WITH SUPPORTS
ANDREI BARAN
Let X be an analytic space, Φ a family of supports on X, and F a coher- entOX-module. Andreotti-Kas [3] and Andreotti-Banica [1] introduced natural inductive limit topologies and projective limit topologies on the cohomology groupsHΦq(X,F). In this paper, we prove that these topologies coincide. Thus, HΦq(X,F) is endowed with a natural topology which in general (if Φ is different fromclorc– the families of all the closed sets ofX, respectively, of the compact sets ofX) may not be of type quotient of Fr´echet-Schwatz or quotient of strong dual of Fr´echet-Schwatz. As an application we state absolute duality results for cohomological invariants with supports, endowed with the natural topologies.
AMS 2010 Subject Classification: 32C35, 32C37, 46A04, 46A13.
Key words: Serre duality, family of supports, Dolbeault cohomology groups with supports, ˇCech cohomology groups with supports.
1. INTRODUCTION
Let X be an analytic space and F a coherent OX-module. It is well- known that the cohomology group Hq(X,F) has a natural topology of type QFS (quotient of Fr´echet-Schwatz) and the cohomology group Hcq(X,F) has a natural topology of type QDFS (quotient of the strong dual of a Fr´echet- Schwatz space), both topologies deduced, ultimately, from the topology of uniform convergence on compact sets on the sections of OX. These topologies appear, for instance, in the absolute duality theorems of Serre-Malgrange.
Introducing natural locally convex topologies onHΦ•(X,F), the cohomol- ogy groups with supports in the family of supports Φ onX, proved more diffi- cult, and was done in a series of papers by A. Andreotti, A. Kas, and C. B˘anic˘a:
[1–4]. These topologies are defined via ˇCech computations: for each suitable open covering one introduces two topologies – an “inductive limit topology”
and a “projective limit topology”. Since in general there is no open mapping theorem for these topologies, a priori, all these topologies may be different (the natural refinement mappings are continuous, but it is not obvious how to show directly that they are open).
MATH. REPORTS15(65),4(2013), 331–342
The purpose of this paper is to prove that, if X is a complex manifold, then all these topologies coincide (see Theorem 5.1). This unique topology will be called the natural topology.
A partial result was obtained in [1], Proposition 4.2 – namely that, for a given open covering, the inductive limit topology and the projective limit topology coincide under the suplimentary condition that the family of supports and its dual are obtained by a 2-point compactification of X.
The idea of the proof of Theorem 5.1 is to compare the topologies defined with ˇCech computations with similar topologies defined using the Dolbeault resolution onX.
The statement of Theorem 5.1 holds with exactly the same proof if X is a complex space with singularities, provided we produce an analogue for the Dolbeault complex on X. This can be found in [7], where, using an “em- bedding atlas” for the analytic space X (i.e. a family of local embeddings of X in complex manifolds) one constructs, for each coherent OX-module. F, a resolution for F whose terms are soft sheaves, as well as topological sheaves, with topologies of type FS.
For interesting examples of families of supports for which Theorem 5.1 applies, other then the family of all closed sets or the family of compact sets, see Examples 2.4 or 2.5.
As an application of Theorem 5.1 we state absolute duality results for cohomological invariants with supports endowed with the natural topologies (Section 6). A particular case of Theorem 6.1 (for locally free OX-modules and families of supports of a special form, linked to the existence of a plurisub- harmonic exhaustion function) was obtained in [8].
2. FAMILIES OF SUPPORTS
Let X be a topological space. Recall that a family of supports Φ on X consists of a family of closed subsets of X s.t.
ifS ∈Φ andS1⊂S is a closed subset then S1∈Φ ifS1, S2∈Φ thenS1∪S2∈Φ.
The family of supports Φ is said to be paracompactifying if every S ∈Φ is paracompact and has a closed neighbourhood which is also in Φ.
