TOPOLOGY OF ALGEBRAIC VARIETIES

FOUAD ELZEIN AND ANDRÁS NÉMETHI

Abstract. Let Y be a normal crossing divisor in the smooth projective algebraic variety X (defined over ℂ) and let U be a tubular neighbourhood of Y in X. We construct homological cycles generating H_{∗}(A, B), where (A, B) is one of the following pairs (Y, ∅), (X, Y), (X, X − Y), (X − Y, ∅) and (∂U, ∅). The construction is compatible with the weights in H_{∗}(A, B, ℚ) of Deligne’s mixed Hodge structure.

I. Introduction

In his fundamental article [L], Leray gives a cohomological version of the theory of residues along a smooth hypersurface in an analytic manifold. The original motivation of the present article was the generalization of the theory of residues, but now along a normal crossing divisior (NCD) Y of a smooth projective algebraic variety X (defined over ℂ). Leray’s construction of the inverse image of a cycle on Y leads to a homological cycle of the boundary of a tubular neihgbourhood of Y. In our case, in order to obtain a cycle as inverse image, we need to impose specific condition on the cycle on Y. We obtain such conditions by duality from Deligne’s mixed Hodge structure (MHS) on the cohomology of algebraic varieties [D].

Once this fact has been well understood, we realised that we needed to develop a complete homological theory of some complexes (compatible with the natural mixed Hodge structures), and to describe the topology/homology of the pairs (Y, ∅), (X, Y), (X, X − Y), (X − Y, ∅) and (∂U, ∅), where U is a tubular neihgbourhood of Y in X. These results are presented in this paper. The generalization of the residue theorems will be published elsewhere.

The reader may have recognized that the reduction to the NCD case uses Hironaka’s desingularization theorem [H] in order to carry our results to smooth open algebraic varieties. In the cohomology theory, the main ingredient is Deligne’s mixed Hodge structure. By duality, one can define the weight filtration in homology with ℚcoefficients. Our ultimate result is the construction of representative homological cycles according to their weight. Such constructions have been used occasionally in examples or some specific situations. Our objective is a systematic treatment giving a general account of the topology of algebraic varieties.

¿From the point of view of the present article, the key argument is the degeneration at rank two of the weight spectral sequence defined by the mixed Hodge complex, valid only for algebraic varieties. This in turn leads us to consider homological complexes with good intersection theory and for this it has been necessary to use the simplicial theory provided by the triangulations of varieties instead of singular chains.

During the preparation of this work, the second author visited Univ. de Nantes in June 1998 and the first author visited MSRI in Berkeley in September 1998. We thank these institutions for their hospitality. Research at MSRI is supported in part by NSF grant DMS-9701755. The second author was partially supported by NSF grant DMS-9622724.

The paper is organized as follows: in the introduction we recall some known facts about the topology of a regular neighbourhood of a NCD. We refer to the articles by Clemens [C] and A’Campo [A]. In the second part of this introduction, we already present the main steps of our construction.

In the second section we consider the complex of “dimensionally transverse” subanalytic chains. This is in fact the complex of the sub-analytic “intersection chains” associated with the natural stratification of Y and zero perversity (in the sense of Goresky–MacPherson [GM]). Moreover, we define and analyze different chain morphisms whose role is absolutely fundamental in the construction of cycles.

Using the usual and the “dimensionally transverse” chains, in section 3 we define several homological double complexes A_{∗∗}(A, B) corresponding to the above pairs of spaces (A, B). They carry natural weight filtrations, giving rise to the corresponding spectral sequences. For example, in the case of (X − Y, ∅) it is possible to show that our spectral sequence is dual to the weight spectral sequence of the logarithmic complex via residues. In this section we prove for all of the considered pairs the degeneration property at rank 2 of the weight spectral sequence. In section 4 we review some of the algebraic properties of a double complex whose spectral sequence degenerates at rank 2.

Using the result of the previous two sections, in section 5 we construct cycles compatible with the weight filtration. Here an additional important ingredient is the construction of chain morphism T ot_{∗}(A_{∗∗}(A, B)) → C_{∗}(A, B) for any pair (A, B). This is based on the results of section 3. (Here T ot_{∗} stays for the total complex, and C_{∗}(A, B) is the complex of sub-analytic chains of (A, B).)

The case of (∂U, ∅) is the most involved, so we separated it in section 6.

I.1. Now, we start a more detailed presentation.

I.1.1. The stratification of Y. Let X be a smooth projective algebraic variety containing a normal crossing divisor Y. Let (Y_{i})_{i∈I} be the decomposition of Y into irreducible components. Here the index set I is a totally ordered set.

The divisor Y has a natural stratification by subspaces Y^{j} consisting of “points of multiplicity ≥ j in Y ”, i.e.:

Y^{j} = ⋃_{α_{1}<···<α_{j}} Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{j}}, with normalization Ỹ^{j} = ∐_{α_{1}<···<α_{j}} Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{j}}.

Sometimes we will use the notation Y_{α_{1},... ,α_{j}} = Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{j}} as well. It is convenient to write Ỹ^{0} = Y^{0} = X.

I.1.2. The tubular neighbourhood of Y. For any variety X and divisor Y, X admits a triangulation compatible with Y, hence there exists a regular (tubular) neighbourhood in X which is a deformation retract of Y (see e.g. [R-S]). In the case of a NCD, the tubular neighbourhood U of Y and a projection from U to Y can have even some additional properties, see [C].

A’ Campo [A] describes such a neighbourhood as follows. The real (oriented) blowup Π_{i} : Z_{i} → X with center Y_{i} provides a differentiable manifold Z_{i} with boundary and a projection Π_{i} on X, inducing a diffeomorphism outside Y_{i}. Moreover, Π_{i}^{−1}(Y_{i}) is an S^{1}-bundle associated to the oriented normal bundle N_{Y_{i}/X}. We can identify the fibers at a point y in Y_{i} with the set of real oriented normal directions to Y_{i}. For example, if Y_{i} is a point in ℂ, then Z_{i} is a half-cylinder with boundary S^{1} over the point. In general, the boundary of Z_{i}, equal to Π_{i}^{−1}(Y_{i}), is diffeomorphic to the boundary of the complement of an open tubular neighbourhood of Y_{i} in X.

Next we consider the fibered product of the projections (Π_{i})_{i∈I} over X. Thus we obtain Π : Z → X, where Z is a manifold with corners whose boundary ∂Z equal to Π^{−1}(Y).

Here Z is homeomorphic to the complement of an open tubular neighbourhood U of Y in X, hence ∂Z is homeomorphic also to the boundary of the closure of U. In fact U is homeomorphic to the mapping cylinder of Π|∂Z : ∂Z → Y.

In order to see this, let’s fix a homeomorphism ϕ_{α} : ∂Z × [0, α] → Z_{α}, where Z_{α} is some collar, and ϕ_{α}|∂Z ×{0} is the inclusion ∂Z ↪ Z_{α}. There is a natural projection p_{α} : Z_{α} → ∂Z given by pr_{1} ∘ϕ_{α}^{−1}, where pr_{1} is the projection on the first factor. (This retract can be extended to a strong deformation retract of the inclusion ∂Z ↪ Z_{α}.) Using this, one can define tubular neighbourhoods of Y in X as follows.

For any ϵ ∈ (0, α], define U_{ϵ} := Π(ϕ_{α}(∂Z × [0, ϵ])). Its boundary ∂U_{ϵ} is clearly Π(ϕ_{α}(∂Z×{ϵ})). Moreover, there is a unique map q_{ϵ} : U_{ϵ} → Y such that q_{ϵ}∘Π = Π∘p_{ϵ}. From the definitions follows that for two neighbourhoods U_{ϵ_{1}} ↪^{i} U_{ϵ_{2}}, for 0 < ϵ_{1} < ϵ_{2}, one has q_{ϵ_{2}} ∘ i = q_{ϵ_{1}}. This shows, that one can construct a uniform family of tubular neighbourhoods U_{ϵ} of Y in X, all of them homeomorphic to the mapping cylinder of Π|∂Z, with all the natural compatibilities. This construction is rather satisfactory in the discussion of topological invariants.

But, in fact, Z has a natural semi-analytic structure (see II.3.1), and the above homeomorphism ϕ_{α} can be chosen as a sub-analytic homeomorphism as well (in fact, even as a piecewise analytic isomorphism with respect to the natural stratification). For the proof of this last statement, see [P]. Therefore, all the homeomorphisms, discussed in the previous paragraphs, can be considered as sub-analytic homeomorphisms.

I.1.3. The construction of the double complexes A_{∗∗}(A, B). For any p ≥ 0, we denote the complex of the usual (respectively the “dimensionally transverse”) subanalytic chains (cf. II.1) defined on Ỹ^{p} by (C_{∗}(Ỹ^{p}), ∂) (respectively by (C_{∗}^{⋔}(Ỹ^{p}), ∂)). One has the following chain morphisms:

⋯ C_{∗−4}^{⋔}(Ỹ^{2}) ←^{∩} C_{∗−2}^{⋔}(Ỹ^{1}) ←^{∩} C_{∗}^{⋔}(Ỹ^{0})
↓j_{1}
C_{∗}(Ỹ^{0}) ←^{i} C_{∗}(Ỹ^{1}) ←^{i} C_{∗}(Ỹ^{2}) ⋯

Then define (modulo some shift of indexes): A_{∗∗}(X, X − Y) = (C_{∗}^{⋔}(Ỹ^{p}), ∂, ∩)_{p≥1}, A_{∗∗}(X−Y) = (C_{∗}^{⋔}(Ỹ^{p}), ∂, ∩)_{p≥0}, A_{∗∗}(Y) = (C_{∗}(Ỹ^{p}), ∂, i)_{p≥1}, A_{∗∗}(X, Y) = (C_{∗}(Ỹ^{p}), ∂, i)_{p≥0}, and finally A_{∗∗}(∂U) by the cone of the morphism j_{1} in the above diagram where we replace Ỹ^{0} by U. The weight filtration W_{∗} of a double complex A_{∗∗} is defined by W_{s} := ⊕_{p≤s}A_{pq}.

I.1.4. The construction of the cycles. First, using the geometry of the pair (X, Y) (e.g. the projection Π and intersections corresponding to the stratification), for any pair (A, B) we construct a quasi-isomorphism m_{A,B} : T ot_{∗}(A_{∗∗}(A, B)) → C_{∗}(A, B). Then any representative c_{st} of an element [c_{st}] ∈ ker(d^{1}|E_{st}^{1}) is completed to a chain c_{st}^{∞} = c_{st}+c_{s−1,t+1}+⋯ ∈ Z_{st}^{∞} with D(c_{st}^{∞}) = 0, where D is the differential of the associated total complex T ot_{∗}(A_{∗∗}) (and c_{pq} ∈ A_{pq}). Therefore, m_{A,B} associates with any [c_{st}] a closed cycle m_{A,B}(c_{s,t}^{∞}) of dimension k = s + t.

For the various pairs (A, B) considered above, we have the following result.

I.1.5. Theorem.

a) The first term (E_{st}^{1}, d^{1}) of the weight spectral sequence can be explicitly determined from the homology of the spaces Ỹ^{p} and from the various normal bundles of the components of Ỹ^{p} in Ỹ^{p−1}.

b) E_{st}^{r} ⟹ H_{s+t}(A, B, ℤ), and induces a weight filtration on the integer homology. Moreover E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(A, B, ℚ) (the last considered in Deligne’s MHS).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at rank 2.

d) The above construction provides all the cycles of the pair (A, B) (modulo boundary) according to their weights. More precisely, the homology class [m_{A,B}(c_{s,t}^{∞})] of dimension k = s + t is well-defined modulo W_{−t−1}H_{k}(A, B, ℚ), and these type of classes generate W_{−t}H_{k}(A, B, ℚ).

Here we refer the reader to theorem III.0 and propositions V.2.2, VI.2.3 and VI.2.4 for precise statements.

I.1.6. Example. Suppose Y = Y_{1}∪Y_{2} has two components with smooth intersection Y_{1,2} = Y_{1} ∩ Y_{2}. Two homology classes [a_{i}] ∈ H_{k}(Y_{i}), i = 1, 2, satisfying [a_{1}] ∩ [Y_{1,2}] = [a_{2}] ∩ [Y_{1,2}] in H_{k−2}(Y_{1,2}) give rise in ∂U to a k + 1 dimensional cycle (generating a homology class of weight −k −2). Indeed, first we can assume that the representative a_{i} is transversal to Y_{1,2} in Y_{i}. Due to the condition about [a_{i}] ∩ [Y_{1,2}], there exists a chain a_{1,2} in Y_{1,2} such that ∂a_{1,2} = a_{2} ∩ Y_{1,2} − a_{1} ∩ Y_{1,2}. Since dim a_{i} = k, we have:

dim a_{i} ∩ Y_{1,2} = k − 2, dim a_{1,2} = k − 1, dim Π^{−1}a_{i} = k + 1, dim Π^{−1}a_{1,2} = k + 1.

Moreover, by the very construction, Π^{−1}(a_{1} + a_{2} + a_{1,2}) has no boundary. It is the wanted closed cycle in ∂U.

Now, consider the case when Y has three irreducible components Y_{i} (i = 1, 2, 3) with Y_{1,2,3} ≠ ∅. Similarly as above, we want to lift some closed cycles a_{i} into ∂U. The first obstruction is [a_{i} ∩ Y_{i,j}] = [a_{j} ∩ Y_{i,j}] in the homology of Y_{i,j} (for any pair i ≠ j). Using this, we create the new chains a_{i,j} in Y_{i,j} as above. Now, if we want to lift these new chains and glue them together, we face the second obstruction provided by the triple intersection Y_{1,2,3}. The main point is that there exists a good choice of a_{i,j} such that this new obstruction is trivial; its triviality is equivalent with the vanishing of the second differential d_{2} of the spectral sequence. Basically, this is the main message of the present paper.

II. Topological Preliminaries

II.1. Geometric chains.

II.1.1. Preliminary remarks. In some cases it is not absolutely evident how can we dualize a result established in cohomology. For example, if we want to compute the homology of Y, then Deligne spectral sequence in cohomology has a very natural analog in homology. On the other hand, if we want to find the homological analogue of the cohomological theory of X − Y, then we have to realize that there is no obvious homological candidate. Actually, the E_{1}-term of the spectral sequence associated with the log complex (Ω_{X}^{∗}(log Y), W) can be easily dualized; but we want (and need to) dualize the whole spectral sequence, in particular we have to construct the E^{0} term on X. This is crucial in the construction of cycles in X − Y as well.

