j_PanrX^an : P^anrX^an,→ P^an and jP^anrC^an: P^anrC^an,→ P^an

for the open immersions, and apply (5.1.1) to obtain the exact sequence 0−→(jP^anrX^an)_!F_|PanrX^an−→(jP^anrC^an)_!F_|P^anrC^an−→ lim←−

M >N

IC^NanF /I_X^ManF −→

−→ coker (F → F_/X^an)−→ coker (F → F/C^an)−→ · · · Taking fibers at x∈ C^an, we obtain the exact sequence

0−→( lim←−

M >N

IC^NanF /I_X^ManF )x−→ coker (F → F_/X^an)x=

= coker (Fx→(F_/X^an)_x)−→ coker (F → F/C^an)_x= coker (Fx→(F/C^an)_x) .

Now, for a Stein semianalytic compact neighbourhood K of x in P^an, Γ(K,F ) is a module of finite type over the noetherian ring Γ(K,OP^an). For a coherentOP^an-IdealJ , and K as before, we denote by \Γ(K,F )Γ(K,J )

the Γ(K,J )-adic completion of Γ(K, F ). Then, by [BS, VI.2.2 (i)], (F/C^an)x= lim

−→K

Γ(K,\F )_Γ(K,ICan) , (F_/X^an)x= lim

−→K

Γ(K,\F )_Γ(K,I

Xan). Then

ker (coker (Fx→(F/X^an)_x)−→ coker (Fx→(F/C^an)_x)) = lim−→

K

ker ( \Γ(K,F )Γ(K,I_X^an)/Γ(K,F ) −→ \Γ(K,F )Γ(K,ICan)/Γ(K,F )) = 0 .

 5.2. The canonical isomorphism

j_!^an(Ω^•_Wan)_/Xan−−−→ lim←−

M >N

(I_C^Nan^−•Ω_P^•an

I_X^Man−•) induces a canonical morphism of hypercohomology groups

H_DR,cⁱ (X^an)−−−→ lim←−

M >N

Hⁱ(X^an,IC^Nan−•Ω_P^•an

I_X^Man−•) .

Via the GAGA isomorphisms lim←−

M >N

H^•(X^an,I_C^Nan^−•Ω_P^•an

I_X^Man−•)←−−− lim^∼= ←−

M >N

H^•(X,I_C^N−•Ω_P^•I_X^M−•) we obtain a canonical morphism

(5.2.1) αⁱ_c : H_DR,cⁱ (X^an)−−−→ HDR,cⁱ (X) .

5.3. Theorem. The canonical morphism αⁱ_c is an isomorphism for any i.

Proof.

H_DR,c^• (X^an) = H^•(X^an, j_!^an(Ω_W^• an)_/Xan)

∼

−−−→=

(5.1.2)H^•(X^an, lim

←−N

I_C^Nan^−•(Ω_P^•an)_/X^an)

∼= H^•(X^an, lim

M >N←−

(I_C^Nan^−•Ω_P^•an

I_X^Man−•))

∼=

−−−→ H^•(X^an, lim

M >N←−

(Ω_P^•an

I_X^Man−•→ Ω^•P^an

IC^Nan−•)tot)

∼=

−−−→(1.3.1)

lim←−

M >N

H^•(X^an, (Ω^•_Pan

I_X^Man−•→ ΩP^•an

IC^Nan−•)_tot)

∼= lim

M >N←−

H^•(X^an,IC^Nan−•Ω_P^•an

I_X^Man−•)

∼=

←−−−GAGA lim

M >N←−

H^•(X,I_C^N−•Ω^•_PI_X^M−•) = H_DR,c^• (X)

 5.3.1. We point out the following alternative proof of theorem 5.3, valid if P is supposed to be smooth.

We have a morphism of exact sequences of abelian sheaves on X^an

0−−−−−−→ CX^an−−−−−−−−→ C_Xan −−−→ C_Can −−−→ 0

 y

 y

 y 0−−−→ j^an! (Ω_W^• an)_/Xan −−−→ (ΩP^•an)_/X^an −−−→ h^an_∗ (Ω_P^•an)_/Can

where as usual C_Xan = j^an_! C_Xan and C_Can = h^an_∗ C_Can, as sheaves on X^an. All vertical arrows are quasi-isomorphism by the Formal Analytic Poincar´e lemma and exactness of j_!^an and h^an_∗ . In particular

H_DR,c^• (X^an) = H^•(X^an, j^an_! (Ω_W^• an)/X^an)−→ H^{∼= •}(X^an, ((Ω_P^•an)_/X^an→ h^an_∗ (Ω_P^•an)_/C^an)tot) which considerably simplifies the above proof.

5.3.2. We remark moreover that in the analytic case Rlim←−

M >N

IC^Nan−•Ω^•_Pan

I_X^Man−• ∼= lim

M >N←−

IC^Nan−•Ω_P^•an

I_X^Man−•

while in the algebraic case this is false; in fact in general lim

←−

(1)

M >NI_C^N−•Ω_P^•I_X^M−•6= 0.