The dual of Φ is the family of supports Ψ ={T ⊂X|T closed andT∩S compact for all S∈Φ}. A pair of families of supports (Φ,Ψ) is a dual pair of families of supports if each is the dual of the other.
Example2.1. The family of all closed subsets ofXis a a family of supports that we denote by “cl”.
Example 2.2. The family of all compact subsets ofX is a family of sup- ports that we denote by “c”.
Example 2.3. If S ⊂ X is a closed subset then the family of all closed subsets ofS is a family of supports.
Example 2.4. LetK ⊂X be a compact set. Then on XK one defines the families of supports.
Φ ={S ⊂XK |S is relatively compact inX}
Ψ ={T ⊂XK |T is closed in X}
Example 2.5. Let f : X → Y be a continuous mapping between two topological spaces andV ⊂Y an open set. On f−1(V) one defines the families of supports
c(f, V) ={S⊂f−1(V)|S closed andf(S) is relatively compact inV} Ψ(f, V) ={T ⊂f−1(V)|T closed andf|T :T →V is a proper mapping}
Remark that if X is paracompact then the families of supports in ex- amples 2.1, 2.2, 2.4, 2.5 are paracompactifying, while the family of supports in Example 2.3 is not. The pair (cl, c) is a dual pair of families of supports.
Similarly, the families of supports defined in examples 2.4 and 2.5 form dual pair of families of supports.
3.TOPOLOGIES ON ΓΦ(X,F)
Let X be a topological space and F a topological sheaf on X (i.e. for all open set U ⊂ X F(U) is a locally convex topological vector space, and all restrictions are continuous). If S ⊂ X is a closed subset then we consider ΓS(X,F) as a topological vector subspace of Γ(X,F).
Let Φ be a family of supports on X. The “inductive limit topology” on ΓΦ(X,F) is defined via the isomorphism:
lim−→
S∈Φ
ΓS(X,F)→ΓΦ(X,F)
Let (Φ,Ψ) be dual pair of families of supports on X. For T ∈ Ψ set Φ(T) = {F ⊂ X | F closed and F ∩T compact}. The “projective limit topology” on ΓΦ(X,F) is defined via the isomorphism:
ΓΦ(X,F)→ lim←−
T∈Ψ
lim−→
F∈Φ(T)
ΓF(X,F) Proposition 3.1. The identity mapping:
id: lim−→
S∈Φ
ΓS(X,F)→ lim←−
T∈Ψ
lim−→
F∈Φ(T)
ΓF(X,F) is continuous.
Proof. Follows easily from the commutative diagram:
ΓS(X,G) −−−−→iS lim−→
F∈I(T)
ΓS(X,G)
y
x
lim−→
S∈Φ
ΓS(X,G) −−−−→id lim←−
T∈Ψ
lim−→
F∈I(T)
ΓS(X,G) since the inclusion iS is continuous for allS ∈Φ
In the sequel, all the topological spaces will be assumed paracompact and locally compact and the family of supports will be assumed paracompactifying.
4. CECH COHOMOLOGY WITH SUPPORTSˇ
Let X be an analytic space and Φ a family of supports on X. If U = (Ui)i∈I is an open covering ofX and S ∈Φ set:
I(S) ={i∈I|Ui∩S6=∅}, U(S) = (Ui)i∈I(S), andSU(1) = [
i∈I(S)
Ui
To simplify notation we write S(1) instead of SU(1)if the covering is clear from the context. One says that the coveringU is adapted to Φ if, for allS∈Φ, S(1) ∈Φ. Let N(U) be the nerve of U and N(U, q) the subset of simpleces of lengthq in N(U).
If F is anOX-module let
0→ F → C0(U,F)→ C1(U,F)→...
be the ˇCech complex of F relative to U. We setCΦq(U,F) = ΓΦ(X,Cq(U,F)) and let HΦ•(U,F) be the cohomology of the complex CΦ•(U,F).