Since the Gysin differential, in the cohomological E_{1}-term of X −Y, dualizes to the intersection of cycles, we need to work with chains with good intersection properties with respect to the stratification defined by Y. Similarly as in the case of the intersection homology groups, we have several options to define our chain complex. In the original definition of the intersection homology groups, Goresky and MacPherson used geometric chains with some restrictions provided by the perversities [GM]. On the other hand, H.King recovered these groups using singular chains [K]. We will follow here the first option (in fact, we will use sub-analytic geometric chains). In order to have a good intersection theory, we need some kind of transversality property. Here again we have several possibilities. Our choice asks only a “dimensional transversality” of the chains. This has the big advantage that the complex of these chains coincides with the complex of zero perversity chains of Goresky–MacPherson; but has the disadvantage, that in the intersections of the cycles we have to handle an intersection multiplicity problem.

Finally, our choice for sub-analytic chains is motivated by the fact that these chains are stable with respect to the real blowing up along Y (in contrast with the P.L. chains).

II.1.2. Good class of subsets and chains. Before we start the precise definition of our geometric chains, let us review briefly the general theory. We will follow the presentation from [G], pages 146-155.

Fix a manifold M. The group of geometric chains is defined in three steps. First, one defines a good class C of subsets of M. This should satisfy the following properties:

(1) if a subset S of M is in C, then M has a Whitney stratification such that S is a union of strata, and each stratum is in C;

(2) the class C is closed under unions, intersections and differences;

(3) the closure of a subset in C is in C.

In the second step, one defines the geometric prechains (relative to C). Basically, they can be represented as ∑_{α} m_{α}S_{α}, where each S_{α} is a closed subset from C (with a fixed orientation) and m_{α} is an integer (the “multiplicity of S_{α}”). A geometric chain is an equivalence class of geometric prechains with respect to a natural equivalence relation. (For details, see [G].)

In this paper we will use sub-analytic chains associated with our complex analytic manifolds: if M is a complex analytic manifold, and we fix a real analytic structure on M, then the class of all sub–analytic subsets of M form the good class of subsets C.

II.1.3. The definition of the (geometric) chains. Now, we define our chains. In some of the definitions we follow closely the paper [GM] of Goresky and MacPherson (where the case of the P.L. geometric chains is considered). We will regard our manifold X as a pseudomanifold of dimension 2n, where n is the complex dimension of X. We consider its natural stratification X_{2n−2p+1} = X_{2n−2p} = Y^{p} for p ≥ 1, hence Σ = Y. Similarly as above Ỹ^{0} = Y^{0} = X.

By [H2], X admits a (canonical) sub–analytic triangulation, compatible with the stratification. In fact, any two sub–analytic triangulation admits a common refinement (see [H2] 2.4).

For the next definitions, we fix an integer p ≥ 0. Then Ỹ^{p} is again a pseudomanifold. This stratification will be denoted by {Ỹ_{2n−2p−2r}^{p}}_{r≥0}.

If T is a sub–analytic triangulation, let C_{∗}^{T} (Ỹ^{p}) be the chain complex of simplicial chains of Ỹ^{p} with respect to T. By definition, a chain of Ỹ^{p} is an element of C_{∗}^{T} (Ỹ^{p}) for some sub–analytic triangulation T, however one identifies two chains c ∈ C_{∗}^{T} and c′ ∈ C_{∗}^{T′} if their canonical images in C_{∗}^{T′′} coincide, for some common refinement T′′ of T and T′. The group of all chains is denoted by C_{∗}(Ỹ^{p}). Obviously, there is a natural boundary operator ∂ which makes C_{∗}(Ỹ^{p}) a complex.

Notice that C_{∗}(Ỹ^{p}) is exactly the group of sub-analytic geometric chains discussed in II.1.2. Indeed, if we fix an arbitrary closed sub-analytic subset S of Ỹ^{p}, then by [H2] there is a sub-analytic triangulation T which makes S an element of C_{∗}^{T} (Ỹ^{p}) (with all multiplicities one).

If ξ ∈ C_{k}^{T}, then the support |ξ| of ξ is the union of the closures of those k–simplices σ for which the coefficient of ξ is non–zero. Actually, the support of ξ is independent on T, and it is a k–dimensional sub-analytic subset. We emphasize (even if this is clear in most of the cases, since X is compact), that in all our discussions, we will deal with chains with compact supports. (See also [B], I.) This remark is essential, when we will consider the sheaf-versions of our complexes.

The homology of the complex C_{∗}(Ỹ^{p}) is the usual homology H_{∗}(Ỹ^{p}) (cf. [G], page 155). Moreover, for any triangulation T, the natural morphism C_{∗}^{T} (Ỹ^{p}) → C_{∗}(Ỹ^{p}) is a quasi-isomorphism. Indeed, the triangulation homeomorphism t : |K| → Ỹ^{p} from a simplicial complex |K| to Ỹ^{p} identifies the simplicial homology of |K| and the usual homology of Ỹ^{p}.

II.2. “Dimensionally transverse” chains.

We introduce the following double complex of dimensionally transverse chains in Ỹ^{p}, p ≥ 0. We say that a chain ξ ∈ C_{k}(Ỹ^{p}) is dimensionally transverse, if a) dim(|ξ|) ∩ Ỹ_{2n−2p−2r}^{p} ≤ k − 2r, for any r > 0; and

b) dim(|∂ξ|) ∩ Ỹ_{2n−2p−2r}^{p} ≤ k − 1 − 2r, for any r > 0.

The subgroup of C_{∗}(Ỹ^{p}) consisting of the dimensionally transverse chains is denoted by C_{∗}^{⋔}(Ỹ^{p}). The boundary operator ∂ maps dimensionally transverse chains in dimensionally transverse chains, hence defines a complex with ∂^{2} = 0.

The reader, familiar with intersection homology, immediatelly will realize that the complex C_{∗}^{⋔}(Ỹ^{p}) is exactly the complex IC¯0 ; ∗ (Ỹ^{p}) of intersection chains corresponding to the zero perversity (see [GM]).

II.2.1. Lemma. The natural inclusion j : (C_{∗}^{⋔}(Ỹ^{p}), ∂) → (C_{∗}(Ỹ^{p}), ∂) is a quasiisomorphism. In particular, the homology of (C_{∗}^{⋔}(Ỹ^{p}), ∂) is H_{∗}(Ỹ^{p}).

Proof. This follows from section 4.3 of [GM] (because the Poincaré map H^{2n−2p−k}(Ỹ^{p}) → H_{k}(Ỹ^{p}) is an isomorphism, provided by the smoothness of Ỹ^{p}). Actually, since Ỹ^{p} is smooth, and the intersection homology group is independent on the stratification (3.2 in [GM]), all the intersection homology groups are the same and equal to the usual homology.

II.2.2. The intersections with the strata. We want to define the intersection ξ ∩ Y_{α} ∈ C_{k−2}^{⋔}(Y_{α_{1},... ,α_{p}} ∩Y_{α}) for any chain ξ = ξ_{α_{1}<···<α_{p}} ∈ C_{k}^{⋔}(Y_{α_{1},... ,α_{p}}), where α ∉ {α_{1}, …, α_{p}}. (Here, if p = 0 then Y_{α_{1},... ,α_{p}} denotes X.) First notice the following fact (cf. [GM], page 138).

II.2.3. Fact. If C ⊂ Ỹ^{p+1} is a (k–2)–dimensional sub-analytic subset of Ỹ^{p+1} and if D ⊂ C is a (k–3)–dimensional sub-analytic subset, then there is a one-to-one correspondence between chains β ∈ C_{k−2}(Ỹ^{p+1}) such that |β| ⊂ C, |∂β| ⊂ D, and between homology classes β̃ ∈ H_{k−2}(C, D). Furthermore, ∂β corresponds to the class ∂_{∗}(β̃) in H_{k−3}(D) under the connecting homomorphism ∂_{∗} : H_{k−2}(C, D) → H_{k−3}(D).

In the definition of the intersection ξ ∩ Y_{α}, we face the problem of the notion of an “intersection multiplicity”. This is clarified in the work of Lefschetz, however we will work in the spirit of [GM]. It is clear that |ξ| ∩ Y_{α} has dimension ≤ k − 2, and this intersection satisfies the transversality restrictions of the new space. We have to determine the coefficients of the simplices which support the intersection.

Using the above Fact, the intersection ξ ↦ ξ ∩ Y_{α} is completely determined by the following composition:

H_{k}(|ξ|, |∂ξ|)
≈ ↑ ∩[Y_{α_{1},... ,α_{p}}]
H^{2n−2p−k}(Y_{α_{1},... ,α_{p}} − |∂ξ|, Y_{α_{1},... ,α_{p}} − |ξ|)
↓ i^{∗}
H^{2n−2p−k}(Y_{α_{1},... ,α_{p}} ∩ Y_{α} − |∂ξ|, Y_{α_{1},... ,α_{p}} ∩ Y_{α} − |ξ|)
≈ ↓ ∩[Y_{α_{1},... ,α_{p}} ∩ Y_{α}]

H_{k−2}(|ξ| ∩ Y_{α}, |∂ξ| ∩ Y_{α}).

Above, the first map is the (inverse) of the cap product with the fundamental class of Y_{α_{1},...α_{p}}, as it is presented in the Appendix of [GM], page 162; the second map is the restriction i^{∗}, where i is the natural inclusion; and finally, the third map is again the cap product with the fundamental class of Y_{α_{1},...α_{p}} ∩ Y_{α}. The cap products are isomorphisms (cf. [loc.cit.]).

Then the intersection ξ ↦ ξ ∩ Y_{α} is defined as follows; ξ determines an element in H_{k}(|ξ|, |∂ξ|) via Fact, and the image of that element by the above composition determines ξ ∩ Y_{α}, again by the above Fact. Remark. It is important to note that |ξ| ∩ Y_{α} is not necessarily equal to |ξ ∩ Y_{α}| since the chains are not necessarily transversal in the differential sense (cf. II.3.7).

Next, we define a (boundary) operator ∩ : C_{k}^{⋔}(Ỹ^{p}) → C_{k−2}^{⋔}(Ỹ^{p+1}). If ξ ∈ C_{k}^{⋔}(Ỹ^{p}), we write ξ_{α_{1}<···<α_{p}}^{k} for the corresponding components of ξ corresponding to the decomposition of Ỹ^{p} (the index k emphasizes the dimension, and sometimes it is omitted). Then we define:

∩(ξ_{α_{1}<···<α_{p}}^{k}) = ∑_{α_{1}<···<α_{i}<α<···<α_{p}}(−1)^{k+i} ξ_{α_{1}<···<α_{p}}^{k} ∩ Y_{α}.

II.2.4. Lemma. a) ∩^{2} = 0 and b) ∂ ∩ + ∩ ∂ = 0.

Proof. a) Let α < β and α, β ∉ {α_{1}, …, α_{p}}, where α_{1} < ⋯ < α_{i} < α < α_{i+1} < ⋯ < α_{i+j} < β < ⋯ < α_{p}. ¿From the properties of the intersection (see, e.g. [GM], page 144), one obtains that (ξ_{α_{1}<···<α_{p}} ∩ Y_{α}) ∩ Y_{β} = (ξ_{α_{1}<···<α_{p}} ∩ Y_{β}) ∩ Y_{α}. Then the sum of the following two contributions in ∩^{2} is zero:

∩_{Y_{β}}(∩_{Y_{α}}ξ_{α_{1}<...<α_{p}}^{k}) = (−1)^{k+i} ∩_{Y_{β}} (ξ_{α_{1}<···<α_{p}}^{k} ∩ Y_{α}) = (−1)^{j+1}(ξ_{α_{1}<···<α_{p}}^{k} ∩ Y_{α}) ∩ Y_{β}

∩_{Y_{α}}(∩_{Y_{β}}ξ_{α_{1}<...<α_{p}}^{k}) = (−1)^{k+i+j} ∩_{Y_{α}} (ξ_{α_{1}<···<α_{i+j}<···<α_{p}}^{k} ∩ Y_{β}) = (−1)^{j}(ξ_{α_{1}<···<α_{p}}^{k} ∩ Y_{β}) ∩ Y_{α}.

b) follows from the definition of ∩ and ∂, including the sign (−1)^{k} when dim |ξ| = k.

The fact that the homology of (C_{∗}^{⋔}(Ỹ^{p}), ∂) is exactly H_{∗}(Ỹ^{p}) (cf. Lemma II.2.1) and part (b) of II.2.4 show that ∩ induces an operator H_{∗}(Ỹ^{p}) → H_{∗−2}(Ỹ^{p+1}).

II.2.5. Lemma. For any p ≥ 0, the operator H_{∗}(Ỹ^{p}) → H_{∗−2}(Ỹ^{p+1}) induced by ∩ is

∩[ξ_{α_{1}<···<α_{p}}^{k}] = ∑_{α_{1}<···<α_{i}<α<···<α_{p}}(−1)^{k+i} [ξ_{α_{1}<···<α_{p}}^{k}] ∩ [Y_{α}]

where ∩[Y_{α}] denotes the homological Gysin map, or transfer map i_{!}, where i : Y_{α_{1},... ,α_{p}}∩ Y_{α} ↪ Y_{α_{1},... ,α_{p}} is the natural inclusion. More precisely, i_{!} = P D ∘ i^{∗} ∘ P D, where P D denotes the Poincaré Dualities in the corresponding spaces (cf. e.g. [Br], page 368).

Proof. The result follows from the definition of the intersection ξ ∩ Y_{α}.

II.2.6. The Poincaré duality map. We will need later the Poincaré isomorphism between homology of Y and cohomology of X with support in Y, so we adopt here to our situation the construction of [GM], pages 139-140, inspired by the classical construction which provides the Poincaré Duality for manifolds (see e.g. [Br], page 338).

Similarly as above, for any triangulation T, compatible with the stratification, one can consider the chain complex of simplicial cochains (C_{T}^{∗}(Ỹ^{p}), δ) of Ỹ^{p}. Here C_{T}^{i} (Ỹ^{p}) = Hom(C_{i}^{T} (Ỹ^{p}), ℤ). Let T′ be the first barycentric subdivision of T, and let σ̂ denote the barycentre of the simplex σ ∈ T. Let T_{i} be the i-skeleton of T, thought of as a subcomplex of T′. It is spanned by all vertices of σ̂ such that dim σ ≤ i. Let D_{i} be the i-coskeleton spanned, as a subcomplex of T′, by all the vertices σ̂ such that dim σ ≥ i. There are canonical simplex preserving deformation retracts:

II.2.7. X − |T_{i}| → |D_{i+1}| and X − |D_{i+1}| → |T_{i}|.

