Weight filtration on the cohomology of complex analytic spaces

We extend Deligne's weight filtration to the integer cohomology of complex analytic spaces (endowed with an equivalence class of compactifications). In general, the weight filtration that we obtain is not part of a mixed Hodge structure. Our purely geometric proof is based on cubical descent for resolution of singularities and Poincar\'e-Verdier duality. Using similar techniques, we introduce the singularity filtration on the cohomology of compactificable analytic spaces. This is a new and natural analytic invariant which does not depend on the equivalence class of compactifications and is related to the weight filtration.


Introduction
The weight filtration was introduced by Deligne [Del71], [Del74] following Grothendieck's yoga of weights, and as the key ingredient of mixed Hodge theory. This is an increasing filtration defined functorially on the rational cohomology of every complex algebraic variety, expressing the way in which its cohomology is related to cohomologies of smooth projective varieties. In Deligne's approach, the weight filtration on the cohomology of a singular complex algebraic variety X, supposed compact to simplify, is the filtration induced by a smooth hypercovering X • → X of X. Indeed, the induced spectral sequence E p,q 1 (X; A) = H q (X p ; A) ⇒ H p+q (X; A) defines a filtration on H n (X; A) for any given coefficient ring A. Using Hodge theory, Deligne proved that when A = Q, the above spectral sequence degenerates at the second stage and that the filtration on the rational cohomology is well defined and functorial.
In [GS96], Gillet and Soulé gave an alternative proof of the well-definedness of the weight filtration using smooth hypercoverings and algebraic K-theory. Their more geometric approach allowed them to obtain the result with integer coefficients, at least for compact support cohomology. For a general coefficient ring A, the above spectral sequence does not necessarily degenerate at the second stage. However, they proved that, from the second stage onwards, the corresponding spectral sequence for cohomology with compact supports is a well defined algebraic invariant of the variety. An analogous construction yielded a weight filtration on algebraic K-theory with compact supports (see also the work of Pascual-Rubió [PR09]).
Based on cubical hyperresolutions (see [GNPP88]), Guillén and Navarro-Aznar [GN02] developed a general descent theory which allows to extend contravariant functors compatible with elementary acyclic squares (smooth blow-ups) on the category of smooth schemes, to functors on the category of all schemes in such a way that the extended functor is compatible with acyclic squares (abstract blow-ups).
Totaro observed in his ICM lecture [Tot02], that using the results on cohomological descent of [GN02], the weight filtration is well defined on the cohomology with compact supports of any complex or real analytic space with a given equivalence class of compactifications, for which Hironaka's resolution of singularities holds. Following this idea, and using Poincaré duality for real manifolds with Z 2 -coefficients, McCrory and Parusinski obtained the weight filtration for real algebraic varieties on Borel-Moore Z 2 -homology in [MP11], and on compactly supported Z 2 -homology in [MP12]. Similar results for compactly supported Z 2 -cohomology of real varieties appear in [LP14], where Limoges and Priziac prove that products in cohomology are compatible with the weight filtration.
Let X be a (compactificable) complex analytic space. Every cubical hyperresolution X • → X induces a spectral sequence converging to a filtration of H n (X; A), which we call singularity filtration. In this note we prove that from the E 2 -term onwards, this spectral sequence does not depend on the choice of the hyperresolution (Theorem 4.1). This result is a corollary of the cohomological descent of [GN02] and Poincaré-Verdier duality. In the same vein, in Theorem 5.1 we obtain a generalization of the weight filtration on the cohomology H n (X; A), when X is endowed with an equivalence class of compactifications (see Definition 2.8). In particular, if X is a complex algebraic variety, the weight filtration is well-defined on H(X; A). For smooth manifolds, the singularity filtration is trivial, while for compact analytic spaces, it coincides with the weight filtration.
To obtain these results we give an analytic version of the extension criterion of functors of [GN02] (see Theorem 2.3). The analytic setting differs from the algebraic setting appearing in loc.cit., mainly due to the weaker formulation of Chow-Hironaka's Lemma and certain finiteness issues, and it may find applications to study other topological invariants. For instance, it should allow to define a Hodge and a weight filtration on the rational homotopy type of complex analytic spaces, extending the filtrations obtained by Morgan [Mor78], to the analytic setting. We will present this multiplicative theory elsewhere.
It is easy to see that the previous results are also valid for the cohomology with Z 2 coefficients of real algebraic varieties. Other cohomology theories such as Borel-Moore homology or cohomology with compact supports can also be studied using parallel techniques, allowing to recover the quoted results of Gillet-Soulé and McCrory-Parusinski, and the results announced by Totaro concerning the weight filtration.