If the coveringU is adapted to Φ one checks that there is an isomorphism:
(4.1) CΦq(U,F) = lim−→
S∈Φ
Cq(U(S),F) = lim−→
S∈Φ
Y
α∈N(U(S),q)
F(Uα)
where, as usual, if α = (i0, ..., iq) then Uα =U0 ∩...∩Uq. Remark that if U is a Stein covering of X and F is Stein-acyclic (e.g. F coherent OX-module) then HΦq(U,F) is algebraically isomorphic with HΦq(X,F). Assume now that F is a topological OX-module (e.g. F a coherent OX-module or, for X a complex manifold, F a sheaf of (p, q)-differential forms with coefficients in a coherent OX-module). The isomorphism 4.1 induces on CΦ•(U,F) an induc- tive limit topology – we denote it by (CΦ•(U,F),“lim−→”). One checks that the differentials of the ˇCech complex are continuous for these topologies. Denote
by (HΦ•(U,F),“lim−→”) the corresponding cohomology groups endowed with the induced topologies.
If (Φ,Ψ) is a dual pair of families of supports and the coveringUis adapted to both Φ and Ψ one has another natural way of computingCΦq(U,F):
(4.2) CΦq(U,F) = lim←−
T∈Ψ
Ccq(U(T),F) = lim←−
T∈Ψ
M
β∈N(U(T),q)
F(Uβ)
IfF is a topologicalOX-module the isomorphism 4.2 induces onCΦ•(U,F) a projective limit topology – we denote it by (CΦ•(U,F),“lim←−”). One checks that the differentials of the ˇCech complex are continuous for these topologies.
Denote by (HΦq(U,F),“lim←−”) the corresponding cohomology groups endowed with the induced topologies.
The two topologies actually coincide:
Proposition 4.1. Let X be an analytic space and F a topological OX- module. If (Φ,Ψ) is a dual pair of families of supports and the covering U is adapted to both Φ and Ψthen the identity map:
(CΦq(U,F),“lim−→”)→(CΦq(U,F),“lim←−”) is a topological isomorphism.
The proposition will follow from a general lemma on topological vector spaces. To state the lemma we need an ad-hoc definition. For an index set I, a family of subsetsS ⊂ P(I), is called an inductive system of subsets ofI if all subsets of α∈ S belong to S and the union of two elements ofS is in S. The dual inductive system of S is the inductive systemT ={β ⊂I |β∩α finite, for all α∈ S}.Remark that if U = (Ui)i∈I is a covering of X and Φ a family of supports on X then (I(S)S∈Φ) is an inductive system of subsets of I.
Lemma 4.2. Let I be an index set and S,T two inductive systems of subsets of I, each the dual of the other, and let (Fi)i∈I a familly of locally convex topological vector spaces. Then the identity map:
lim−→
α∈S
Y
i∈α
Fi −→ lim←−
β∈T
M
i∈β
Fi
is a topological isomorphism, where on both terms one considers the canonical topologies deduced from the topologies of the spaces Fi.
Proof. One checks easily that the two terms describe the same subset in Q
i∈I
Fi. The continuity follows from the commutative diagrams:
lim−→
α∈S
Q
i∈α
Fi id
−−−−→ lim←−
β∈T
L
i∈β
Fi
x
y
πβ
Q
i∈α
Fi ρ
α
−−−−→β L
i∈β
Fi
since the morphismsραβ are obviously continuous, whereραβ((vi)i∈α) = ((wi)i∈β), withwi =vi for i∈α∩β and wi = 0,otherwise.
To check that id is open let W be a circled convex 0-neighbourhood in lim−→
α∈S
QFi. Set Wi = W ∩Fi and β(W) = {i ∈ I |Wi 6= Fi}. Since Q
i∈α
Fi is endowed with the product topology, α∩β(W) is finite for all α and hence, β(W)∈ T. Moreover,
W = [(W ∩ Y
i∈β(W)
Fi)× Y
i /∈β(W)
Fi]∩lim−→
α∈S
Y
i∈α
Fi
From this, it follows immediately thatπβ(W)(W) is open in L
i∈β(W)
Fi and W =πβ(W)−1 πβ(W)(W) which ends the proof of the lemma.