Now, identify C_{T}^{i} (X) with ⊕_{dim σ=i}H^{i}(σ, ∂σ) = H^{i}(|T_{i}|, |T_{i−1}|), and define pd : C_{T}^{i} (X) → C_{2n−i}^{T′}(X) by the following composition:

H^{i}(|T_{i}|, |T_{i−1}|)

↓ ∩[X]

H_{2n−i}(X − |T_{i−1}|, X − |T_{i}|)

≈ ↓ (deformation retract) H_{2n−i}(|D_{i}|, |D_{i+1}|) ↓

H_{2n−i}(|T′_{2n−i}|, |T′_{2n−i−1}|)

Actually, H_{2n−i}(|T′_{2n−i}|, |T′_{2n−i−1}|) = C_{2n−i}^{T′}(X), but we have even something more. Since the image of any chain by this composition is supported by union of |D_{i}|’s, and any |D_{i}| is dimensionally transverse, one obtains a homomorphism pd : C_{T}^{i} (X) → C_{2n−i}^{T′,⋔}(X). Similarly, for any Y_{α_{1},... ,α_{p}}, one can define a homomorphism:

pd : C_{T}^{i} (Y_{α_{1},... ,α_{p}}) → C_{2n−2p−i}^{T′,⋔}(Y_{α_{1},... ,α_{p}}).

By [GM] (7.2), this is a chain map:

II.2.8. ∂ ∘ pd = pd ∘ δ.

II.2.9. Lemma. a) Fix α ∉ {α_{1}, …, α_{p}}. Then the following diagram is commutative:

C_{T}^{i} (Y_{α_{1},... ,α_{p}}) ⟶^{pd}  C_{2n−2p−i}^{T′,⋔}(Y_{α_{1},... ,α_{p}})
↓i^{∗}  ↓∩
C_{T}^{i} (Y_{α_{1},... ,α_{p}} ∩ Y_{α}) ⟶^{pd} C_{2n−2p−i−2}^{T′,⋔}(Y_{α_{1},... ,α_{p}} ∩ Y_{α})

b) The above diagram (and II.2.8) provides at homology level a commutative diagram:

H^{i}(Y_{α_{1},... ,α_{p}}) ⟶^{P D}  H_{2n−2p−i}(Y_{α_{1},... ,α_{p}})
↓i^{∗}  ↓∩
H^{i}(Y_{α_{1},... ,α_{p}} ∩ Y_{α}) ⟶^{P D} H_{2n−2p−i−2}(Y_{α_{1},... ,α_{p}} ∩ Y_{α})

where the horizontal maps are the Poincaré Duality isomorphisms (cf. also with II.2.5).

Proof. Use the definition of ∩ and pd and II.2.7.

II.3. The chain correspondence Π_{⋆}^{−1}.

II.3.1. Triangulation of Z . Let us indicate quickly that the real blow up Π : Z → X of Y in X can be realized as a semi-analytic subset of a real vector space. Since X is compact, there is no difference between bounded semi-analytic and subanalytic simplices [H2, Def. 3.3, page 179]. The variety X is locally, analytically, isomorphic to a ball in ℂ^{n}, such that the variety Y_{J} is defined by z_{1} · . . . · z_{j} = 0 where j = |J|, then the variety Z is isomorphic to the semi-analytic subset of a real vector space: T^{n} × (ℝ^{+})^{n} with the the product structure of (ℝ^{+})^{n} and the n-dimensional torus. The projection Π is written, in the coordinates θ_{k} on the torus, ρ_{k} on the reals and z_{k} on ℂ^{n} for k = 1, ..., n, as z_{k} ∘ Π = ρ_{k} e^{iθ_{k}}. We have

Π^{−1}(Y_{J} − Y^{j+1}) ≃ T^{j} × T^{n−j} × (ℝ^{∗})^{n−j} ≃ T^{j} × (ℂ^{∗})^{n−j}

where we use in the last isomorphism the correspondence defined by z_{k} = ρ_{k}e^{iθ_{k}}, k = j + 1, ..., n to describe Y_{J} − Y^{j+1}.

Define the strict transform of a simplex Δ on Ỹ^{r} as the closure of the inverse image in Z of the complement of Y^{r+1} in Δ.

Let Δ_{α} be a semi-analytic (resp. sub-analytic) simplex contained in the domain of a chart on X and defined by a set of equalities and inequalities of analytic functions f_{α,λ}(x., y.)_{λ∈Λ} in the real coordinates (x., y.) of ℂ^{n} ; its inverse image is obtained by replacing (x., y.) with ρ_{.}(cos(θ.), sin(θ.)) in the defining equations. The strict transform of the simplices Δ_{α} in X are semi-analytic subsets of Z (resp. sub-analytic).

They define in Z a decomposition in a finite system of sub-analytic sets, so that we can use Hironaka’s existence theorem [H2, page 180] to define a triangulation by sub-analytic simplices, subordinated to this decomposition.

II.3.2. In this subsection we present our main topological tool: a correspondence denoted Π_{⋆}^{−1} which gives closed chains in Z, respectively in its boundary ∂Z, as “strict inverse” of chains on X. But first we have to fix some orientation conventions regarding the fibers Π^{−1}(y^{o}) for different points y^{o} ∈ Y.

We denote the oriented boundary of a disc in the complex plane by S^{1} = ∂D (where we consider the natural orientation as a boundary). For any point y^{o} ∈ Y^{1} − Y^{2}, the circle Π^{−1}(y^{o}) appears as the boundary of the complement of a disc in ℂ, hence it is −S^{1}. If y^{o} ∈ Y^{p}−Y^{p+1}, the situation is similar. In a local model, when Y is defined by {y_{1} … y_{p} = 0} ⊂ ℂ^{p} where y^{o} stays for the origin. Let U be defined by {y | min |y_{i}| ≤ 1} ⊂ ℂ^{p} and the component Y_{α_{i}} by y_{i} = 0. Then Π^{−1}(y^{o}), set-theoretically, is the tori S_{1}^{1} × … × S_{p}^{1}, where S_{i}^{1} is defined by {y_{i} : |y_{i}| = 1}. Then, we fix the orientation of Π^{−1}(y^{o}) as given by the product orientation (−S_{1}^{1}) × ⋯ × (−S_{p}^{1}). Moreover, if σ is a contractible C^{∞}-submanifold in Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{p}} − Y^{p+1} (α_{1} < ⋯ < α_{p}), then on the set-theoretical inverse image Π^{−1}(σ) we define the product orientation σ × (−S_{α_{1}}^{1}) × ⋯ × (−S_{α_{p}}^{1}).

With this notations, the following holds.

II.3.3. Lemma. Fix indices α_{1} < ⋯ < α_{i} < α < α_{i+1} < ⋯ < α_{p} and let σ ⊂ Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{p}} be an oriented ℂ^{∞} sub-manifold with boundaries such that σ and ∂σ intersect Y_{α} transversally, but σ ∩ Y_{β} = ∅ for any β ∉ {α, α_{1}, …, α_{p}}. (The C^{∞} transversal intersection is considered in Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{p}}.) Then

∂(Π^{−1}σ) = Π^{−1}(∂σ) + (−1)^{dim σ+i}Π^{−1}(σ ∩ Y_{α}).

Proof. In a neighbourhood of an intersection point p ∈ (σ ∩ Y_{α}) (respectively(∂σ) ∩ Y_{α}), σ has a product structure D × T where T is a ball (respectively a half ball) in Y_{α} ∩ Y_{α_{1}} ∩ ⋯ ∩ Y_{α_{p}}, and D is a real 2-disc transversal to Y_{α}. We denote the boundary of D by ∂D. Let D_{η} = D − {small open disc of radius η and origin 0}; i.e. ∂D_{η} = ∂D−S_{η}^{1}. Then we have the diffeomorphism Π^{−1}(D×T) ≈ Π^{−1}(D_{η}×T). Since Π^{−1}(D_{η} ×T) = D_{η} ×T ×Ω, where Ω = (−1)^{p}S_{α_{1}}^{1} ×⋯×S_{α_{p}}^{1}, we get: ∂Π^{−1}(D ×T) = ∂Π^{−1}(D_{η} × T) = ∂D × T × Ω − S_{η}^{1} × T × Ω + D_{η} × ∂T × Ω =

= (−1)^{dim T +i}T × (−1)^{p+1}S_{α_{1}}^{1} × ⋯ × S_{α}^{1} × ⋯ × S_{α_{p}}^{1} + ∂(D × T) × Ω

= (−1)^{i+dim σ}Π^{−1}(σ ∩ Y_{α}) + Π^{−1}∂σ.

II.3.4. Next, we want to lift an arbitrary sub-analytic chain. First notice that for each sub-analytic subset S ⊂ Ỹ^{r}, the strict transform S̃ := cl(Π^{−1}(S − Y^{r+1})) (i.e. the closure of the inverse image of the complement of Y^{r+1} in S) is a sub-analytic subset of Z. Indeed, the strict transform can be constructed using a real analytic isomorphism and the permitted operations listed in II.1.2 (2-3) (cf. also with II.3.1).

Fix again the integer p ≥ 0, and for any chain ξ ∈ C_{k}^{⋔}(Ỹ^{p}) we want to construct a chain Π_{p}^{−1}(ξ) in C_{k+p}(∂Z) if p ≥ 1, respectively in C_{k}(Z) if p = 0. By the above remark, the inverse image Π^{−1}(|ξ|) of the support of ξ is a (k + p)–dimensional sub– analytic subset of Z. Similarly as above, in the construction we will use Fact II.2.3.

Like in the case of Leray’s cohomological residue, we have the following morphism of relative homology:

Π_{p,rel}^{−1} : H_{k}(|ξ|, |∂ξ| ∪ (|ξ| ∩ Y^{p+1})) ⟶ H_{k+p}(Π^{−1}(|ξ|), Π^{−1}(|∂ξ|) ∪ Π^{−1}(|ξ| ∩ Y^{p+1})).

Since Π^{−1}(|ξ|) → |ξ| is an oriented fiber bundle over |ξ| − Y^{p+1}, with fibers (S^{1})^{p}, the classical spectral sequence argument leads to an isomorphism. Indeed, we use deformation retract tubular neighbourhoods to thicken |ξ| ∩ Y^{p+1} and its inverse image Π^{−1}(|ξ| ∩ Y^{p+1}), then the excision theorem to reduce to the fiber bundle case and show that the relative morphism above is an isomorphism.

Next, we use the morphism

i : H_{k}(|ξ|, |∂ξ|) ⟶ H_{k}(|ξ|, |∂ξ| ∪ (|ξ| ∩ Y^{p+1}))

induced by the inclusion (|ξ|, |∂ξ|) ⟶ (|ξ|, |∂ξ| ∪ (|ξ| ∩ Y^{p+1})). Notice that i is an isomorphism since codim|ξ|∩Y^{p+1} is 2 in |ξ| and in the long homology exact sequence of the pair H_{i}(|∂ξ| ∪ (|ξ| ∩ Y^{p+1}), |∂ξ|) = H_{i}(|ξ| ∩ Y^{p+1}, |∂ξ| ∩ Y^{p+1}) = 0 if i = k or k − 1.

Now, the image of ξ, via the composed map Π_{p,rel}^{−1} ∘i, determines completely Π_{p}^{−1}(ξ) (via Fact). By definition, this is the application ξ ↦ Π_{p}^{−1}(ξ). Sometimes the index p will be replaced by ⋆.

In a simple way, this chain Π_{p}^{−1}(ξ) can be defined as follows. Write ξ as a finite sum ∑ a_{σ} σ, where σ are the simplices in the support of ξ with non-zero coefficients. Let σ^{0} be σ −Y^{p+1}. Then the closure cl(Π^{−1}(σ^{0})) of Π^{−1}(σ^{0}) is a (k +p)–dimensional sub–analytic set. Then define Π_{p}^{−1}(ξ) by ∑ a_{σ} cl(Π^{−1}(σ^{0})).

The next result generalizes the above lemma.

II.3.5. Proposition. Fix the integer p ≥ 0 and indices α_{1} < ⋯ < α_{p}. For any α ∉ {α_{1}, …, α_{p}}, set i(α) = i if α_{1} < ⋯ < α_{i} < α < α_{i+1} < ⋯ < α_{p}. Then for any ξ ∈ C_{k}^{⋔}(Y_{α_{1},... ,α_{p}}):

∂(Π_{p}^{−1}ξ) = Π_{p}^{−1}(∂ξ) + ∑_{α} (−1)^{k+i(α)} Π_{p+1}^{−1}(ξ ∩ Y_{α}).

Above, the sum is over all α with α ∉ {α_{1}, …, α_{p}}. The equality is considered in C_{∗}(∂Z) if p ≥ 1, and in C_{∗}(Z) if p = 0.

Proof. The contribution Π_{p}^{−1}(∂ξ) is clear. Next we want to determine the coefficients of the simplices which lie in Π^{−1}(|ξ| ∩ Y_{α}). It is enough to work modulo Y_{α} ∩ (∪_{β}Y_{β}), where the union is over all β ∉ {α, α_{1}, …, α_{p}}, since it intersects |ξ| in codimension 4 and its inverse image intersects Π^{−1}(|ξ|) in codimension 2.

Consider the composition:

H_{k+p}(Π^{−1}(|ξ|), Π^{−1}(|∂ξ|) ∪ Π^{−1}(|ξ| ∩ Y^{p+1}))
↓ ∂
H_{k+p−1}(Π^{−1}(|∂ξ| ∪ (|ξ| ∩ Y^{p+1})), Π^{−1}(|∂ξ| ∪ (|ξ| ∩ ∪_{β}Y_{β})))
≈ ↓ e
H_{k+p−1}(Π^{−1}(|ξ| ∩ Y_{α}), Π^{−1}(|∂ξ| ∩ Y_{α}) ∪ Π^{−1}(|ξ| ∩ Y_{α} ∩ ∪_{β}Y_{β}))).

Again, the union ∪_{β} is over all β ∉ {α, α_{1}, …, α_{p}}. If A = Π^{−1}(|ξ|), B = Π^{−1}(|∂ξ| ∪ (|ξ| ∩ ∪_{β}Y_{β}))), and C = Π^{−1}(|ξ| ∩ Y_{α}), then the first map is the boundary operator H_{k+p}(A, B ∪ C) → H_{k+p−1}(B ∪ C, B), and the second is the excision isomorphism H_{k+p−1}(B ∪ C, B) → H_{k+p−1}(C, B ∩ C).

The composed map e ∘ ∂ and the map ξ ↦ ξ ∩ Y_{α} are connected by the following diagram, which commutes up to sign:

H_{k}(|ξ|, |∂ξ|) ⟶^{Π_{p}^{−1}} H_{k+p}(Π^{−1}(|ξ|), Π^{−1}((|∂ξ|) ∪ (|ξ| ∩ Y^{p+1})))

↓ ∩Y_{α} ↓ e ∘ ∂

H_{k−2}(|ξ|∩Y_{α}, |∂ξ|∩Y_{α}) ⟶^{Π_{p+1}^{−1}} H_{k+p−1}(Π^{−1}(|ξ|∩Y_{α}), Π^{−1}((|∂ξ|∩Y_{α})∪(|ξ|∩Y_{α} ∩∪_{β}Y_{β})))

The commutativity of the diagram (up to a sign), basically comes from the fact that for a manifold with boundary, the Lefschetz duality identifies the boundary operator in homology with the restriction map (to the boundary) in cohomology (see e.g. [Br], page 357). The above sign is universal, depends only on the orientation conventions. Therefore, it can be determined using C^{∞}-transversal chains, as in lemma II.3.3.