In Section 1 we show that the category of filtered complexes over an abelian category admits a cohomological descent structure with respect to the class of weak equivalences given by E r -quasiisomorphisms: morphisms of filtered complexes inducing a quasi-isomorphism at the r-stage of the associated spectral sequence (see Theorem 1.13).
In Section 2 we establish an extension criterion of functors from smooth to singular complex analytic spaces (Theorem 2.3) as well as a relative version (Theorem 2.9).
In Section 3 we study the behavior of the cohomology functor with respect to acyclic squares of analytic spaces. We then study the Gysin complex of a smooth compactification U ֒→ X with D = X − U a normal crossings divisor.
In Sections 4 and 5 we use the results of the previous sections to define the singularity and weight filtrations respectively, on the cohomology with coefficients in an arbitrary ring A, of compactificable complex analytic spaces (Theorem 4.1 and Theorem 5.1).

Cohomological descent structures on the category of filtered complexes
The extension criterion of functors of [GN02] is based on the assumption that the target category is a cohomological descent category, a variant of the triangulated categories of Verdier. This is essentially a category D endowed with a saturated class of weak equivalences E, and a simple functor s sending every cubical codiagram of D to an object of D and satisfying certain axioms analogous to those of the total complex of a double complex. The simple functor can be viewed as the homotopy limit, and allows to define realizable homotopy limits for diagrams indexed by finite categories (see [Rod12]).
In this section we show that the category of filtered complexes over an abelian category admits a cohomological descent structure, where the weak equivalences are given by E r -quasiisomorphisms.
We first recall some features of cubical codiagrams and cohomological descent categories. We refer to [GN02] for the precise definitions.
1.1. Given a set {0, · · · , n}, with n ≥ 0, the set of its non-empty parts, ordered by the inclusion, defines the category n . Likewise, any non-empty finite set S defines the category S . Every injective map u : S → T between non-empty finite sets induces a functor u : S → T defined by u (α) = u(α). Denote by Π the category whose objects are finite products of categories S and whose morphisms are the functors associated with injective maps in each component.  is an object of Π and X is a functor X : → D. A morphism (X, ) → (Y, ′ ) between cubical codiagrams is given by a pair (a, δ) where δ : ′ → is a morphism of Π and a : δ * X → Y is a natural transformation.
Denote by CoDiag Π D the category of cubical codiagrams of D.
Definition 1.4. A cohomological descent category is given by a cartesian category D provided with an initial object 1, together with a saturated class of morphisms E of D which is stable by products, called weak equivalences, and a contravariant functor s : CoDiag Π D → D, called the simple functor. The data (D, E, s) must satisfy the axioms of Definition 1.5.3 of [GN02]. Objects weakly equivalent to the initial object 1 are called acyclic.
1.5 (Φ-rectified functors). If D is a cohomological descent category, then for each cubical index category ∈ Π, the simple functor induces a functor Ho(D ) → Ho(D). In certain situations, we are interested in cubical diagrams in Ho(D). In general we do not have a simple functor Ho(D) → Ho(D). The notion of Φ-rectified functor corresponds, roughly speaking, to functors F : C → Ho(D) which are defined on all cubical diagrams in the form F : C → Ho(D ), so that we can take the composition C → Ho(D ) → Ho(D) (see 1.6 of [GN02]). For our purposes, it suffices to note that every functor F : C → D induces a Φ-rectified functor F : C → Ho(D).
Let A be an abelian category. The primary example of a cohomological descent category is given by the category of complexes C + (A), with weak equivalences being quasi-isomorphisms and the simple functor s defined via the total complex. We adapt this structure to filtered complexes and obtain a family of cohomological descent structures.
Denote by FA the category of filtered objects of A, and by C + (FA) the category of complexes over objects of FA.
Denote by E r the class of E r -quasi-isomorphisms. Let s > r and consider the functor E s : C + (FA) → C + (A) defined by sending a filtered complex to the E s -stage of its associated spectral sequence. Since it sends morphisms of E r to isomorphisms, there is a functor E s : Deligne introduced the décalage of a filtered complex and proved that its associated spectral sequences are related by a shift of indexing. This proves to be a key tool in the study of filtered complexes and their cohomological descent properties.
