Du Bois complexes and Du Bois pairs

166 5. INJECTIVITY AND VANISHING THEOREMS

filtr´es d’op´erateurs diff´erentiels d’ordre ≤1] and Remark 5.3.2below).

We put

Ω⁰_X = Gr⁰_FΩ^•_X.

There is a natural map (Ω^•_X, σ) → (Ω^•_X, F). It induces OX → Ω⁰_X. If OX → Ω⁰_X is a quasi-isomorphism, then X is said to have Du Bois singularities. We sometimes simply say that X is Du Bois. Let Σ be a reduced closed subvariety of X. Then there is a natural map ρ : (Ω^•_X, F) → (Ω^•_Σ, F) in D_diff,coh^b (X). By taking the cone of ρ with a shift by one, we obtain a filtered complex (Ω^•_X,Σ, F) in D_diff,coh^b (X).

Note that (Ω^•_X,Σ, F) was essentially introduced by Steenbrink in [St, Section 3]. We put

Ω⁰_X,Σ = Gr⁰_FΩ^•_X,Σ.

Then there are a mapJΣ →Ω⁰_X,Σ, whereJΣ is the defining ideal sheaf of Σ on X, and the following commutative diagram

JΣ //

OX //

OΣ +1 //

Ω⁰_X,Σ ^//Ω⁰_X ^//Ω⁰_Σ ^{+1 //}

in the derived category D_coh^b (X) (see also Remark 5.3.4 below).

For completeness, we include the definitions of the derived cate- gories D_coh^b (X), D_diff,coh^b (X), and so on.

Remark 5.3.2 (Derived categories). Let X be a variety. Then D(X) denotes the derived category ofOX-modules and D^b_coh(X) is the full subcategory of D(X) consisting of complexes whose cohomologies are all coherent and vanish in sufficiently negative and positive degrees.

For the details, see [Har1].

Let us consider the category C_diff(X). Each object of C_diff(X) is a triple (K^•, d, F) consisting of a complex (K^•, d) of OX-modules and a decreasing filtration F onK^• such that

(i) K^• is bounded below,

(ii) the filtrationF is biregular, that is, for each componentKⁱ of K^•, there exist integers m and n such that F^mKⁱ = Kⁱ and FⁿKⁱ = 0,

(iii) d is a differential operator of order at most one and preserves the filtrationF, and

(iv) Gr^p_F(d) : Gr^p_F(Kⁱ)→Gr^p_F(Kⁱ⁺¹) is OX-linear for any integers pand i.

5.3. DU BOIS COMPLEXES AND DU BOIS PAIRS 167

LetDdiff(X) be the derived category of the category Cdiff(X). For the details, see [Du]. In this situation, D^b_diff,coh(X) is the full subcategory of Ddiff(X) consisting of (K^•, d, F) such that Gr^p_F(K^•) is an object of D^b_coh(X) for every p.

By using the theory of mixed Hodge structures on cohomology with compact support, we have the following theorem.

Theorem 5.3.3. Let X be a variety and let Σ be a reduced closed subvariety of X. We put j :X−Σ,→X. Then we have the following properties.

(1) The complex (Ω^•_X,Σ)^an is a resolution of j_!CX^an−Σ^an. (2) If in addition X is proper, then the spectral sequence

E₁^p,q =H^q(X,Ω^p_X,Σ)⇒H^p+q(X^an, j_!CX^an−Σ^an) degenerates at E₁, where Ω^p_X,Σ = Gr^p_F Ω^•_X,Σ[p].

From now on, we will simply write X (resp. OX and so on) to express X^an (resp. OX^an and so on) if there is no risk of confusion.

Proof. Here, we use the formulation of [PS, §3.3 and §3.4]. We assume that X is proper. We take cubical hyperresolutionsπ_X :X_• → X and π_Σ : Σ_• →Σ fitting in a commutative diagram.