The above proposition together with the definition of the operator ∩ : C_{k}^{⋔}(Ỹ^{p}) → C_{k−2}^{⋔}(Ỹ^{p+1}), have the following corollary.

II.3.6. Corollary. ∂Π_{⋆}^{−1} = Π_{⋆}^{−1}(∂ + ∩).

II.3.7. Example. Let ξ ∈ C_{∗}^{⋔}(X). In general, the three sets Π^{−1}(|ξ| ∩ Y) (here Π^{−1} denotes the set-theoretical inverse image), |Π_{⋆}^{−1}(ξ)| ∩ ∂Z and |∂Π_{⋆}^{−1}(ξ)| are all distinct. To see this, set X = ℂ^{2}, Y = {0}×ℂ (Y^{2} = ∅). Fix coordinates z_{j} = x_{j}+iy_{j} (j = 1, 2 ) in ℂ^{2}. Let ξ be a chain with support (x_{1} − 1)^{2} + y_{1}^{2} + x_{2}^{2} − 1 = y_{2} = 0, and coefficient one.

Then |ξ| is a 2-dimensional real sphere with |ξ| ∩ Y = (0, 0), hence Π^{−1}(|ξ| ∩ Y) = S^{1}. On the other hand, the intersection chain ξ ∩ Y = 0 (even if |ξ| ∩ Y ≠ ∅). Actually, ξ ∩ Y is the chain supported by |ξ| ∩ Y with coefficient the intersection multiplicity of |ξ| and Y, which is zero. Therefore, ∂Π_{⋆}^{−1}(ξ) = Π_{⋆}^{−1}∂ξ = 0, hence |∂Π_{⋆}^{−1}(ξ)| = ∅. Finally, the reader is invited to verify that |Π_{⋆}^{−1}(ξ)| ∩ ∂Z is a half circle in S^{1} = Π^{−1}(0, 0).

II.3.8. Remark. In the above discussion, corresponding to the integer p = 0, one can replace the space X by the (compact) tubular neighbourhood U = U_{α} = Π(Z_{α}) (cf. I.1.2). This means that the complex C_{∗}(X) is replaced by C_{∗}(U), and C_{∗}^{⋔}(X) by C_{∗}^{⋔}(U). Notice that ∩ : C_{∗}^{⋔}(U) → C_{∗−2}^{⋔}(Ỹ^{1}) is well-defined and still satisfies ∩∂ + ∂∩ = 0. Moreover, one has a map Π_{⋆}^{−1} : C_{∗}^{⋔}(U) → C_{∗}(Z_{α}), defined similarly as Π_{⋆}^{−1} : C_{∗}^{⋔}(X) → C_{∗}(Z), which satisfies ∂Π_{⋆}^{−1} = Π_{⋆}^{−1}(∂ + ∩).

II.4. The complex of sheaves. C_{∗}(Ỹ^{p}) has a natural sheafification C∗ ; Ỹ^{p} which is fine ([GM2] page 97, or [B] page 33). Actually, the complex C∗ ; Ỹ^{p} is quasi-isomorphic to the dualizing complex D∗ ; Ỹ^{p} (cf. [GM2] page 97, or [B] page 33).

What we will need in this paper, is a different construction, we will sfeafify the dual complex, and we will obtain a quasi-isomorphism with ℤỸ^{p}.

For any open set V ⊂ Ỹ^{p}, let C_{k}(V, Ỹ^{p}) be the subgroup of chains ξ ∈ C_{k}(Ỹ^{p}) with compact support |ξ| in V. Obviously, for any open pair V ⊂ W ⊂ Ỹ^{p}, there is a natural inclusion C_{k}(V, Ỹ^{p}) → C_{k}(W, Ỹ^{p}). Now, define the dual C^{k}(V, Ỹ^{p}) by Hom_{ℤ}(C_{k}(V, Ỹ^{p}), ℤ). Then for any V ⊂ W as above, the “restriction” C^{k}(W, Ỹ^{p}) → C^{k}(V, Ỹ^{p}) defines a presheaf C^{k}(Ỹ^{p}) on Ỹ^{p}, satisfying the condition (S2) (i.e. it is “conjuctive” in the terminology of [Br]). Let C̄^{k}(Ỹ^{p}) be the associated sheaf with global sections C̄^{k}(Ỹ^{p}); and let C_{0}^{k}(Ỹ^{p}) be the subgroups of elements of C^{k}(Ỹ^{p}) with empty support. Then

0 → C_{0}^{k}(Ỹ^{p}) → C^{k}(Ỹ^{p}) → C̄^{k}(Ỹ^{p}) → 0

is exact (cf. [Br] page 22). Moreover, C^{∗}(Ỹ^{p}) and C_{0}^{∗}(Ỹ^{p}) form complexes, and H^{∗}(C_{0}^{∗}(Ỹ^{p})) = 0 (by subdivision argument, see [Br2], page 26 in the case of singular chains).

Therefore, the complexes C^{∗}(Ỹ^{p}) and C̄^{∗}(Ỹ^{p}) are quasi-isomorphic. On the other hand, for any open V, there is an augmentation map C_{0}(V, Ỹ^{p}) → ℤ which give rise to a resolution 0 → ℤỸ^{p} → C̄^{∗}(Ỹ^{p}). The sheaf C̄^{k}(Ỹ^{p}) is a module over the ring of ℤ-constructible functions on Ỹ^{p}. Indeed, for any constructible function f and φ ∈ C^{k}(Ỹ^{p}) one can define f · φ ∈ C^{k}(Ỹ^{p}) by (f · φ)(σ) = f(σ̂)φ(σ), where σ̂ is the barycenter of σ. This shows that the above resolution is a resolution of fine sheaves.

III. Double complexes and their spectral sequences

We construct in this section various homological complexes giving rise to various homology groups with their weight filtration. The cases (Y, ∅) and (X, Y) are wellknown, but we will need them in the construction of the double complex of ∂U, so we include them in our presentation as well. The new result is the construction of the complexes of the pairs (X − Y, ∅), (X, X − Y) and (∂U, ∅), in which cases we will use the dimensionally transverse cycles. The most involved case is ∂U, which is separated in section VI. In the next paragraphs, as an introductory guide, we will stress the case X − Y.

The main new object is homological double complex of X − Y:

A_{s,t}(X − Y) := C_{t+2s}^{⋔}(Ỹ^{−s}), s ≤ 0, t + 2s ≥ 0

with D = ∂ + ∩ as the differential of the total complex Tot_{∗}(A_{∗∗}(X − Y)).

Our aim, reformulated for the case of X − Y, is to prove that the homology of the above total complex is the homology of X − Y, and its weight filtration provides Deligne’s weight filtration on H_{∗}(X − Y, ℚ). More precisely: III.0. Theorem. Let Y = ⋃_{i∈I} Y_{i} be a NCD in a smooth proper algebraic variety X over ℂ and Π : Z → X the projection as above. The generalized Leray inverse image defines a quasi-isomorphism

Π_{⋆}^{−1} : Tot_{∗}(A_{∗∗}(X − Y)) ⟶ C_{∗}(Z).

In other words, a collection of dimensionally transversal geometric chains {c_{r}}_{r≥p}, where c_{r} is a chain on Ỹ^{r} satisfying ∂c_{p} = 0 and ∂c_{r+1} = c_{r} ∩ Ỹ^{r+1} for any r ≥ p (p ≥ 0), defines a cycle c = ∑_{r≥p} Π_{r}^{−1}c_{r} in Z. Moreover, their homology classes generate H_{∗}(Z, ℚ).

Geometrically this means the following: the highest nontrivial weight in H_{k}(X − Y, ℚ) is −k, and for any p ≥ 0, W_{−k−p}H_{k}(X − Y, ℚ) is generated by classes of cycles e, with a chain decomposition e = ∑_{r≥p} e_{r} such that each e_{r} is the closure of a “fiber bundle” over a c_{r} − Y^{r+1} with fiber (S^{1})^{r}, where c_{r} are chains in Ỹ^{r} and c_{r} − Y^{r+1} is considered in Y^{r}. This is realized geometrically in a small tubular neighbourhood of Y^{r}. The fact that the collection {c_{r}}_{r} satisfies the above condition is equivalent with the fact that {e_{r}}_{r} fit together without boundary, forming a closed cycle.

Above we identified H_{∗}(C_{∗}(Z)) and H_{∗}(X − Y). For more details, see III.4 and V.2.

This result can be obtained by duality with Deligne’s logarithmic differential complex. Here we follow a purely topological path:

1) We start with the basic Mayer–Vietoris resolution for the NC-divisor Y.

2) We introduce the relative homology of X mod Y using the classical cone construction, called here mixed cone, in order to carry the construction with weight filtrations (a diagonalisation process for the weights is needed in order to stay in the category of mixed Hodge complexes [D] so to get mixed Hodge structures on their (co)homology by the basic result of Deligne).

3) We give a topological construction of Poincaré duality between the cohomology of Y and the homology of the pair (X, X − Y).

4) The above duality extends to a duality between the cohomology of the pair (X, Y) and the homology of X − Y. This provides the wanted result.

Moreover we obtain the degeneration of the weight spectral sequence on the total complex of dimensionally transversal chains.

Notations and preliminary remarks. For any double complex (A_{∗∗}, ∂, δ), we denote its total complex by (T ot_{∗}(A_{∗∗}), D), where T ot_{k}(A_{∗∗}) = ⊕_{s+t=k}A_{st} and D = ∂ + δ. (Here the degree of ∂ is (0, −1), of δ is (−1, 0).) The weight filtration of A_{∗∗} is defined by W (A_{∗∗})_{s} := ⊕_{p≤s}A_{pq}. The homological spectral sequence associated with the weight filtration W is denoted by E_{∗∗}^{r}. Recall that E_{st}^{1} = H_{t}(A_{s∗}, ∂) and d^{1} is induced by δ. However, we will violate the weight notation on the ∞-term, and we will use Deligne’s convention: on E_{st}^{∞} the weight is −t (instead of s); i.e. image{H_{i}(T ot_{∗}(W_{s}) → H_{i}(T ot_{∗}(A))} is W_{s−i}H_{i}(T ot_{∗}(A)).

The dual double complex of A_{∗∗} is B_{st} = Hom_{ℤ}(A_{st}, ℤ) with the corresponding dual maps. Its weight filtration is W (B)_{s} = {φ ∈ A^{∗} : φ(W (A)_{−s−1}) = 0}, or equivalently, W (B)_{−s} = ⊕_{p≥s}B_{pq}.

For the definition and properties of “cohomological mixed Hodge complexes” see [D.III]. In the next proofs the notion of “mixed cone” will be important, this corresponds to the mapping cone in the category of mixed Hodge complexes. For details, see [D. III], page 21, or [E], page 49.

III.1. The homological double complex of Y (the homology of the NCD).

Consider the double complex A_{s,t}(Y) := C_{t}(Ỹ^{s+1}) (with s ≥ 0 and t ≥ 0) together with the natural operators:

∂ : C_{k}(Ỹ^{p}) → C_{k−1}(Ỹ^{p}) and i : C_{k}(Ỹ^{p}) → C_{k}(Ỹ^{p−1}).

Here i is defined as follows. If ⊕_{α_{1}<···<α_{p}}c_{α_{1},... ,α_{p}} ∈ C_{k}(Ỹ^{p}), then i(⊕_{α}c_{α}) = ⊕_{β}d_{β} if:

d_{α_{1},... ,α_{p−1}} = ∑_{α_{1}<···<α_{i}<α<···<α_{p−1}} (−1)^{k+i} i_{α_{1},... ,α_{p−1};α}(c_{α_{1},... ,α_{i},α,... ,α_{p−1}}),

where i_{α_{1},... ,α_{p−1};α} is the natural inclusion Y_{α_{1},... ,α_{i},α,... ,α_{p−1}} ↪ Y_{α_{1},... ,α_{p−1}}.

III.1.1. Lemma. a) i^{2} = 0, and b) i∂ + ∂i = 0.

In particular, D := i + ∂ is a differential of the total complex T ot_{∗}(A_{∗∗}(Y)). The weight filtration {W (A)_{s}}_{s} provides a spectral sequence over ℤ. Here are some of its properties.

III.1.2. Proposition.

a) E_{st}^{1} = H_{t}(Ỹ^{s+1}) and d^{1} : E_{st}^{1} = H_{t}(Ỹ^{s+1}) → H_{t}(Ỹ^{s}) = E_{s−1,t}^{1} is i_{∗} induced by i.

b) E_{st}^{r} ⟹ H_{s+t}(Y, ℤ), inducing a weight filtration on H_{∗}(Y, ℤ). Actually, E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(Y, ℚ) (in Deligne’s weight notation).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at level two, i.e. d^{r} ⊗ 1_{ℚ} = 0 for r ≥ 2.

Proof. Let n : Ỹ^{p} → Y be the natural map. Recall that the K_{ℚ}^{·} term in Deligne’s cohomological mixed Hodge complex associated with the space Y is the “Mayer-Vietoris resolution” n_{∗}ℚỸ^{·}:

0 → n_{∗}ℚỸ^{1} → n_{∗}ℚỸ^{2} → · · · .

This is considered with its “bête” filtration W_{−s}(n_{∗}ℚỸ^{·}) = σ_{≥s}(n_{∗}ℚỸ^{·}).

Consider now the dual double complex B_{∗∗} of A_{∗∗}. Then (n_{∗}ℚỸ^{·}, W) and (B_{∗∗}, W)⊗ 1_{ℚ} are quasi-isomorphic. This follows from the discussion II.4. Therefore, their spectral sequence (for r > 0) are isomorphic. This gives a) and b). Finally notice that by a result of Deligne, the weight spectral sequence (over ℚ) of a mixed Hodge complex degenerates at rank two, which provides c).

For a different approach and proof, see [McC], where it is proved that the filtered dualizing complex (i_{∗}D_{Y}^{∗} , τ_{≤}) and (i_{∗}n_{∗}ℚỸ , σ_{≥}) are Verdier dual (here i : Y → X stays for the inclusion).

III.2. The homological double complex of (X, Y).

Starting with Ỹ^{0} = X, we consider now the double complex A_{s,t}(X, Y) := C_{t}(Ỹ^{s}) (with s ≥ 0 and t ≥ 0) together with the natural operators: ∂ : C_{k}(Ỹ^{p}) → C_{k−1}(Ỹ^{p}) and i : C_{k}(Ỹ^{p}) → C_{k}(Ỹ^{p−1}) as above. Again, D := i + ∂ is a differential of the total complex T ot_{∗}(A_{∗∗}(X, Y)).