The Propsition follows now applying the lemma with I = N(U, q), S = (N(U(S), q)S∈Φ),T = (N(U(T), q)T∈Ψ), and Fα = F(Uα) for all α∈ N(U, q).
Corollary 4.3. Under the hypothesis of the proposition, the topologies (HΦ•(U,F),“lim−→”) and(HΦ•(U,F),“lim←−”) coincide.
5.THE COMPARISON THEOREM
Let X be a complex manifold and F a coherent OX-module. Since the sheaves EXp,q of (p, q)-differential forms on X, are OX-flat (see [9]), it follows that the Dolbeault complex:
0−→ F −→ EX0,0⊗OX F −→...−→ EX0,n⊗OXF −→0
is a resolution of F. To simplify notation in the sequel, ifX is clear from the context, we shall write Ep,q forEXp,q and ⊗for⊗OX.
Assume now that Φ is a paracompactifying family of supports. Since the terms of the Dolbeault resolution are soft sheaves, it follows that the groups HΦ•(X,F) can be computed as the cohomology of the complex ΓΦ(X,E0,•⊗F).
Moreover, the terms of the resolution are topological sheaves (for an open set U ⊂ X, Γ(U,Ep,q ⊗ F) is a Fr´echet-Schwartz space). Hence, the terms
of ΓΦ(X,E0,•⊗ F) can be endowed with the “inductive limit topology” (see Section 3); the differentials of the complex are continuous with respect to these topologies. Let (HΦ•(X,F),“lim−→”) denote the cohomology groups endowed with the induced topologies.
Similarly, let (Φ,Ψ) be a dual pair of families of supports onX.The terms of ΓΦ(X,E0,•⊗ F) can be endowed with the “projective limit topology” (see Section 3); the differentials in the complex are continuous with respect to these topologies. Let (HΦ•(X,F),“lim←−”) denote the cohomology groups endowed with the induced topologies.
Theorem 5.1 (Comparison of topologies). Let X be a complex manifold and F a coherent OX-module.
1. If Φ is a family of supports on X and U = (Ui)i∈I is a locally finite covering ofX with Stein, relatively compact open sets adapted to Φ then the natural morphism
(HΦq(U,F),“ lim−→”)→(HΦq(X,F),“ lim−→”) is a topological isomorphism.
2. If (Φ,Ψ) dual pair of families of supports and U = (Ui)i∈I is a locally finite covering of X with relatively compact Stein open sets adapted to bothΦandΨthen all the morphisms in the diagram below are topological isomorphisms:
(5.1)
(HΦq(U,F),“lim−→”) −−−−→ (HΦq(X,F),“lim−→”)
y
y
(HΦq(U,F),“lim←−”) −−−−→ (HΦq(X,F),“lim←−”) Proof. 1. Consider the diagram:
(5.2)
x
x
x
0−−−−→ΓΦ(X,E0,1⊗ F)−−−−→CΦ0(U,E0,1⊗ F)−−−−→CΦ1(U,E0,1⊗ F)−−−−→
x
x
x
0−−−−→ΓΦ(X,E0,0⊗ F)−−−−→CΦ0(U,E0,0⊗ F)−−−−→CΦ1(U,E0,0⊗ F)−−−−→
x
x
CΦ0(U,F) −−−−→ CΦ1(U,F) −−−−→
x
x
0 0
where on all terms we consider the “lim−→” topologies. The columns of the diagram, with the exception of the first one, are acyclic; the cohomology of the first column is HΦ•(X,F); the rows of the diagram, with the exception of the bottom one, are acyclic; the cohomology of the bottom row is HΦq(U,F). To prove our claim it is enough to check that the differentials in the double complex (CΦq(U,E0,p⊗ F))p,q are topological homomorphisms and that the topologies on the terms of the first column, as well as on those of the bottom row are induced by the double complex (see e.g. [5] Lemme 4.6, p. 327).