Definition 1.8. The décalage DecK of a filtered complex K is the filtered complex defined by Note that s 0 and s 1 correspond to the filtered total complexes defined via the convolution with the trivial and the bête filtrations respectively, introduced by Deligne in [Del74]. By forgetting the filtrations on s r we recover the simple functor s of complexes.
Proposition 1.11. Let K be a codiagram of filtered complexes. Then for r ≥ 0, Proof. The category C + (FA) is complete. Furthermore, since the décalage has a left adjoint defined by the shift of a filtration (see [CG13]), it commutes with pull-backs. It also commutes with r-translations (K, W ) → (K[r], W (−r)). We have: This gives an isomorphism of the filtrations Dec(W (r + 1)) and (DecW )(r) on s(K). Proposition 1.12. Let K be a codiagram of filtered complexes. For r ≥ 0, there is a chain of quasi-isomorphisms E * ,q r (s r (K)) ∼ ←→ sE * ,q r (K).
Proof. For r = 0 we have an isomorphism E * ,q 0 (s 0 (K)) ∼ = sE * ,q 0 (K). Assume inductively that the proposition is true for r − 1. We then have a chain of quasi-isomorphisms , where the first and last quasi-isomorphisms follow from Proposition 1.9 and the isomorphisms follow from Proposition 1.11 and the induction hypothesis respectively.
Proof. Consider the functor E r : C + (FA) → C + (A) defined by sending every filtered complex to the r-stage of its associated spectral sequence. Then E r = E −1 r (E), where E denotes the class of quasi-isomorphisms of C + (A). Furthermore, by Proposition 1.12, the complexes E r (s r (K)) and sE r (K) are isomorphic in the derived category D + (A) = C + (A)[E −1 ], for every codiagram K in C + (FA). This isomorphism is compatible with the morphisms µ and λ. By Proposition 1.7.2 of [GN02] the triple (C + (A), E, s) is a cohomological descent category. Hence by Proposition 1.5.12 of loc.cit., this lifts to a cohomological descent structure for the triple (C + (FA), E r , s r ).

Extension criterion of functors for analytic spaces
The extension criterion of functors of [GN02] allows to extend certain cohomological type functors defined on smooth schemes, to all algebraic schemes, using resolution of singularities.
Let An(C) denote the category of complex analytic spaces that are reduced, separated and of finite dimension. Denote by Man(C) the full subcategory of smooth manifolds.
It is an elementary acyclic square if, in addition, all the objects in the diagram are in Man(C), and f is the blow-up of X along Y . In the latter case, the map f is said to be an elementary proper modification.
Remark 2.2. In the analytic setting, we still have Hironaka's resolution of singularities. However, in order to provide an extension criterion valid for analytic spaces, we need to address certain issues concerning finiteness. The first of this issues is Chow-Hironaka's Lemma ( [Hir64], 0.5), stating that every proper birrational map of irreductible schemes factors as a composition of a finite sequence of blow-ups with smooth centers. This result allows the passage from acyclic squares to elementary acyclic squares in the hypotheses of the extension criterion. In the analytic setting, the factorization is made trought the composition of a possibly infinite sequence of blow-ups, which is locally finite. This is a consequence of Hironaka's Flattening Theorem [Hir75].
The second issue concerns the finiteness of ν(X) = (n, c n (X), · · · , c 0 (X)), where c i (X) is the number of irreductible components of dimension i, of a variety X of dimension n, which contain the singular points of X. If X is an algebraic variety, then c i (X) is finite for all i. However, for an analytic space this may not be the case. For compactificable analytic spaces, these two drawbacks disappear.
Denote by Man(C) (resp. An(C)) the full subcategory of Man(C) (resp. An(C)) of compactificable analytic spaces. The following is an analytic version of the extension criterion of functors defined over smooth schemes (see Theorem 2.1.10 of [GN02]).  such that: (1) If X is an object of Man(C), then F ′ (X) ∼ = F (X).
(2) If X • is an acyclic square of An(C), then sF ′ (X • ) is acyclic. In addition, the functor F ′ is essentially unique.
Proof. With the notations of 2.1.10 of [GN02], this is equivalent to prove that the inclusion functor Man(C) → An(C) verifies the extension property. It suffices to replace M ′ = Sm(k) by Man(C) and M = Sch(k) by An(C) in the proof of Theorem 2.1.5 of loc.cit., which by Remark 2.2 is valid for compactificable analytic spaces.
To prove the invariance of the weight filtration we will use a relative version of the above result.
Let An(C) 2 denote the category of pairs (X, U ) where X is an analytic space and U is an open subset of X such that D = X − U is a closed analytic subspace of X.