If we introduce the double complex A_{∗∗}(X) of X defined by A_{s∗}(X) = C_{∗}(X) if s = 0 and = 0 otherwise, then A_{s∗}(X, Y) = A_{s∗}(X) ⊕ A_{s−1,∗}(Y), and A_{∗∗}(X, Y) can be interpreted as the Cone(ĩ) of ĩ: A_{∗∗}(Y) → A_{∗∗}(X), where ĩ|A_{s∗}(Y) = 0 if s ≠ 0, and ĩ|A_{0∗}(Y) = i : C_{∗}(Ỹ^{1}) → C_{∗}(X).

III.2.1. Proposition.

a) E_{st}^{1} = H_{t}(Ỹ^{s}) and d^{1} is i_{∗}, induced by i.

b) E_{st}^{r} ⟹ H_{s+t}(X, Y, ℤ), inducing a weight filtration on H_{∗}(X, Y, ℤ). E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(X, Y, ℚ) (in Deligne’s weight notation).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at level two, i.e. d^{r} ⊗ 1_{ℚ} = 0 for r ≥ 2.

Proof. The proof is similar as in the case of III.1.2. In this case, K_{ℚ}^{·} is

0 → n_{∗}ℚỸ^{0} → n_{∗}ℚỸ^{1} → · · · .

Actually this (and the whole cohomologically mixed Hodge complex of (X, Y)) can be constructed as a mixed cone of the complexes of X, respectively of Y. This is compatible with the construction of A_{∗∗}(X, Y).

III.3. The homological double complex of (X, X − Y) or (U, ∂U).

Here U is the “tubular neighbourhood” of Y. We define

A_{s,t}(X, X − Y) := C_{t+2(s−1)}^{⋔}(Ỹ^{−(s−1)}),

with s ≤ 0 and t+2(s−1) ≥ 0. Then ∂ and ∩ act as ∂ : A_{s,t} → A_{s,t−1} and ∩ : A_{s,t} → A_{s−1,t}, hence D = ∂ + ∩ is the differential of the total complex Tot_{∗}(A_{∗∗}(U, ∂U)). Corollary II.3.6 reads as:

III.3.1. Corollary. Π_{⋆}^{−1} : (T ot_{∗}(A_{∗∗}(X, X − Y)), D) → (C_{∗−1}(∂Z), ∂) is a morphism of complexes i.e. ∂Π_{⋆}^{−1} = Π_{⋆}^{−1}D.

Moreover, II.2.1, II.2.5 and II.2.8 and Poincaré duality imply the following result.

III.3.2. Proposition.

a) E_{s+1,t}^{1} = H_{t+2s}(Ỹ^{−s}), and d_{1} is the transfer map i_{!}.

b) E_{st}^{r} ⟹ H_{s+t}(X, X − Y, ℤ), and E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(X, X − Y, ℚ).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at level two, i.e. d^{r} ⊗ 1_{ℚ} = 0 for r ≥ 2.

d) The Poincaré duality map pd (cf. II.2.6) induces an isomorphism of spectral sequences (for any r ≥ 1) between the cohomological spectral sequnence of Y and the above homological spectral sequence of (X, X − Y), which provides exactly the ismorphism ∩[X] : Gr_{2n−t}^{W}H^{2n−s−t}(Y) → Gr_{−t}^{W}H_{s+t}(X, X − Y).

e) Π_{⋆}^{−1} from III.3.1 induces the boundary operator H_{∗}(X, X−Y, ℤ) = H_{∗}(U, ∂U, ℤ) → H_{∗−1}(∂U, ℤ).

Proof. The main result here is the degeneration of the spectral sequence at level 2, which follows via duality d) from the NCD case.

Fix a triangulation T and let T′ be its first barycentric subdivision. Then the Poincaré duality map (cf. II.2.6) can be organized in the following morphism of double complexes.

Let A_{st}^{T}(Y) = C_{t}^{T} (Ỹ^{s+1}), which form a double complex with ∂ and i, similarly as in III.1. Set the double complex B_{st}^{T}(Y) = Hom(A_{st}^{T}(Y), ℤ) with dual morphisms δ and i^{∗}. Let A_{s+1,t}^{T′}(X, X − Y) = C_{t+2s}^{⋔,T′}(Ỹ^{−s}) with boundary morphisms ∂ and ∩ (similarly as A_{∗∗}(X, X − Y) defined above). Then pd : B_{−s,2n−t}^{T}(Y) → A_{st}^{T′}(X, X − Y) satisfies ∂ ∘ pd = pd ∘ δ (cf. II.2.8) and ∩ ∘ pd = pd ∘ i^{∗} (cf. II.2.9).

Now, A_{∗∗}(Y) is quasi-isomorphic to A_{∗∗}^{T}(Y) (i.e. their spectral sequences are the same for r ≥ 1), and the later is dual to B_{∗∗}^{T}(Y). Using II.2.9, pd induces an isomorphism at the level of the E^{1} term, hence it is an quasi-isomorphism, and it induces isomorphism at the level of any E^{r} (r ≥ 2). On the other hand, E^{r}(A_{∗∗}^{T′}(X, X − Y)) = E^{r}(A_{∗∗}(X, X − Y)) for r ≥ 1. Hence the result follows.

III.4. The homological double complex of X − Y.

Define A_{s,t}(X − Y) := C_{t+2s}^{⋔}(Ỹ^{−s}), with s ≤ 0 and t + 2s ≥ 0. Then D = ∂ + ∩ is a differential of the total complex Tot_{∗}(A_{∗∗}(X − Y)).

Similarly to the case of the pair (X, Y), we can define A_{∗∗}^{⋔}(X) by A_{s∗}^{⋔}(X) = C_{∗}^{⋔}(X) if s = 0 and = 0 otherwise. Then A_{s∗}(X −Y) = A_{s∗}^{⋔}(X)⊕A_{s+1,∗}(X, X −Y). Actually, A_{∗−1,∗}(X − Y) is the cone of the morphism ∩ : A_{∗∗}^{⋔}(X) → A_{∗∗}(X, X − Y), where ∩ is the intersection for s = 0 and zero otherwise.

Hence the Poincaré duality in III.3.2 extends to:

III.4.1. Proposition.

a) E_{s,t}^{1} = H_{t+2s}(Ỹ^{−s}), and d_{1} is the transfer map i_{!}.

b) E_{st}^{r} ⟹ H_{s+t}(X − Y, ℤ), and E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(X − Y, ℚ).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at level two, i.e. d^{r} ⊗ 1_{ℚ} = 0 for r ≥ 2.

d) The Poincaré duality map pd (cf. II.4.6) induces an isomorphism of spectral sequences (for any r ≥ 1) between the cohomological spectral sequnence of (X, Y) and the above homological spectral sequence of X − Y, which provides exactly the ismorphism ∩[X] : Gr_{2n−t}^{W}H^{2n−s−t}(X, Y) → Gr_{−t}^{W}H_{s+t}(X − Y).

III.4.2. Remark. Notice that by corollary II.3.6 and III.4.1

Π_{⋆}^{−1} : (T ot_{∗}(A_{∗∗}(X − Y)), D) → (C_{∗}(Z), ∂)

is a quasi-isomorphism of complexes.

Notice also that H_{∗}(C_{∗}(Z)) = H_{∗}(X − Y). Indeed, if Z_{α} is a small collar of Z, then H_{∗}(Z) = H_{∗}(Z − Z_{α}) = H_{∗}(Z − ∂Z) = H_{∗}(X − Y).

The above morphism can be extended to the pair (X, X−Y) as follows. Consider now the complex (Ker_{∗+1}, ∂) defined as the kernel of the morphism Π_{∗} : (C_{∗}(∂Z), ∂) → (C_{∗}(Y), ∂) induced by Π. In fact, this complex coincide with the kernel of the morphism Π_{∗} : (C_{∗}(Z), ∂) → (C_{∗}(X), ∂) since Π induces an isomorphism over X − Y.

III.4.3. Corollary.

a) Π_{⋆}^{−1} : A_{s,t}(X−Y) → C_{s+t}(Z) maps A_{s+1,t}(X, X−Y) ⊂ A_{s,t}(X−Y) in Ker_{s+t+1}.

b) There is a natural commutative diagram whose vertical arrows are quasi-isomorphisms.

0 → T ot_{∗}(A_{∗+1,∗}(X, X − Y)) → T ot_{∗}(A_{∗∗}(X − Y)) → T ot_{∗}(A_{∗∗}^{⋔}(X)) → 0
↓Π_{⋆}^{−1}  ↓Π_{⋆}^{−1}  ↓j_{1}
0 →  Ker_{∗+1}  →  C_{∗}(Z)  →  C_{∗}(X)  → 0

The horizontal lines induce the long homology exact sequence of the pair (X, X − Y).

Proof. The image of any cycle ξ ∈ C_{k}^{⋔}(Ỹ^{p}), with p ≥ 1, via the composite map

C_{k}^{⋔}(Ỹ^{p}) ⟶^{Π_{p}^{−1}} C_{k+p}(∂Z) ⟶^{Π_{∗}} C_{k+p}(Y)

has k-dimensional support. Actually, this support is the image of |ξ| by the natural map Ỹ^{p} → Y. Hence it supports no cycle in C_{k+p}(Y).

IV. Review of the spectral sequence associated with a double complex

IV.1. Let A = ⊕_{s,t}A_{s,t} be a finite homological double complex with operators

[figure: A_{s−1,t}; δ; A_{s,t}; δ^{2} = 0, ∂^{2} = 0, δ ∂ + ∂ δ = 0; A_{s,t−1}; ∂]

Let Tot_{k} A = ⊕_{s+t=k}A_{s,t} and D = d + δ be the associated total complex. Define the (weight) filtration by W_{ℓ}A = ⊕_{s≤ℓ}A_{s,t}. By general theory, there is a spectral sequence (E_{s,t}^{r}, d_{r})_{r≥1}, converging to the graded homology of A. We put E_{s,t}^{0} = A_{s,t}; then

E_{s,t}^{1} = Ker(∂ : A_{s,t} → A_{s,t−1})/Im(∂ : A_{s,t+1} → A_{s,t})

and d^{1} is induced at the homology level by δ. Let

\begin{aligned}Z_{s,t}^{r} & = \{c ∈ W_{s} Tot_{s+t} A : Dc ∈ W_{s−r}A\} \\ Z_{s}^{r} & = \{c ∈ W_{s}A : Dc ∈ W_{s−r}A\} \\ Z_{s,t}^{∞} & = \{c ∈ W_{s} Tot_{s+t} A : Dc = 0\} \\ Z_{s}^{∞} & = \{c ∈ W_{s}A : Dc = 0\}. \end{aligned}

Then

\begin{aligned}E_{s,t}^{r} & = Z_{s,t}^{r}/(Z_{s−1}^{r−1} + DZ_{s+r−1}^{r−1}) \\ E_{s,t}^{∞} & = Z_{s,t}^{∞}/(Z_{s−1}^{∞} + DA ∩ W_{s}A). \end{aligned}

In particular, E_{s,t}^{r} is generated by the class of elements

c_{st} + c_{s−1,t+1} + ⋯ + c_{s−r+1,t+r−1} ∈ Z_{st}^{r} (c_{pq} ∈ A_{pq})

which satisfy

(∗_{r}) ∂c_{st} = 0; ∂c_{s−1,t+1} + δc_{st} = 0; …; ∂c_{s−r+1,t+r−1} + δc_{s−r+2,t+r−2} = 0.

The map d^{r} can be defined as:

d_{s,t}^{r}[c_{st} + c_{s−1,t+1} + ⋯ + c_{s−r+1,t+r−1}] = [δc_{s−r+1,t+r−1}]

For r = 1, this means that c_{st} ∈ A_{pq} with ∂c_{st} = 0. For r = 2, Z_{st}^{2} is generated by elements of type c_{st}+c_{s−1,t+1} such that ∂c_{s,t} = 0, ∂c_{s−1,t+1}+δc_{s,t} = 0, which generates kerd^{1} on E^{1}. For r = ∞, E_{st}^{∞} is generated by class of cycles of type

c_{st} + c_{s−1,t+1} + ⋯ + c_{s−r,t+r} + ⋯

satisfying

(∗_{∞})  ∂c_{st} = 0; ∂c_{s−r,t+r} + δc_{s−r+1,t+r−1} = 0 for any r ≥ 1.

IV.1.1. Now, assume that d^{2} = d^{3} = ⋯ = 0 which means E_{st}^{2} ≡ E_{st}^{∞}. Then consider

Ker(d_{1} : E_{st}^{1} → E_{s−1,t}^{1}) → E_{st}^{∞}.

This shows that for any c^{2} = c_{st} + c_{s−1,t+1} satisfying (∗_{2}), there is a class c^{∞} = c′_{st} + c′_{s−1,t+1} + ⋯ with (∗_{∞}), such that [c^{2}] = [c^{∞}] in E_{st}^{∞}. More precisely, we show that we can choose the cycle c^{∞} with c′_{st} = c_{st}.

IV.1.2. Proposition.

a) Consider a class [c_{st}] ∈ Ker[d_{1} : E_{st}^{1} → E_{s−1,t}^{1}] and fix a representative c_{st} ∈ A_{st} with ∂c_{st} = 0 and δc_{st} ∈ ∂A_{s−1,t+1} i.e., we can define a cycle c^{2} = c_{st} +c_{s−1,t+1} ∈ Z_{st}^{2}. If d^{2} = 0, then c^{2} can be replaced by c̃_{2} = c_{st} + c̃_{s−1,t+1} such that [c^{2}] = [c̃^{2}] in E_{st}^{2} and c̃^{2} can be completed to a cycle c̃^{3} = c_{st} + c̃_{s−1,t+1} + c̃_{s−2,t+2} ∈ Z_{st}^{3}.

The fact that d^{2} = 0 is equivalent to the fact that this can be done for any cycle c^{2}.

b) More generally, consider an element c^{r} = ∑_{i=0}^{r−1} c_{s−i,t+i} ∈ Z_{s,t}^{r} and assume that d^{r} = 0. Then c^{r} can be replaced by c̃^{r} = c_{st} + ∑_{i=1}^{r−1} c̃_{s−i,t+i} ∈ Z_{s,t}^{r} with [c^{r}] = [c̃^{r}] ∈ E_{s,t}^{r} such that c̃^{r} can be completed to c̃^{r+1} = c̃^{r} + c̃_{s−r,t+r} ∈ Z_{st}^{r+1}.

Again, in the case of an arbitrary spectral sequence, if any c_{r} can be completed to c̃^{r+1} as above, then d^{r} = 0.

c) In particular, if d^{2} = d^{3} = ⋯ = 0, then

i) any representative c_{st} (with ∂c_{st} = 0 and δc_{st} ∈ ∂A_{s−1,t+1}) of a class in Ker(d_{1}|E_{st}^{1}) can be completed to a cycle

c_{s,t}^{∞} = c_{s,t} + ∑_{i≥1} c_{s−i,t−i} ∈ Z_{st}^{∞};

ii) if c_{st}^{∞} and c′_{st}^{∞} are two liftings of c_{st} as in (i) , then c_{st}^{∞} − c′_{st}^{∞} ∈ Z_{s−1}^{∞}; iii) if c_{st} and c′_{st} are representatives of the same class of Ker(d_{1}|E_{st}^{1}) (i.e. c_{st} − c′_{st} ∈ ∂B_{s,t+1}) and we complete c_{st} and c′_{st} to c_{st}^{∞} and c′_{st}^{∞}, respectively, as in (i), then

c_{st}^{∞} − c′_{st}^{∞} ∈ DA ∩ W_{s} + Z_{s−1}^{∞}.