a. The inclusion “ΓΦ(X,E0,p⊗ F) →CΦ0(U,E0,p⊗ F)” is a topological homo- morphism. Continuity follows from the commutative diagram:
lim−→
S∈Φ
ΓS(X,E0,p⊗ F) −−−−→ lim−→
S∈Φ
C0(U(S),E0,p⊗ F) x
x
ΓS(X,E0,p⊗ F) −−−−→ C0(U(S),E0,p⊗ F) Let L:CΦ0(U,E0,p⊗ F) → ΓΦ(X,E0,p⊗ F), L((si)i∈I) = P
i
ρisi where (ρi)i∈I is a C∞-partition of unity subordinate to the covering U. One checks that L is a right inverse for the inclusion. To finish the proof of a. we show that L is continuous. It is enough to check that for each S ∈Φ the restriction of L,LS:C0(U(S),E0,p⊗ F)→ΓS(1)(X,E0,p⊗ F) is continuous. Since U is locally finite LS|Uk : C0(U(S),E0,p⊗ F) → Γ(Uk,E0,p⊗ F) is a finite sum P
ρisi, hence continuous; it follows that (LS|Uk)k∈I : C0(U(S),E0,p⊗ F) → Q
k∈I
Γ(Uk,E0,p ⊗ F) is continuous;
since (LS|Uk)k∈I coincides with LS when we restrict the target space to the topological subspace ΓS(1)(X,E0,p⊗ F) ofQ
Γ(Uk,E0,p⊗ F), we are done.
b. The horizontal differentials “d:CΦq(U,E0,p⊗ F)→CΦq+1(U,E0,p⊗ F)” are topological homomorphisms. The proof is similar with that for a. One defines the operator:
L: lim−→
S∈Φ
Y
α∈N(U(S),q+1)
Γ(Uα,E0,p⊗ F)→ lim−→
S∈Φ
Y
α∈N(U(S),q)
Γ(Uα,E0,p⊗ F) by L((si0,...iq+1)) = (ti0,...iq), with ti0,...iq = P
i∈I
ρisii0,...iq, where (ρi)i∈I is aC∞-partition of unity subordinate to the coveringU; then, one verifies that d(L◦d(t)−t) = 0 (i.e. L is a homotopy operator), and that L is continuous.
c. The inclusion “CΦq(U,F) →CΦq(U,E0,0⊗ F)” and the vertical differentials
“CΦq(U,E0,p⊗ F)→CΦq(U,E0,p+1⊗ F)” are topological homomorphisms.
The statement follows immediately from the following general lemma:
Lemma 5.2. Let (ui : Ai → Bi)i∈I be a family of topological homomor- phisms between locally convex topological vector spaces. If S is an inductive systems of subsets of I, then the canonical morphism:
u: lim−→
α∈S
Y
i∈α
Ai → lim−→
α∈S
Y
i∈α
Bi
is a topological homomorphism, where on both terms one considers the canon- ical topologies deduced from the topologies of the spaces Ai and Bi.
Proof. The continuity ofu is clear from the continuity of the morphisms ui. To finish the proof set
Aα=Y
i∈α
Ai,A= lim−→
α∈S
Aα,Bα=Y
i∈α
Bi ,B = lim−→
α∈S
Bα
Let W be a circled convex 0-neighbourhood in A. It is sufficient to find a 0- neighbourhoodV ⊂B such thatu(A)∩V ⊂u(W). For this, letWα =W∩Aα, Wi =W ∩Ai, andβ(W) = {i∈I |Wi 6=Ai}. The definition of the topology on the spacesAα implies thatα∩β(W) is finite for all α∈ S, and hence,
W = [(W ∩ Y
i∈β(W)
Ai)× Y
i /∈β(W)
Ai]∩A
For i ∈ β(W) choose Vi ⊂ Bi a circled convex 0-neighbourhood s.t.