5.4. Comparison of algebraic and analytic Poincar´e dualities. We will show that the pairings (3.5.4) (and therefore also the duality morphisms (3.1.2)) are compatible with the algebraic-analytic comparison maps. Notation is as in (3.1); in particular, P is smooth.

5.4.1. Proposition. Under the assumptions of (3.1), we have a canonical morphism of pairings of spectral sequences

Ext^•

Ω_P^•(M^•, Ω^•_P) × H^•(X,M^•) −−−−−→ H^•

X(P, Ω^•_P)





 y





 y





 y Ext^•_Ω•

P an

(M^•an, Ω_P^•an)× H^•(X^an,M^•an)−−−−−→ H^•_X^an(P^an, Ω_P^•an) . 30

Proof. LetI^••be a resolution ofM^•as in (3.5.2). LetM^{(M )•an}−→ J^{(M )••}be an injective resolution ofM^{(M )•an}in the category of C_(X^• an

)^{(M )}_{P an}-Modules. We obtain a resolution

(5.4.2) M^•an∼= “lim

←−”

M

M^{(M )•an}−→ “lim←−”

M

J^{(M )••}=:J^••

ofM^•anin ProC (P^an). There is a canonical morphism of complexes of objects of ProC (P^an),I^••an−→ J^••

(which is an isomorphism of D(ProC (P^an))). On the other hand, if I^•(Ω_P^•an) is an injective resolu-tion of Ω^•_Pan as a CP^•an-Module, then we have a canonical morphism of complexes of objects of C (P^an), E^•(Ω^•_P)^an−→ I^•(Ω_P^•an) (again an isomorphism of D(C (P^an))). Therefore we have the following diagram of (pairings of) double complexes

Hom^••Ω^•_{P an}(I^••an, E^•(Ω^•_P)^an)⊗ I^••an−−−→ (Γ_XE^•(Ω_P^•))^an

 y

 y Hom^••Ω^•_{P an}(I^••an, I^•(Ω^•_Pan)) ⊗ I^••an−−−→ Γ_X^anI^•(Ω_P^•an)

x



 y

Hom^••Ω_{P an}^• (J^••, I^•(Ω_P^•an)) ⊗ J^•• −−−→ Γ_X^anI^•(Ω_P^•an) . We notice that the canonical morphisms of bicomplexes

Hom^••Ω^•_{P an}(I^••an, I^•(Ω_P^•an))−−−→ Hom^•Ω^•_{P an}(M^•an, I^•(Ω_P^•an))←−−− Hom^••Ω^•_{P an}(J^••, I^•(Ω_P^•an)) induce isomorphisms of the associated spectral sequences (see for example [HL, 4.2]). Therefore, taking the associated diagram of pairings of spectral sequences and composing with the canonical GAGA morphism of pairings of spectral sequences induced by

Hom^••Ω_P^•(I^••, E^•(Ω_P^•)) ⊗ I^•• −−−→ Γ_XE^•(Ω_P^•)

 y

 y

 y

ε_∗Hom^••Ω^•_{P an}(I^••an, E^•(Ω^•_P)^an)⊗ ε_∗I^••an−−−→ ε_∗Γ_XE^•(Ω_P^•)^an,

we complete the proof. 

5.4.3. Theorem. For any i, the canonical comparison isomorphisms αi of (5.0.4) and αⁱ_c of (5.2.1) are compatible with Poincar´e duality, i.e. they fit in a commutative diagram

H_i^DR(X) ⊗ HDR,cⁱ (X) −−−→ C

α_iok ok^αic k

H_i^DR(X^an)⊗ HDR,cⁱ (X^an)−−−→ C . Proof. We apply the previous proposition, withM^•= “lim

←−”

Mj^{(M )}_! Ω^•

X_W^{(M )}. We obtain a commutative diagram

H_i^DR(X) ⊗ HDR,cⁱ (X) −−−→ H²ⁿ

X(P, Ω_P^•) = H₀^DR(X)

αiok ok^αic ok^α0

H_i^DR(X^an)⊗ HDR,cⁱ (X^an)−−−→ H²ⁿ_X^an(P^an, Ω^•_Pan) = H₀^DR(X^an).

We consider the algebraic trace map Tr : H²ⁿ

X(P, Ω_P^•)−→ C and the map Tr^an: H²ⁿ

X^an(P^an, Ω_P^•an)−→ C uniquely defined by fitting in the commutative diagram

H²ⁿ

X(P, Ω_P^•) −−−→ C^Tr





 y

∼=

H²ⁿ

X^an(P^an, Ω^•_Pan)−−−→

Tr^an

C

Composing with Tr and Tr^an, we obtain the diagram of the statement. 

Nel documento Poincar´ e Duality for Algebraic De Rham Cohomology. (pagine 27-31)