V. Constructing rational cycles

The aim of the present section is to use the additional degeneration property of the weight spectral sequences and the above review in order to give a construction of cycles which represent W_{−t}H_{s+t}(A, B), where (A, B) = (Y, ∅), (X, Y), (X, X − Y), (X − Y, ∅) (The case (∂U, ∅) is again separated in section VI.). Moreover, we find topological properties and characterizations of these cycles. We will keep the notations of the previous sections. All the homology groups are considered with rational coefficients.

V.1. Cycles in (X, X − Y).

V.1.1. First notice that H_{∗}(X, X − Y) = H_{∗}(U, ∂U). In this subsection we will construct relative cycles in (U, ∂U), i.e chains ξ supported in U with |∂ξ| ⊂ ∂U.

For any (s, t) with s + t = k fix c_{st} ∈ A_{st}(X, X − Y) with ∂c_{st} = 0 and ∩c_{st} ∈ im∂. Then it can be completed to a cycle c_{s,t}^{∞} = c_{st} +c_{s−1,t+1} +⋯ with Dc_{s,t}^{∞} = 0. Consider Π_{⋆}^{−1}c_{s,t}^{∞} ∈ C_{k−1}(∂Z). Then by III.3.1 one has ∂Π_{⋆}^{−1}c_{s,t}^{∞} = 0. Fix a sub-analytic homeomorphism ϕ_{α} : ∂Z × [0, α] → Z_{α}, where Z_{α} is a collar of ∂Z ⊂ Z (cf. I), and write U = Π∘ϕ_{α}(∂Z×[0, α]). For any chain ξ ∈ C_{∗}(∂Z), one can associate in a natural way a new chain ξ × [0, α] ∈ C_{∗+1}(∂Z × [0, α]). Then Π_{∗} ∘ (ϕ_{α})_{∗}(Π_{⋆}^{−1}c_{s,t}^{∞} × [0, α])) ∈ C_{k}(U) has boundary supported in ∂U = Π ∘ ϕ_{α}(∂Z × {α}). For simplicity, we will denote this relatice cycle by c_{s,t}^{rel}.

Notice that for p ≥ 1, |c_{s,t}^{rel}| ∩ Y^{p} = |c_{s′,k−s′}|, where s′ := min{s, −p + 1}. More precisely, if p ≥ 1−s, the intersection |c_{s,t}^{rel}|∩Y^{p} is |c_{−p+1,k+p−1}|, which has dimension k − p − 1. For p = −s, the intersection |c_{s,t}^{rel}| ∩ Y^{p} is |c_{s,t}|, which has dimension k − p − 2.

V.1.2. Proposition.

a) Fix k = s + t. To any homology class [c_{st}] in H_{t+2(s−1)}(Ỹ^{−s+1}) with [c_{st}] ∩ Ỹ^{−s+2} = 0, the above construction gives a relative cycle c_{s,t}^{rel}. Their classes generate W_{−t}H_{k}(X, X − Y) = W_{−t}H_{k}(U, ∂U) (and they are well-defined modulo W_{−t−1}H_{k}).

b) W_{−t}H_{k}(X, X − Y) = {[c] : c a relative cycle in X with |∂c| ⊂ X − Y such that dim |c| ∩ Y^{p} = k − p − 1 for p ≥ 1 − k + t, and < k − p − 1 for p = −k + t}. In these intersection conditions we always assume that p ≥ 1, otherwise the assumption is considered empty. In particular, W_{−k}H_{k}(X, X − Y) = H_{k}(X, X − Y).

The above formula is also equivalent to the following characterization:

W_{−t}H_{k}(X, X − Y) = {[c] : dim |c| ∩ Y^{−k+t} < 2k − t − 1.}

This shows that the homeomorphism type of the pair (X, Y) (or even of (U, ∂U)) determines completely the weight filtration of H_{∗}(X, X − Y).

V.1.3. The support filtration. Let U^{p} ⊂ X be a small regular neighbourhood of Y^{p} in X (cf. [R-S], ch. 3). Then for U sufficiently small (with respect to U^{p}), and p ≥ 1, one can consider the group

S_{p}H_{k}(U, ∂U) := im ( H_{k}(U^{p}, U^{p} ∩ ∂U) → H_{k}(U, ∂U) ).

It is not difficult to see that the (decreasing) filtration {S_{p}H_{k}(U, ∂U)}_{p} is independent on the choice of neighbourhoods U^{p}; it will be called the “support filtration” of H_{k}(U, ∂U).

In the above construction, |c_{s,t}^{rel}| ⊂ U^{p} (where p = 1 − s and s + t = k), therefore:

W_{−k−p+1}H_{k}(U, ∂U) ⊂ S_{p}H_{k}(U, ∂U).

In general, the inclusion is strict.

V.2. Cycles in X − Y.

V.2.1. Here we will use the identification H_{∗}(X − Y) = H_{∗}(Z). Similarly as above, for any (s, t) with s + t = k fix c_{st} ∈ A_{st}(X − Y) with ∂c_{st} = 0 and ∩c_{st} ∈ im∂. Then it can be completed to a cycle c_{s,t}^{∞} = c_{st} + c_{s−1,t+1} + ⋯ with Dc_{s,t}^{∞} = 0. Consider Π_{⋆}^{−1}c_{s,t}^{∞} ∈ C_{k}(Z) (cf. III.4.2).

V.2.2. Proposition.

a) The cycles Π_{⋆}^{−1}c_{s,t}^{∞} generate W_{−t}H_{k}(Z) = W_{−t}H_{k}(X − Y) (and they are welldefined modulo W_{−t−1}).

b) W_{−k}H_{k}(X −Y) = H_{k}(X −Y). For t < −k, W_{−t}H_{k}(X −Y) can be characterized by the exact sequence of the pair (X, X − Y):

H_{k+1}(X) → H_{k+1}(X, X − Y) → H_{k}(X − Y) → H_{k}(X).

Above H_{i}(X) is pure of weight −i, and the weight filtration of H_{k+1}(X, X − Y) is characterized in the previous subsection.

For a more detailed and geometrical presentation of the cycles in X − Y, see theorem III.0.

The above result shows that the homeomorphism type of the pair (X, Y) determines completely the weight filtration of H_{∗}(X − Y). On the other hand, it is well known that one cannot recover the weight filtration from the homeomorphism type (even the analytic type) of X − Y. For a counterexample, see e.g. [S-S], (2.12).

V.2.3. The support filtration. Similarly as above, let U^{p} be a regular neighbourhood of Y^{p} in X for any p ≥ 1, and for p = 0 take U^{0} = X. Define for any p ≥ 0:

S_{p}H_{k}(X − Y) := im ( H_{k}(U^{p} ∩ (X − Y)) → H_{k}(X − Y) ).

This provides the decreasing “support filtration” S_{∗}H_{k}(X − Y). Then:

W_{−k−p}H_{k}(X − Y) ⊂ S_{p}H_{k}(X − Y),

and the inclusion, in general, is strict.

V.3. Cycles in Y.

V.3.1. Consider (A_{∗∗}(Y), ∂, i) and fix a pair (s, t) with s + t = k. If one wants to construct a closed cycle in Y, it is natural to start with a chain c_{st} ∈ C_{t}(Ỹ^{s+1}) (with ∂c_{st} = 0) and tries to extent it. The first obstruction is i(c_{st}) ∈ im∂, i.e. the existence of c_{s−1,t+1} with i(c_{st})+∂c_{s−1,t+1} = 0. Then the second obstruction is i(c_{s−1,t+1}) ∈ im∂, and so one. The remarkable fact is that once the first obstruction is satisfied then all the others are automatically satisfied. This follows from the degeneration of the spectral sequence (III.1.2), and it is a consequence of the algebraicity of Y; in a simple topological context it is not true.

If c_{st} ∈ A_{st}(Y) = E_{st}^{0} satisfies ∂c_{st} = 0 and i(c_{st}) ∈ im∂, then it can be completed to a cycle

c_{st}^{∞} = c_{st} + c_{s−1,t+1} + ⋯ + c_{0,k}

with Dc_{st}^{∞} = 0 (see section IV). Let n : Ỹ^{p} → Y be again the natural projection for any p ≥ 1. Then the correspondence c_{st} ↦ n_{∗}(c_{0,k}) ∈ C_{∗}(Y) defines a closed cycle in Y. (We invite the reader to verify that ∂n_{∗}(c_{0,k}) = 0.)

V.3.2. Proposition. Fix k = s + t. To any homology class [c_{st}] ∈ H_{t}(Ỹ^{s+1}) with i[c_{st}] = 0 one can construct a cycle

c_{st}^{∞} = c_{st} + c_{s−1,t+1} + ⋯ + c_{0,k}

with Dc_{st}^{∞} = 0. The homology classes [n_{∗}c_{0,k}] ∈ H_{k}(Y) generate W_{−t}H_{k}(Y). Any other completion c_{st}^{∞} of c_{st} provides the same homology class modulo W_{−t−1}H_{k}(Y).

Actually, the chains c_{s−i,t+i} can be recovered from the intersections |c_{0,k}| ∩ Y^{s−i+1}. Indeed, for any i with s > i ≥ 0, and write p = s − i + 1:

|n_{∗}c_{0,k}| ∩ Y^{p} = n(|c_{p−1,k−p+1}|), whose dimension is k − p + 1.

On the other hand, if p ≥ s + 2, then in the homology class [c_{st}] one can take a representative c_{st} which is in a general position with respect to the Y^{p} (i.e. they are transversal), therefore:

|n_{∗}c_{0,k}| ∩ Y^{p} = |n_{∗}c_{st}| ∩ Y^{p} whose dimension is t − 2(p − s − 1) < k − p + 1.

V.3.3. Corollary. The weight filtration of H_{k}(Y) can be characterized by:

W_{−t}H_{k}(Y) = {[c] : dim |c| ∩ Y^{p} = k − p + 1 for p ≤ k − t + 1,
and < k − p + 1 for p = k − t + 2.}

Actually, this is equivalent to the following characterization:

W_{−t}H_{k}(Y) = {[c] : dim |c| ∩ Y^{k−t+2} < t − 1.}

This can be rewritten in the language of intersection homology as follows. For any integer s ≥ 0, consider the perversity \underline{s} defined by \underline{s}(2i) = i for 0 ≤ i ≤ s, and \underline{s}(2i) = s for i ≥ s. (Since we have no stratum with odd codimension, \underline{s}(2i + 1) is unimportant.) Notice that \underline{s} is not a perversity in the sense of [GM] (i.e. does not satisfy \underline{s}(2) = 0), it is a generalized perversity (see. e.g. [K] or [H-S]).

V.3.4. Corollary. W_{−k+s}H_{k}(Y) = im (IH\underline{s} ; k(Y) → H_{k}(Y)).

The above corollary in fact says that Deligne’s weight filtration and the Zeeman filtration (or “support filtration”) coincide. This fact was conjectured by MacPherson, and verified by C. McCrory in [McC] (see also [GNPP] and [H-S]). One of the main consequences of this fact is that the weight filtration of Y is completely topological.

In the last part of this subsection, we present a different construction for the case Y.

V.3.5. Milnor’s construction. Sometimes it is convenient to replace Y by another space, called the “geometric realization” of the “semi-simplicial object” {Ỹ^{p}}_{p}. The construction is done by Milnor [M] (see also [S], [D] and [A]).

This new space is homotopic to Y. It can be constructed as follows. Consider the natural (normalization) map n : Ỹ^{1} → Y. For any points y ∈ Y with y ∈ Y_{α_{1},... ,α_{j}} and y ∉ Y_{β} for β ∉ {α_{1}, ⋯, α_{j}}, the set n^{−1}(y) consists of j points {y_{k}}_{k}. Obviously, if for any y, we identify in Ỹ^{1} the points {y_{k}}_{k}, then the quotient-space is exactly Y. But we want a slightly different construction: for any y, we glue to Ỹ^{1} a (j − 1)– simplex Δ_{y} = [y_{1}, …, y_{j}], in a compatible way. The new space SY will have the following properties. The map sn : SN → Y, defined by sn(Δ_{y}) = y is continuous, it extends n (i.e. sn|Ỹ^{1} = n), and it is a homotopy equivalence.

Actually, SY can be written as

SY = ∐_{p≥1} Ỹ^{p} × Δ_{p−1} / ∼,

where Δ_{p−1} is the (p − 1)–simplex, and ∼ is a suitable identification. If for any 0 ≤ j ≤ p − 1, i_{j} : Ỹ^{p} → Ỹ^{p−1} denote the natural maps induced by the inclusions Y_{α_{1},... ,α_{p}} ↪ Y_{α_{1},... ,α̂_{j+1},··· ,α_{p}}, and ∂_{j} : Δ_{p−2} → Δ_{p−1} are the face maps, then we identify:

(i_{j}y, x) ∼ (y, ∂_{j}x), for any 0 ≤ j ≤ p − 1, y ∈ Ỹ^{p}, and x ∈ Δ_{p−2}.

For more details, see [M], [G] or [A] (with even some pictures on page 239).

One of the advantages of the space SY is the existence of a morphism sn^{−1} : C_{k}(Ỹ^{p}) → C_{k+p−1}(SY). This is defined as follows. For any k–simplex σ we will denote by the same symbol σ the chain with support σ and coefficient one. We will define sn^{−1}(σ) by the chain

[σ × Δ_{p−1}] ⊂ ∐ Ỹ^{p} × Δ_{p−1} / ∼ .

(Here [σ ×Δ_{p−1}] can be covered by simplices, and we take all of them with coefficient one; this is the wanted chain.) This generates a linear map sn^{−1}.

V.3.6. Lemma. The above map induces a morphism of complexes:

ns^{−1} : (T ot_{∗}(A_{∗∗}(Y)), D) → (C_{∗}(SY), ∂)

which is a quasi-isomorphism.

Proof. For the first part notice that

∂(σ × Δ_{p−1}) = ∂σ × Δ_{p−1} + (−1)^{k}σ × ∑_{i}(−1)^{i}∂_{i}Δ_{p−1} = ∂σ × Δ_{p−1} + i(σ) × Δ_{p−2},

i.e. ∂(sn^{−1}(σ) = sn^{−1}(∂σ) + sn^{−1}(i(σ)) (cf. the identification ∼ above). The second part follows from the above lemma III.1.2 (and its proof).