Vi ∩ ui(Ai) ⊂ ui(Wi). Then V = [co( S
i∈β(W)
Vi) × Q
i /∈β(W)
Bi] ∩B is the 0-neighbourhood in B we are looking for (as usual, co() denotes the convex hull).
2. Consider now the “lim←−” topologies in diagram 5.2. One checks easily that the inclusion ΓΦ(X,E0,p⊗ F) → CΦ0(U,E0,p⊗ F) is continuous. Now, consider the composition:
lim−→
S∈Φ
ΓS(X,E0,p⊗ F)−→id lim←−
T∈Ψ
lim−→
F∈I(T)
ΓF(X,E0,p⊗ F)−→v CΦ0(U,E0,p⊗ F) According to Proposition 3.1id is continuous, and v is continuous when on the last space one considers the “lim←−” topology. Since the “lim−→” topol- ogy and “lim←−” topology onCΦ0(U,E0,p⊗F) coincide (see Proposition 4.1), 1.a. implies that the composition v◦idis a topological homomorphism.
Hence, both id and v are topological homomorphisms. Thus, all mor- phisms in diagram 5.1 are topological isomorphisms.
Corollary 5.3. Let X be a complex manifold and F a coherent OX-module. If (Φ,Ψ) is a dual pair of families of supports on X and U = (Ui)i∈I is a locally finite covering of X with relatively compact Stein open sets adapted to bothΦ and Ψthen the “lim−→” topology and “lim←−” topology on ΓΦ(X,E0,•⊗ F) coincide.
The topology in Theorem 5.1 is the natural topology on HΦq(X,F).
Remark 5.4.LetF• ∈D+coh(X)i.e. F• is a complex ofOX-module boun- ded below and with coherent cohomology. Using the semi-simplicial techniques of [6] one can prove like in Theorem 5.1 that the inductive limit topology and the projective limit topology onHΦq(X,F•) coincide.
Remark 5.5.Consider the setting of Theorem 5.1. If F∗ is the cosheaf associated to the coherent OX-module F then, as for the cohomology groups with supports, on the homology groupsHΦq(U,F∗) andHΦq(X,F∗) one defines an inductive limit topology and a projective limit topology (for details see [3]
or [1]). Using Lemmas 4.2 and 5.2 one proves an analogue of Theorem 5.1 for homology with supports. Thus, the homology group H Φq(X,F∗) is endowed with a natural topology.
6.DUALITY RESULTS
SinceExtqΦ(X;F, ωX)'HΦq(X, RHom(F, ωX)), Remarks 5.5 or 5.4 show that the inductive limit topology and the projective limit topology on the in- variantsExtqΦ(X;F, ωX) coincide (for details on the definition of the topologies on ExtqΦ see [3] or [1]); here, ωX denotes as usual the sheaf of maximal de- gree holomorphic differential forms on the complex manifold X. Similarly, if X is a complex space with singularities and KX is the dualizing complex on X, then the inductive limit topology and the projective limit topology on ExtqΦ(X;F, KX) coincide. Thus, the duality results of [1] (e.g. Theorem 5.6) yield dualities with respect to the natural topologies:
Theorem 6.1. Let X be a complex n-dimensional manifold and (Φ,Ψ) a dual pair of families of supports on X admitting adapted coverings. As- sume that F is a coherent OX-module. Then there is a topological duality between the separated associated toHΦq(X,F) and Extn−qΨ (X;F, ωX) when on both spaces one considers the natural topology. Moreover, HΦq(X,F) is Haus- dorff iff Extn−q+1Ψ (X;F, ωX) is Hausdorff.