Σ_•

πΣ

//X_•

πX

Σ _ι ^// X

Let Hdg(X) :=Rπ_X∗Hdg^•(X_•) be a mixed Hodge complex of sheaves on X giving the natural mixed Hodge structure on H^•(X,Z) (see [PS, Definition 5.32 and Theorem 5.33]). We can obtain a mixed Hodge complex of sheaves Hdg(Σ) := Rπ_Σ∗Hdg^•(Σ_•) on Σ analo- gously. Roughly speaking, by forgetting the weight filtration and the Q-structure of Hdg(X) and considering it in D_diff,coh^b (X), we obtain the Du Bois complex (Ω^•_X, F) of X (see [GNPP, Expos´e V (3.3) Th´eor´eme]). We can also obtain the Du Bois complex (Ω^•_Σ, F) of Σ analogously. By taking the mixed cone of Hdg(X) → ι_∗Hdg(Σ) with a shift by one, we obtain a mixed Hodge complex of sheaves on X giving the natural mixed Hodge structure on H_c^•(X −Σ,Z) (see [PS, 5.5 Relative Cohomology]). Roughly speaking, by forgetting the weight filtration and the Q-structure, we obtain the desired filtered complex (Ω^•_X,Σ, F) in D^b_diff,coh(X). WhenX is not proper, we take completions of X and Σ ofX and Σ and apply the above arguments to X and Σ.

Then we restrict everything toX. The properties (1) and (2) obviously

168 5. INJECTIVITY AND VANISHING THEOREMS

hold by the above description of (Ω^•_X,Σ, F). By the above construction and description of (Ω^•_X,Σ, F), we know that the map JΣ → Ω⁰_X,Σ in D^b_coh(X) is induced by natural maps of complexes.

Remark 5.3.4. Note that the Du Bois complex Ω^•_X is nothing but the filtered complex Rπ_X∗(Ω^•_X•, F). For the details, see [GNPP, Ex- pos´e V (3.3) Th´eor´eme and (3.5) D´efinition]. Therefore, the Du Bois complex of the pair (X,Σ) is given by

Cone^•(Rπ_X∗(Ω^•_X•, F)→ι_∗Rπ_Σ∗(Ω^•_Σ•, F))[−1].

By the construction of Ω^•_X, there is a natural map a_X : OX → Ω^•_X which induces OX → Ω⁰_X in D^b_coh(X). Moreover, the composition of a^an_X : OX^an → (Ω^•_X)^an with the natural inclusion CX^an ⊂ OX^an in- duces a quasi-isomorphism CX^an −→' (Ω^•_X)^an. We have a natural map a_Σ :OΣ →Ω^•_Σ with the same properties as a_X and the following com- mutative diagram.

OX //

aX

OΣ aΣ

Ω^•_X ^//Ω^•_Σ

Therefore, we have a natural map b : JΣ →Ω^•_X,Σ such that b induces JΣ → Ω⁰_X,Σ in D^b_coh(X) and that the composition of b^an : (JΣ)^an → (Ω^•_X,Σ)^an with the natural inclusion j_!CX^an−Σ^an ⊂ (JΣ)^an induces a quasi-isomorphism j_!CX^an−Σ^an −→' (Ω^•_X,Σ)^an. We need the weight fil- tration and the Q-structure in order to prove the E1-degeneration of Hodge to de Rham type spectral sequence. We used the framework of [PS, §3.3 and §3.4] because we had to check that various diagrams related to comparison morphisms are commutative (see [PS, Remark 3.23]) for the proof of Theorem 5.3.3 (2) and so on.

Let us recall the definition of Du Bois pairs by [Kv5, Definition 3.13].

Definition 5.3.5 (Du Bois pairs). With the notation of5.3.1 and Theorem 5.3.3, if the map JΣ → Ω⁰_X,Σ is a quasi-isomorphism, then the pair (X,Σ) is called aDu Bois pair.

By the definitions, we can easily check the following useful propo- sition.

Proposition5.3.6. With the notation of5.3.1and Theorem5.3.3, we assume that both X and Σ are Du Bois. Then the pair(X,Σ) is a Du Bois pair, that is, JΣ →Ω⁰_X,Σ is a quasi-isomorphism.

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 169

Let us recall the following well-known results on Du Bois singular- ities.

Theorem 5.3.7. Let X be a normal algebraic variety with only quotient singularities. Then X is Du Bois. Note that X has only rational singularities.

Theorem 5.3.7 follows from, for example, [Du, 5.2. Th´eor`eme], [Kv1], and so on.

Lemma 5.3.8. Let X be a variety with closed subvarieties X1 and X₂ such that X = X₁ ∪X₂. Assume that X₁, X₂, and X₁ ∩X₂ are Du Bois. Note that, in particular, we assume that X₁∩X₂ is reduced.

Then X is Du Bois.

For the proof of Lemma 5.3.8, see, for example, [Schw, Lemma 3.4].

Although it is dispensable, we will use the notion of Du Bois com- plexes for the proof of the Hodge theoretic injectivity theorem: Theo- rem 5.4.1.

5.4. Hodge theoretic injectivity theorems