Then III.1.2 and IV.1.2 read as:

V.3.7. Proposition.

a) If c_{st} ∈ A_{st}(Y) = E_{st}^{0} satisfies ∂c_{st} = 0 and i(c_{st}) ∈ im∂, then it can be completed to a cycle

c_{st}^{∞} = c_{st} + c_{s−1,t+1} + ⋯ + c_{0,k}

with Dc_{st}^{∞} = 0. This means (cf. III.1.3) that ∂ns^{−1}c_{st}^{∞} = 0, generating a homology class in H_{k}(SY) = H_{k}(Y).

b) The homology classes [ns^{−1}c_{st}^{∞}] generate W_{−t}H_{k}(Y). Any other completion c_{st}^{∞} of c_{st} provides the same homology class modulo W_{−t−1}H_{k}(Y).

The cycle ns^{−1}(c_{st}^{∞}) constructed above can be projected via sn into Y. In this case, all the chains c_{s−i,t+i} (for s > i ≥ 0) provide “degenerate” chains in Y (with support of dimension t + i). This shows that the correspondence c_{st} ↦ sn^{−1}(c_{st}^{∞}) ∈ C_{∗}(SY) can be replaced by c_{st} ↦ n_{∗}(c_{0,k}) ∈ C_{∗}(Y).

V.4. Cycles in (X, Y).

V.4.1. The construction is similar to the previous case. Take c_{st} ∈ A_{st}(X, Y) with ∂c_{st} = 0 and i(c_{st}) ∈ im∂. Then complete to c_{st}^{∞} = c_{st} + ⋯ + c̃_{0,k}, where k = s + t. Here c̃_{0,k} ∈ C_{k}(X) with ∂c̃_{0,k} = −i(c_{1,k−1}), hence |∂c̃_{0,k}| ⊂ Y provided that s > 0.

V.4.2. Proposition.

a) The relative cycles c̃_{0,k} (associated with c_{st} as above) generate W_{−t}H_{k}(X, Y), and are well-defined modulo W_{−t−1}H_{k}(X, Y).

b) W_{−t}H_{k}(X, Y) = {[c] : c a relative cycle in X with |∂c| ⊂ Y such that dim |c|∩ Y^{p} = k − p for p ≤ k − t, and < k − p for p = k − t + 1}. This is equivalent to the following characterization:

W_{−t}H_{k}(X, Y) = {[c] : dim |c| ∩ Y^{k−t+1} < t − 1}.

c) If ∂ : H_{k}(X, Y) → H_{k−1}(Y) is the natural map, then W_{−t}H_{k}(X, Y) = ∂^{−1}W_{−t}H_{k−1}(Y). In particular, W_{∗}H_{∗}(X, Y) is topological. Moreover, it can also be characterized by

W_{−t}H_{k}(X, Y) = {[c] : dim |∂c| ∩ Y^{k−t+1} < t − 1}.

V.4.3. Let i′ be the composite of the natural maps C_{∗}(SY) → C_{∗}(Y) → C_{∗}(X) (induced by sn and the inclusion). Then there is the following commutative diagram of complexes:

0 → T ot_{∗}(A_{∗∗}(X)) → T ot_{∗}(A_{∗∗}(X, Y)) → T ot_{∗}(A_{∗−1,∗}(Y)) → 0
↓1  ↓1 ⊕ sn^{−1}  ↓sn^{−1}
0 →  C_{∗}(X)  →  Cone_{∗}(i′)  →  C_{∗−1}(SY)  → 0

It is clear that the second line provides the long homology exact sequence of the pair (X, Y), and the vertical arrows in the diagram are quasi-isomorphisms.

V.5. The homology of Y^{p} and (X, X − Y^{p}).

V.5.1. We end this section with the discussion of some connections between the weight filtration and the homology of the spaces Y^{p} and (X, X − Y^{p}). The proofs are standard or similar to the proofs already presented, therefore we will omit them. If A_{∗∗} is a double complex, we denote by σ_{s≤i}A_{∗∗} the subcomplex of A_{∗∗} defined by (σ_{s≤i}A_{∗∗})_{st} = A_{st} for s ≤ i and zero otherwise. σ_{s≥i+1}A_{∗∗} is the quotient double complex A_{∗∗}/σ_{s≤i}A_{∗∗}.

V.5.2. We start with the double complex of Y. For any p ≥ 1:

H_{k}(T ot_{∗}(σ_{s≥p−1} A_{∗∗}(Y))) = H_{k−p+1}(Y^{p}).

Moreover, W_{∗}(σ_{s≥p−1}A_{∗∗}(Y)) induces the weight filtration on this space. The exact sequence:

0 → σ_{s≤p−2}A_{∗∗}(Y) → A_{∗∗}(Y) → σ_{s≥p−1}A_{∗∗}(Y) → 0

provides the identity:

W_{p−2−k}H_{k}(Y) = Ker ( H_{k}(Y) →^{b} H_{k−p+1}(Y^{p}) ),

where b is the (“Mayer-Vietoris”) boundary map (associated with the closed covering Y = ∪_{i}Y_{i}).

Similar identities are valid for A_{∗∗}(X, Y), in particular, for p ≥ 1:

W_{p−1−k}H_{k}(X, Y) = Ker ( H_{k}(X, Y) →^{b′} H_{k−p}(Y^{p}) ),

where b′ is the composite of the boundary operator and b.

In the case of (X, X − Y), for p ≥ 1, one has:

H_{k}(T ot_{∗}(σ_{s≤−p+1} A_{∗∗}(X, X − Y))) = H_{k+p−1}(X, X − Y^{p}),

and W_{∗}(σ_{s≤−p+1}A_{∗∗}(X, X −Y)) induces the weight filtration on this space. The exact sequence:

0 → σ_{s≤−p+1}A_{∗∗}(X, X − Y) → A_{∗∗}(X, X − Y) → σ_{s≥−p+2}A_{∗∗}(X, X − Y) → 0

provides:

W_{−p+1−k}H_{k}(X, X − Y) = im ( H_{k+p−1}(X, X − Y^{p}) →^{b} H_{k}(X, X − Y) ),

where b again is a (“Mayer-Vietoris”) boundary map (associated with the open covering (X, X − Y_{i})_{i}).

Similar property holds for A_{∗∗}(X − Y), in particular, for p ≥ 1:

W_{−p−k}H_{k}(X − Y) = im ( H_{k+p}(X, X − Y^{p}) →^{b′} H_{k}(X − Y) ),

where b′ is the composite of b and the boundary map.

The above properties of the pairs (Y, ∅) and (X, X − Y), respectively of (X, Y) and (X − Y, ∅) correspond by the Poincaré Duality. This follows also from the commutativity of some diagrams like the following:

H^{2n−k}(Y) ⟶^{P D} H_{k}(X, X − Y)
↑b^{∗}  ↑b
H^{2n−k−p+1}(Y^{p}) ⟶^{P D} H_{k+p−1}(X, X − Y^{p})

Above, P D denotes the Poincaré Duality ∩[X] (induced by pd).

VI. The homology of ∂U.

VI.1. The homological double complex of ∂U.

VI.1.1. The space ∂U appears in a natural way in two homological exact sequences. One of them is the pair (U, ∂U). The homological exact sequence of this pair, and the isomorphism H_{∗}(U) = H_{∗}(Y) suggests that a possible double complex for ∂U should satisfy

0 → A_{∗+1,∗}(U, ∂U) → A_{∗∗}(∂U) → A_{∗∗}(Y) → 0.

Here, the double complexes of (U, ∂U), by definition is the same as the double complex of (X, X − Y). Now, using this and the double complex of Y, one can try to define the double complex of ∂U by

A_{s,t}(∂U) = \{ \begin{matrix}C_{t+2s}^{⋔}(Ỹ^{−s}) & for s ≤ −1 \\ C_{t}(Ỹ^{s+1}) & for s ≥ 0. \end{matrix}

But now we are confronted with the definition of the arrows of the double complex. Actually, we have all the vertical arrows, and all the horizontal arrows corresponding to s ≤ −1 and s ≥ 0. But we need to define a map C_{∗}(Ỹ^{1}) → C_{∗−2}^{⋔}(Ỹ^{1}) with some nice properties. First of all, this map should be compatible with the other arrows, in the sense that the whole complex should form a double complex. On the other hand, we expect (since we know the cohomological E_{1} term of the corresponding spectral sequence) that this map should induce at the homology level the “intersection matrix”; more precisely, the map ⊕_{α}c_{α} ↦ ⊕_{α}d_{α}, where for c_{α} ∈ H_{k}(Y_{α}) one gets d_{α} = ∑_{β} c_{β} ∩ [Y_{α}]. Here, if α ≠ β, then c_{β} ∩ [Y_{α}] is provided by the transfer map i_{!} : H_{k}(Y_{β}) → H_{k−2}(Y_{α,β}) composed with i_{∗} : H_{k−2}(Y_{α,β}) → H_{k−2}(Y_{α}) induced by the inclusion. If α = β, then c_{α} ∩ [Y_{α}] is the cap product by [Y_{α}].

Therefore, the wanted map C_{∗}(Ỹ^{1}) → C_{∗−2}^{⋔}(Ỹ^{1}) should be some kind of intersection, but we realize immediatelly that we face serious obstructions: we have to intersect cycles which are not “transversal”, and the image should be special “transversal” cycle. Even if we try to modify our complexes, similar type of obstruction will survive. The explanation for this is the following. The cap product c_{α} ∩ [Y_{α}] cannot be determined only from the spaces {Ỹ^{p}}_{p≥1}, we need the Poincaré dual of the spaces Y_{α} in X, hence we need also to consider the space X (or at least U) in our double complex.

So, we have to think about ∂U as the boundary of Z, and we have to consider the pair of spaces (Z, ∂Z). Notice that H_{∗}(Z, ∂Z) = X_{∗}(X, Y) and H_{∗}(Z) = H_{∗}(X − Y), so it is natural to ask for for a double complex with:

0 → A_{∗+1,∗}(X, Y) → A_{∗∗}(∂U) → A_{∗∗}(X − Y) → 0.

The construction is done in the next subsection.

VI.1.2. The construction of A_{∗∗}(∂U). Consider the following diagram of complexes:

⋯ C_{∗−4}^{⋔}(Ỹ^{2}) ←^{∩} C_{∗−2}^{⋔}(Ỹ^{1}) ←^{∩} C_{∗}^{⋔}(Ỹ^{0})
↓j_{1}
C_{∗}(Ỹ^{0}) ←^{i} C_{∗}(Ỹ^{1}) ←^{i} C_{∗}(Ỹ^{2}) ⋯

The first line corresponds to the double complex A_{∗∗}(X −Y) (where the column s = 0 is C_{∗}^{⋔}(Ỹ^{0}) = C_{∗}^{⋔}(X)), and the second line is the double complex A_{∗∗}(X, Y). Notice that j_{1} can be considered as a morphism of double complexes, hence the usual cone construction provides the double complex A_{∗∗}(∂U) = {A_{s∗}(∂U)}_{s}:

⋯ C_{∗−4}^{⋔}(Ỹ^{2}) ←^{∩} C_{∗−2}^{⋔}(Ỹ^{1}) ←^{∩} C_{∗}^{⋔}(Ỹ^{0})
⊕  ↙^{j_{1}}  ⊕
C_{∗}(Ỹ^{0}) ←^{i} C_{∗}(Ỹ^{1}) ←^{i} C_{∗}(Ỹ^{2}) ⋯
s = −2  s = −1  s = 0  s = 1

VI.1.3. Proposition.

a) The E^{1} term of the spectral sequence associated with (A_{∗∗}(∂U), W) is:

⋯ H_{∗−4}(Ỹ^{2}) ←^{∩} H_{∗−2}(Ỹ^{1}) ←^{∩} H_{∗}(Ỹ^{0})
⊕  ↙^{id}  ⊕
H_{∗}(Ỹ^{0}) ←^{i_{∗}} H_{∗}(Ỹ^{1}) ←^{i_{∗}} H_{∗}(Ỹ^{2}) ⋯
s = −2  s = −1  s = 0  s = 1

where ∩ is given in II.2.5, e.g. d ∈ H_{k}(X):

∩(d) = ⊕_{β} (−1)^{k}d ∩ [Y_{β}] ∈ ⊕_{β}H_{k−2}(Y_{β}) = H_{k−2}(Ỹ^{1});

and i_{∗} is induced by i (cf. III.1), e.g. for ⊕_{α}c_{α} ∈ ⊕_{α}H_{k}(Y_{α}) = H_{k}(Ỹ^{1}):

i_{∗}(⊕_{α}c_{α}) = ∑_{α}(−1)^{k} i_{α}(c_{α}) ∈ H_{k}(X).

The above E^{1} term is quasi-isomorphic to the complex:

⋯ H_{∗−4}^{⋔}(Ỹ^{2}) ←^{∩} H_{∗−2}^{⋔}(Ỹ^{1}) ←^{I} H_{∗}(Ỹ^{1}) ←^{i_{∗}} H_{∗}(Ỹ^{2}) ⋯,

where I := ∩ ∘ i_{∗}, i.e.

I(⊕_{α}c_{α}) = ⊕_{β} (∑_{α} i_{α}(c_{α}) ∩ [Y_{β}] ).

b) E_{st}^{r} ⟹ H_{s+t}(∂U, ℤ) and E_{st}^{∞} ⊗ ℚ = Gr_{−t}^{W}H_{s+t}(∂U, ℚ).

c) E_{∗∗}^{∗} ⊗ ℚ degenerates at level two, i.e. d^{r} ⊗ 1_{ℚ} = 0 for r ≥ 2.

Proof. For a) and b) use II.2.1 and II.2.5 and the corresponding definitions. c) follows from the results of the previous subsections and from the construction of the mixed cone.

Now, we would like to construct a morphism of complexes T ot_{∗}(A_{∗∗}(∂U)) → C_{∗}(∂U). For this the above complex is not convenient, because of the presence of the global terms C_{∗}^{⋔}(Ỹ^{0}) and C_{∗}(Ỹ^{0}). In the next construction, we replace these complexes by the complexes of chains supported by the close neighbourhood U. More precisely, we define A′_{∗∗}(∂U):

⋯ C_{∗−4}^{⋔}(Ỹ^{2}) ←^{∩} C_{∗−2}^{⋔}(Ỹ^{1}) ←^{∩} C_{∗}^{⋔}(U)
⊕  ↙^{j_{1}} ⊕
C_{∗}(U) ←^{i} C_{∗}(Ỹ^{1}) ←^{i} C_{∗}(Ỹ^{2}) ⋯

Then E^{1}(A′_{∗∗}) is:

⋯ H_{∗−4}(Ỹ^{2}) ←^{∩} H_{∗−2}(Ỹ^{1}) ←^{∩} H_{∗}(U)
⊕  ↙^{id}  ⊕
H_{∗}(U) ←^{i_{∗}} H_{∗}(Ỹ^{1}) ←^{i_{∗}} H_{∗}(Ỹ^{2}) ⋯

which is quasi-isomorphic to E^{1}(A_{∗∗}), hence E^{r}(A_{∗∗}) = E^{r}(A′_{∗∗}) for any r ≥ 2.