Likewise, let Man(C) 2 be the full subcategory of An(C) 2 of those pairs (X, U ) with X smooth and D = X − U a normal crossings divisor in X which is a union of smooth divisors.
is said to be an acyclic square if f : X → X is proper, i : Y → X is a closed immersion, the diagram of the first components is cartesian, f −1 (U ) = U and the diagram of the second components is an acyclic square of An(C).  Let An(C) 2 comp denote the full subcategory of An(C) 2 given by those pairs (X, U ) such that X is compact. Define Man(C) 2 comp similarly. In particular, if (X, U ) ∈ An(C) 2 comp we have that both X and U are objects of An(C). 2.7. Denote by γ : An(C) 2 comp → An(C) the forgetful functor (X, U ) → U , and let Σ be the class of morphisms s of An(C) 2 comp such that γ(s) is an isomorphism. Then γ induces a functor η : An(C) 2 comp [Σ −1 ] −→ An(C). In the algebraic situation, Nagata's Compactification Theorem implies that the functor η alg induces an equivalence of categories This does not hold in the analytic case. However, the localized category An(C) 2 comp [Σ −1 ] is equivalent to the category An(C) ∞ defined as follows. Two compactifications f 1 : X 1 → X ′ 1 and f 2 : X 2 → X ′ 2 of a morphism of analytic spaces f : U → U ′ are said to be equivalent if there exists a third compactification f 3 : X 3 → X ′ 3 which dominates them. A morphism of An(C) ∞ is given by an equivalence class of morphisms of An(C) 2 comp .
An acyclic square in An(C) ∞ is a square induced by an acyclic square of An(C) 2 comp . The following is an analytic version of Theorem 2.3.6 of [GN02].
comp , then sF ′ (X • , U • ) is acyclic. Furthermore, the functor F ′ is essentially unique. The remaining of the proof follows analogously to that of Theorem 2.3.6 of loc. cit., via the equivalence of categories given in 4.7 of loc.cit..

Acyclic squares and Gysin complex
In this section we study the behavior of the cohomology functor with respect to certain acyclic squares of smooth analytic spaces. We then introduce the Gysin complex of a pair (X, U ), where U ֒→ X is a smooth compactification with D = X − U a normal crossings divisor and describe its behavior with respect to elementary acyclic squares.
Let X be a complex analytic space. Given a commutative ring A we will denote by A X the constant sheaf over X associated to A and by H * (X; A) the singular cohomology of X with coefficients in A. For a continuous map f : X → Y we will denote Rf * := f * C • Gdm , where C • Gdm is the Godement resolution.
is exact for all q.
Proof. We have a a Mayer-Vietoris long exact sequence Therefore it suffices to see that the map f * : H q (X; A) → H q ( X; A) is injective. This is a well known consequence of Poincaré-Verdier duality, which gives the existence of a trace morphism f ♯ such that f ♯ f * = 1. We recall the proof. The map f * is induced by a morphism of sheaves A X → Rf * A X . Since f is proper, Rf * = Rf ! , and we have an adjunction Rf * ⊢ f ! . Since X and X are smooth and of the same pure dimension, there is a quasi-isomorphism The trace map f ♯ : Rf * A X → A X is deduced by adjunction from the identity morphism 1 X ∈ Hom(A X , A X ) of Lastly, since f is birrational, the composition f ♯ • f * : A X → Rf * A X → A X is the identity, since it coincides with the identity over an open dense subset of X. Hence f ♯ induces a left inverse of f * , and f * is injective.
We shall also use the following blow-up formula for cohomology. A proof can be found in Theorem VI.4.5 of [FL85], which is an axiomatization of Theorem VII.3.7 of [SGA71].
For all q ≥ 0, there is a commutative acyclic square 3.3 (Gysin complex). Let (X, U ) ∈ Man(C) 2 . We may write D := X − U = D 1 ∪ · · · D N as the union of irreducible smooth divisors meeting transversally. Let D (0) = X and for 0 < p ≤ N let D (p) be the disjoint union of all p-fold intersections D I = D i 1 ∩ · · · ∩ D ip with I = {i 1 , · · · , i p } ⊂ {1, · · · , N }. Since D is a normal crossings divisor, D (p) is smooth. For all q ≥ 0, the Gysin complex G q (X, U ) is the cochain complex defined by  Proof. Let f : Then the morphisms G * IJ (f ) are the components of G q (f ) p : G q (X, U ) p → G q (X ′ , U ′ ) p . It follows from the decomposition property of determinants, that this is a map of complexes (see [GN02], pag. 84).