Theorem 6.2. Let X be a complex analytic space and (Φ,Ψ) a dual pair of families of supports on X admitting adapted coverings. Assume that F is a coherent OX-module. Then there is a topological duality between the
separated associated to HΦq(X,F) and Ext−qΨ (X;F, KX) when on both spaces one considers the natural topology. Moreover, HΦq(X,F) is Hausdorff iff Ext−q+1Ψ (X;F, KX) is Hausdorff.
In particular, let f :X → Y be a morphism of analytic spaces, V ⊂ Y an open set, and F a coherent OX-module (or F ∈Dbcoh(X), i.e. a bounded complex with coherent cohomology). With the families of supports defined in Example 2.5, one has, in the derived category of OY-modules, the isomor- phisms:
RΓc(V, Rf∗(F)) ' RΓc(f,V)(f−1(V),F) RΓ(V, Rf!(DX(F))) ' RΓΨ(f,V)(f−1(V), DX(F))
where, as usual,DX(F) =RHom(F, KX),or, ifX is an n-dimensional complex manifold,RHom(F, ωX[n]). Via the isomorphisms induced on the cohomology, one introduces natural topologies on the cohomology groups of Rf∗(F) and Rf!(DX(F)):
Hcq(V, Rf∗(F)) ' Hc(f,Vq )(f−1(V),F)
Hq(V, Rf!(DX(F))) ' ExtqΨ(f,V)(f−1(V);F, KX)
Corollary 6.3. Let f :X → Y be a morphism of analytic spaces and V ⊂ Y an open set. Assume that F is a coherent OX-module. Then there is a topological duality between the separated associated to Hcq(V, Rf∗(F))and H−q(V;Rf!(DX(F))) when on both spaces one considers the natural topolo- gies. Moreover, Hcq(V, Rf∗(F)) is Hausdorff iff H−q+1(V;Rf!(DX(F))) is Hausdorff.
REFERENCES
[1] A. Andreotti and C. B˘anic˘a, Relative Duality on Complex Spaces I.Rev. Roum. Math.
Pures Appl. 20(1975), 981–1041.
[2] A. Andreotti and C. B˘anic˘a,Relative Duality on Complex Spaces II.Rev. Roum. Math.
Pures Appl. 21(1976), 1139–1181.
[3] A. Andreotti and A. Kas,Duality on Complex Spaces. Annali Sc. Norm. Super. Pisa Cl. Sci. (5)27(1973), 187–263.
[4] C. B˘anic˘a, Homologie et dualite en geometrie analytique complexe. In: Fonctions de Plusieurs Variables Complexes III –Seminaire F. Norguet 1975–1977. Lecture Notes in Math. 670, pp. 1–21, Springer Verlag, 1978.
[5] C. B˘anic˘a and O. St˘an˘asil˘a, M´ethodes alg´ebriques dans la th´eorie globale des espaces complexes. Editura Academiei si Gauthier-Villars Editeurs, 1977.
[6] A. Baran, Dualit´e pour l’hyperext sur les espaces complexes. C.R. Math. Acad. Sci.
Paris302, s´erie 1, (1986), 627–630.
[7] A. Baran,A Semi-simplicial Construction of the Dualizing Complex of an Analytic Space de s´eparation pour l’hyperext. to appear.
[8] C. Laurent-Thi´ebault and J. Leiterer, A Separation Theorem and Serre Duality for the Dolbeault Cohomology. Ark. Mat. 40(2002), 301–321.
[9] B. Malgrange,Ideals of Differentiable Functions. Oxford Univ. Press, 1966.
[10] J.-L. Verdier,Topologie sur les espaces de cohomologie d’un complexe de faisceaux ana- lytiques `a cohomologie coh´erente. Bull. Soc. Math. France99, (1971), 337–343.
Received 11 February 2013 Institute of Mathematics of the Romanian Academy,
P.O.Box 1-764,
RO-014700 Bucharest, Romania andrei.baran@imar.ro