VI.1.4. Now, the advantage of this second double complex lies in the following construction. Let A_{∗∗}(Z_{α}) be the double complex:

\begin{aligned}⋯ & C_{∗−4}^{⋔}(Ỹ^{2}) & ←^{∩} & C_{∗−2}^{⋔}(Ỹ^{1}) & ←^{∩} & C_{∗}^{⋔}(U) & ← & 0 & ⋯ \\ & s = −2 & & s = −1 & & s = 0 & & s = 1 \end{aligned}

Then, there is a natural morphisms of complexes Π_{⋆}^{−1} : T ot_{∗}(A_{∗∗}(Z_{α})) → C_{∗}(Z_{α}) (cf. II.3.8). This can be inserted in the following commutative diagram:

0 → T ot_{∗}(A_{∗+1,∗}(X, X − Y)) → T ot_{∗}(A_{∗∗}(Z_{α})) → C_{∗}^{⋔}(U) → 0
↓Π_{⋆}^{−1}  ↓Π_{⋆}^{−1}  ↓j_{1}
0 →  Ker_{∗+1}  →  C_{∗}(Z_{α})  → C_{∗}(U) → 0

Since, using a sub-analytic homeomorphism Z_{α} ≈ ∂U × [0, α], the second line in the above diagram induces the long homology exact sequence of the pair (U, ∂U).

VI.1.5. Corollary. The vertical arrows in the above diagram are quasi-isomorphisms.

VI.2. Cycles in ∂U.

VI.2.1. The complex A_{∗∗}(∂U) was important from theoretical point of view: its spectral sequence is degenerating at rank two. On the other hand, the complex A′_{∗∗}(∂U) has the same E^{r} term for r ≥ 2, hence the degeneration property is still valid, but has the advantage that it is local: X is replaced by the neighbourhood U of Y. In the above construction we will use this second double complex A′_{∗∗}.

Notice that σ_{s≤−1}A_{∗∗}(X − Y) = A_{∗+1,∗}(X, X − Y) is a subcomplex of A′_{∗∗}(∂U). The corresponding pair provides the exact sequence:

VI.2.2. → H_{k+1}(X, X − Y) ⟶^{∂} H_{k}(∂U) ⟶^{α} H_{k}(Y) ⟶^{δ} H_{k}(X, X − Y) → .

Since the weights of H_{k+1}(X, X − Y) are ≤ −k − 1, and the weights of H_{k}(Y) are ≥ −k, the weight filtration of H_{k}(∂U) is completely determined from the above exact sequence and from the weight filtrations of (X, X − Y) and Y. For l ≤ −k − 1: W_{l}H_{k}(∂U) = ∂(W_{l}H_{k+1}(X, X − Y), and for l ≥ −k: W_{l}H_{k}(∂U) = α^{−1}W_{l}H_{k}(Y).

Notice again that the weight filtration of H_{k}(∂U) is completely determined from the topology of the pair (X, Y). On the other hand, it is impossible to determine it from the diffeomorphism type of ∂U. For counterexamples, see again [S-S], e.g. (3.1). (For this one has to use the fact that the link of on isolated singularity (S, s) can be identified with the boundary ∂U , where U → S is a resolution of the singular point and S is a Stein representative of (S, s).)

Corresponding to the above discussion of the weights, in the construction of the cycles in ∂U we will distinguish two different cases as well.

Fix a pair (s, t) with s + t = k and s ≤ −1. Consider c_{st} ∈ A_{s+1,t}(X, X − Y) ⊂ A_{s,t}(∂U) with ∂c_{st} = 0 and ∩c_{st} ∈ im∂. Complete to c_{s,t}^{∞} with Dc_{s,t}^{∞} = 0. Then (cf. III.3.1) Π_{⋆}^{−1}c_{s,t}^{∞} ∈ C_{k}(∂Z) is closed. Recall that we have a natural identification of ∂Z and ∂U.

VI.2.3. Proposition. For s ≤ −1, the cycles Π_{⋆}^{−1}c_{s,t}^{∞} generate W_{−t}H_{k}(∂U).

Moreover, [Π_{⋆}^{−1}c_{s,t}^{∞}] = ∂[c_{s,t}^{rel}], where c_{s,t}^{rel} is constructed in V.1. In particular, these homology classes inherit all the properties of W_{∗}H_{k+1}(X, X − Y) (including their relationship with the “support” filtration). The details are left to the reader.

Now, assume that s ≥ 0. In this case the construction of the cycles is more involved: we have to lift some cycles from Y to ∂U.

Assume that c′_{st} ∈ A′_{st}(∂U) (s ≥ 0) satisfies ∂c′_{st} = 0 and d_{1}[c′_{st}] = 0. If s = 0 this means that c′_{st} = c′_{0k} = c_{0,k}^{⊥} + c_{0,k}, where c_{0,k}^{⊥} ∈ C_{k}^{⋔}(U) and c_{0,k} ∈ C_{k}(Ỹ^{1}) such that ∂c_{0,k}^{⊥} = ∂c_{0,k} = 0 and i_{∗}(c_{0,k}) + c_{0,k}^{⊥} + ∂γ = 0 for some γ ∈ C_{k+1}(U).

If s > 0, then c′_{st} ∈ A′_{st}(∂U) has “only one component”: c′_{st} = c_{st} ∈ C_{t}(Ỹ^{s+1}) with ∂c_{st} = 0 and i_{∗}c_{st} ∈ im∂.

In both cases c′_{st} can be completed to a cycle

c_{s,t}^{∞} = c_{s,t} + c_{s−1,t+1} + ⋯ + c_{1,k−1} + (c_{0,k} + c_{0,k}^{⊥}) + (γ_{−1,k+1} + c_{−1,k+1}) + c_{−2,k+2} + ⋯

where c_{s,t} ∈ C_{t}(Ỹ^{s+1}) for s ≥ 0, c_{s,t} ∈ C_{t+2s}^{⋔}(Ỹ^{−s}) for s ≤ −1, c_{0,k}^{⊥} ∈ C_{k}^{⋔}(U), and γ_{−1,k+1} ∈ C_{k+1}(U). These chains satisfies the following relations:

∂c_{s,t} = 0;

i_{∗}c_{s−l,t+l} + ∂c_{s−l−1,t+l+1} = 0 for 0 ≤ l ≤ s − 1;

∂c_{0,k}^{⊥} = 0;

i_{∗}c_{0,k} + jc_{0,k}^{⊥} + ∂γ_{−1,k+1} = 0;

∩c_{0,k}^{⊥} + ∂c_{−1,k+1} = 0;

∩c_{l,k−l} + ∂c_{l−1,k−l+1} = 0 for l ≤ −1.

Now, we will separate the chain (for the notations, see VI.1.4):

c̄_{st} := c_{0,k}^{⊥} + c_{−1,k+1} + c_{−2,k+2} + ⋯ ∈ A_{∗∗}(Z_{α}).

Then by the above relations, D(c̄_{st}) = 0, where here D is the differential in A_{∗∗}(Z_{α}). Therefore, Π_{⋆}^{−1}c̄_{st} ∈ C_{k}(Z_{α}) is closed. Obviously, the projection pr : Z_{α} ≈ ∂Z × [0, α] → ∂Z provides a closed cycle pr_{∗}Π_{⋆}^{−1}c̄_{st} ∈ C_{k}(∂Z).

VI.2.4. Proposition. For any s ≥ 0 and t + s = k, the cycles pr_{∗}Π_{⋆}^{−1}c̄_{st} generate W_{−t}H_{k}(∂U).

Notice that in the cycles c̄_{st} we do not see the chains c_{l,k−l} for l ≥ 0, in particular neither c_{s,t}, the chain which generates c̄_{st}. The chain c_{st} is completed to a sequence c_{st} + c_{s−1,t+1} + ⋯ + c_{0,k} with c_{0,k} ∈ C_{k}(Ỹ^{1}). The chain i_{∗}c_{0,k} actually is closed (in U) and supported by Y. Now, this is replaced by the transversal chain c_{0,k}^{⊥} ∈ C_{k}^{⋔}(U) such that i_{∗}c_{0,k} + c_{0,k}^{⊥} + ∂γ = 0 for some γ, and finally c_{0,k}^{⊥} is completed to c̄_{st}. It is really remarkable that the above algebraic construction plays the role of a very geometric operation: it replaces a closed chain supported in Y by another closed chain supported in U, homologous with the original one in U, and dimensionally transversal to the stratification given by Y (i.e. it solves the problem, obstruction mentioned in VI.1.1.)

VI.2.5. Example. Assume that n = 2, and Y is a connected set of curves {Y_{i}}_{i} in X. Set g = ∑_{i} genus(Y_{i}). If Γ is the dual graph of the curves then let c_{Γ} = rankH_{1}(|Γ|) be the number of independent cycles in |Γ|. Let I be the intersection matrix of the irreducible curves Y_{i}. Then it is well-known that rankH_{1}(∂U) = rank Ker I + 2g + c_{Γ}. These three contributions correspond exactly to the weight of H_{1}(∂U).

Indeed, for s = −1, take c_{−1,2} ∈ C_{0}^{⋔}(Ỹ^{1}) as above. A possible chain c_{−1,2} is an arbitrary point P in Y^{1} − Y^{2} with coefficient one. Then c_{−1,2}^{∞} = c_{−1,2} and Π^{−1}(c_{−1,2}) is a circle S^{1}, the loop around Y in a transversal slice at P. These loops γ_{P} generate W_{−2}H_{1} (isomorphic to coker I). If s = 0, consider a closed 1-cycle c_{0,1} in one of the components of Y. This can be changed by a homologous cycle c_{0,1}^{⊥} in U, which has no intersection with Y. Hence it can be contracted to ∂U. These cycles generate W_{−1}H_{1} (so that dim Gr_{−1}^{W}H_{1} = 2g). Notice that the lifting c_{0,1} ↦ c_{0,1}^{⊥} is defined modulo the cycles of type γ_{P}. Finally, consider c_{1,0} ∈ C_{0}(Y^{2}) such that d_{1}[c_{1,0}] = 0. Here d_{1} : H_{0}(Y^{2}) → H_{0}(Ỹ^{1}) and Ker d_{1} ≈ H_{1}(|Γ|). Take c_{0,1} ∈ C_{1}(Ỹ^{1}) such that ic_{1,0} + ∂c_{0,1} = 0. Then i(c_{0,1}) is a cycle in U supported by Y. We replace it by c_{0,1}^{⊥} which has no intersection with Y. They provide the remaining cycles in W_{0}H_{1} so that dim Gr_{0}^{W} H_{1} = c_{Γ}.

Now, we discuss the case H_{2}(∂U) as well. Take a point P ∈ Y^{2} (with coefficient one) corresponding to c_{−2,4}. Then Π^{−1}c_{−2,4} is a torus in ∂U. They generate W_{−4}H_{2}. Notice that W_{−4}H_{2} ≈ coker∩ : H_{2}(Ỹ^{1}) → H_{0}(Y^{2}) has dimension c_{Γ}. Next, take a generic closed 1-cycle in Ỹ^{1} whose image c in Y has no intersection with Y^{2}. The 2-cycle Π^{−1}(c) in ∂U is an S^{1} bundle over c; they generate W_{−3}H_{2}. Finally, consider c_{02} = ∑_{i} m_{i}Y_{i} ∈ C_{2}(Ỹ^{1}) such that ∩[c_{02}] = 0. This means that [c_{02}] · [Y_{j}] = 0 for all j. The dimension of the space generated by these classes [c_{02}] is coker I. Now, each c_{02} is replaced by a transversal 2-cycle c_{02}^{⊥}. Transversality implies that c_{02}^{⊥} ∩ Y_{i} is a 0-cycle in Y_{i}, which by the above assumption is zero-homologous. In particular, c_{02}^{⊥} ∩ Y_{i} = ∂c_{−1,3}. Now, take a very small tubular neighbourhood U′ of Y^{1} − Y^{2} with projection pr : U′ → Y^{1} − Y^{2}. (Here U − U′ stays for Z_{α}.) Then the boundaries of the chains c_{02}^{⊥} − U′ and the S^{1}-bundle (pr|∂U′)^{−1}(c_{−1,3}) can be identified (modulo sign) so they can be glued. They generate the remaining classes in W_{−2}H_{2}.

VI.2.6. Remark. – Purity results. The above example shows that in general all the possible weights (permitted by the spectral sequence) can appear. For example, if ∂U is the boundary (or link) of a 1-parameter family of projective curves over a small disc, then the intersection matrix I has 1-dimensional kernel, hence H_{1}(∂U) can have weight −2, −1 and 0. On the other hand, if ∂U is the boundary of the resolution of a normal surface singularity, then I is non-degenerate, hence the non-trivial weight of H_{1}(∂U) are −1 and 0.

More generally, if S is a projective algebraic variety with unique singular point s ∈ S, and ϕ : X → S is a resolution of this isolated singularity with normal crossing exceptional divisor Y = ϕ^{−1}(s), then the following additional restrictions hold. They can be easily deduced from the corresponding cohomological statements about the weight filtrations of the links of isolated singularities. For a complete proof and details see e.g. [N-A].

1) If k ≤ n − 1 then Gr_{l}^{W} H_{k}(∂U) = 0 for l not in [−k, 0];

2) If k ≥ n then Gr_{l}^{W} H_{k}(∂U) = 0 for l not in [−2n, −k − 1];

3) If k ≥ n then H_{k}(Y) is pure of weight −k and H_{k}(U) → H_{k}(U, ∂U) is injective;

4) If k ≤ n then H_{k}(U, ∂U) is pure of weight −k and H_{k}(U) → H_{k}(U, ∂U) is surjective;

5) Consider the exact sequence (cf. the E^{1} term in VI.1.3):

⋯ H_{k−4}^{⋔}(Ỹ^{2}) ←^{∩} H_{k−2}^{⋔}(Ỹ^{1}) ←^{I_{k}} H_{k}(Ỹ^{1}) ←^{i_{∗}} H_{k}(Ỹ^{2}) · · · .

If k ≥ n then Ker I_{k} = im i_{∗}, if k ≤ n then Ker ∩ = im I_{k}.

Actually, between the above exact sequence and the the exact sequence VI.2.2 there is the following connection: δ is non-trivial only for weight −k and Gr_{−k}^{W}δ can be identified with Î_{k} : H_{k}(Ỹ^{1})/im i_{∗} → Ker ∩.

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Fouad Elzein, Départ. de Math, Univ. de Nantes, 44072 Nantes Cedex 03, FRANCE

  E-mail address: elzein@math.univ-nantes.fr

András Némethi, 100 Math Building, Ohio State University, 231 West 18th. Avenue, Columbus, Ohio, USA 43210-1174

  E-mail address: nemethi@math.ohio-state.edu