If g : (X ′′ , U ′′ ) → (X ′ , U ′ ) is a morphism of Man(C) 2 , then the matrix of multiplicities M f •g of f • g is the product of the multiplicity matrices M f and M g of f and g. The functoriality of G q then follows from the functoriality of the determinants.
is acyclic for all q.
Proof. We adapt the proof of Proposition 5.9 of [GN02] in the motivic setting (see also [MP12], Sections 5 and 6).
Assume that Y D. We proceed by induction on the number N of smooth irreducible components of D. If N = 0 then G q (X, U ) = H q (X; A) is concentrated in degree 0 and the sequence ( * ) becomes that of Proposition 3.1. Assume From the definition of the Gysin complex we obtain an exact sequence We then have a short exact sequence are acyclic complexes. Therefore the middle complex is acyclic, as desired. This proves (1).
Assume that Y ⊂ D. We proceed by induction over the number of components r of D which contain Y , and the number s of components which do not contain Y .
Assume that (r, s) = (1, 0), so that D is smooth irreducible and Y ⊂ D. Then G q (X, X − D) is the simple of the morphism H q−2 (D; A) → H q (X; A). Denote by D the proper transform of D, and let E = Y ∩ D. Denote by is the simple of the square extend the functor S to a functor An(C) −→ D + 0 (FA-mod), using the cohomological descent structure on C + (FA-mod) associated with the class of E 0 -quasi-isomorphisms. It is easy to see that this is an empty exercise: the extended filtration of the trivial filtration is also trivial. However, if we consider the cohomological descent structure associated with E 1 -quasi-isomorphisms, we obtain a non-trivial filtration which for compact spaces coincides with the weight filtration.
(2) If X is a smooth manifold then S ′ (X) = (S * (X; A), L), where L is the trivial filtration.
(3) For every p, q ∈ Z and every acyclic square of An(C) as in Definition 2.1 there is a long exact sequence (4) If X is a compact complex algebraic variety and A = Q then the filtration induced in cohomology coincides with Deligne's weight filtration after décalage.
Definition 4.2. Let X be a compactificable complex analytic space. The singularity spectral sequence is the spectral sequence associated with the filtered complex S ′ (X) of Theorem 4.1.
Let L ′ denote the increasing filtration induced on H * (X; A). The singularity filtration L p on H * (X; A) is defined by L p H n (X; A) := L ′ p−n H n (X; A). Corollary 4.3. Let X be a compactificable complex analytic space. Then for every n ≥ 0, its cohomology H n (X; A) with values in any commutative ring A carries a singularity filtration which is functorial for morphisms in An(C) and satisfies: 1) If X is smooth then L is the trivial filtration. 2) If X is a complex projective variety and A = Q then L coincides with Deligne's weight filtration.
Note that by Theorem 4.1, the E 2 -term of the singularity spectral sequence is well-defined. The first term L E 1 , which is well-defined up to quasi-isomorphism, admits a description in terms of resolutions as follows: let X • → X be a resolution of a compactificable complex analytic space X. Then: If X is a projective complex variety and A = Q this corresponds to the analogue formula for the weight filtration appearing in Theorem 8.1.15 of [Del74], (see also IV.3 of [GNPP88]).
Remark 4.4. The same arguments give a filtration L on the homology with compact supports and on the Borel-Moore homology of a variety X. In [Gui87], Deligne's weight filtration W and Zeeman's filtration S are compared in the homology of a compact variety, giving the relation S 2N −i−q ⊂ W i−q on H i (X; Q), where N = dimX. The same proof of [Gui87] gives the relation S 2N −i−q ⊂ L i−q for the singularity filtration on the Borel-Moore homology H BM i (X; A).

Weight filtration
Recall that the canonical filtration τ is defined on any given complex K by truncation: Given (X, U ) ∈ Man(C) 2 comp , let j : U ֒→ X denote the inclusion, and (Rj * A U , τ ) the filtered complex of sheaves given by the direct image of the constant sheaf A U , together with the canonical filtration. Taking the right derived functor of global sections we obtain a Φ-rectified functor W : Man(C) 2 comp −→ D + 1 (FA-mod) with values in the 1-derived category of filtered complexes of A-modules (see Definition 1.7), given by W(X, U ) = RΓ(X, (Rj * A U , τ )). By the properties of the global sections functor and the derived direct image functor, we have H n (W(X, U )) ∼ = H n (U ; A).